A MATHEMATICAL TOUR OF THE

ELECTROMAGNETIC FORCE from

Michael Faraday to Quantum Field Theory There are physical phenomena in everyday life that are taken for granted simply because the explanation of their behavior closely matches the expectations of the observer. For some of these phenomenon, an extensive body of theoretical knowledge exists which matches the experimental observations. The electromagnetic force is one of these phenomenon. The observer can envision empty space filled with electromagnetic waves, and describe these waves and their effects on matter with mathematical precision. Devices can be constructed, based on electromagnetic theory, that confirm our belief that the electromagnetic phenomena are well understood — that is, observations are produced consistent with expectations. With further investigation new questions arise, requiring a reformulation of the theory which supports these observations.

The classical electromagnetic field is described by Maxwell’s equations. From these equations much of the material world can be analyzed. At physical sizes below molecules Maxwell’s description of nature becomes unusable. A quantum mechanical description of the electromagnetic field is required.

Such a description is provided by Quantum Electrodynamics. Starting with the classical description of the radiated electromagnetic field, this book makes use of a simple human experience — the receipt of radio signals — to explore the mathematical foundations of the electrodynamics.

Starting with the earliest experiments in electrostatics, Faraday, Maxwell and Hertzian formulations of the radiated field are described. The theory of antennas and electromagnetic reflection and refraction are explored. All of this material is a prelude to the quantum mechanical description of the electromagnetic field and its interaction with matter. In this description, the quantized field interacts with charged particles through the exchange of a particle which carries the electromagnetic force through free space — the photon. The behavior of this interaction at the quantum mechanical level provides new insight to the complexities of nature.

G. B. Alleman Niwot, Colorado

Copyright © 2001

Why Study Mathematics? I am amused, I said, at your fear of the world, which makes you guard against the appliance of insisting upon useless studies; and I quite admit the difficulty of believing that in every man there is an eye of the soul which, when by other pursuits lost and dimmed, is by there purified and re–illuminated and is more precious for then ten thousand bodily eye, for by it alone is truth seen. — Socrates to Glaucon in Plato’s Republic Book VII

Physics ... is essentially an intuitive and concrete science. Mathematics is only a means for expressing the laws that govern phenomena. — Einstein to Solovine in [Solo79]

Table of Contents

§0. PREFACE.............................................................................................................1

§0.1 Guided Preview..........................................................................................2 §0.2 Advise to the Reader.................................................................................5 §0.3 Historical and Mathematical Endnotes................................................5 §0.4 Electrodynamics Notation .......................................................................6

§1. THE FOUR FORCES OF NATURE .....................................................................1–1 §1.1 The Everyday Force of Gravity ..............................................................1–1 §1.2 Early Astronomy — The History of Theory .........................................1–4 §1.3 The Four Forces of Nature ......................................................................1–7 §1.4 The Particle Zoo.........................................................................................1–9 §1.5 Fundamental Forces in Quantum Chromodynamics ........................1–13 §1.6 Quantum Field Theory.............................................................................1–15 §1.7 Preliminaries to Modern Physics ...........................................................1–16 §1.8 Unifying Principals of Nature ................................................................1–17

§2. CLASSICAL FIELD THEORY..............................................................................2–1 §2.1. Electrodynamics ........................................................................................2–2 §2.2. Electrostatics and Early Experiments..................................................2–2 §2.3. Electromagnetic Interactions .................................................................2–6 §2.4. Unifying Electricity and Magnetism.....................................................2–7

§2.4.1. Lines of Force .......................................................................................2–9 §2.4.2. Beginnings of Field Theory ...............................................................2–12 §2.4.3. Removal of Action at a Distance.......................................................2–13

§2.5. Special Relativity and Electromagnetic Fields...................................2–14 §2.6. Light — Particle or Wave ........................................................................2–16 §2.7. Overview of the Wave Equation.............................................................2–18

§3. MAXWELL'S EQUATIONS..................................................................................3–1 §3.1. Maxwell's 1st Equation — Coulomb's Law...........................................3–3 §3.2. Maxwell's 2nd Equation — Absence of Magnetic Monopoles...........3–5 §3.3. Ampère's Law for Steady State Fields...................................................3–5 §3.4. Maxwell's 3rd Equation — Ampere's Law............................................3–7 §3.5. Maxwell's 4th Equation — Faraday's Law of Induction ...................3–9 §3.6. Newton–Lorentz Force Equation...........................................................3–11 §3.7. Coupling Strength of the Electromagnetic Field................................3–13 §3.8. Continuity Equations...............................................................................3–16 §3.9. Summary of Maxwell’s Equations..........................................................3–17

§4. SOLUTIONS TO MAXWELL'S EQUATIONS ......................................................4–1 §4.1. Vector Algebra Solution to Maxwell’s Equations ...............................4–1 §4.2. Vector Potential Solution to Maxwell’s Equations.............................4–1

Table of Contents

§4.3. Integral Form of Maxwell's Field Equations.......................................4–4 §4.3.1. Green's Function and the Potential Solution ................................4–4 §4.3.2. Field Potential Solutions ...................................................................4–6

§4.4. Traveling Waves........................................................................................4–8 §4.4.1. Displacement Current in the Field Equations..............................4–11

§4.5. Classical Explanations for Force from Fields......................................4–11 §4.6. Summary of Classical Field Theory ......................................................4–12

§5. THE RADIATED FIELD ......................................................................................5–1 §5.1. Plane Waves in Free Space .....................................................................5–2

§5.1.1. Longitudinal Propagation Components .........................................5–4 §5.2. Energy in the Radiated Field .................................................................5–4 §5.3. Poynting's Theorem..................................................................................5–5 §5.4. Vector Potential Description of the Radiated Field............................5–7

§5.4.1. Quasi–Stationary Expansion............................................................5–8 §5.4.2. Multipole Expansion...........................................................................5–9 §5.4.3. Radiation Expansion ..........................................................................5–9

§5.5. Polarization of the Radiated Field.........................................................5–10

§6. ANTENNAS AND RADIATED FIELDS ...............................................................6–1 §6.1. Time-Dependent Fields in Conductors .................................................6–3

§6.1.1. Wave Propagation in a Conduction Media ....................................6–5 §6.2. Electromagnetic Waves Incident on a Conductor ..............................6–9 §6.3. Summary of Maxwell's Classical Field Theory ...................................6–14

§7. PRINCIPLE OF RELATIVITY..............................................................................7–1 §7.1. Origins of Relativity Theory ...................................................................7–2

§7.1.1. Invariance of Newton’s Equations of Motion.................................7–3 §7.2. Velocity of Light and Absolute Motion .................................................7–4 §7.3. The Lorentz Transformation ..................................................................7–11

§7.3.1. The Components of Relativity ..........................................................7–12 §7.3.2. Relativity Principals Formally Stated............................................7–12 §7.3.3. Structure of Space–Time....................................................................7–17

§7.4. Covariant Notation ...................................................................................7–19 §7.4.1. Covariant Transformations...............................................................7–20 §7.4.2. Divergence and Curl in 4 Dimensions ............................................7–22

§7.5. Lorentz Transformation in Covariant Form.......................................7–22 §7.6. Maxwell's Equations in 4–Dimensions .................................................7–23 §7.7. Lorentz Transformation of Maxwell’s Equations ...............................7–27

§8. HAMILTONIAN MECHANICS.............................................................................8–1 §8.1. Newton’s Equations in Lagrangian Form............................................8–1 §8.2. Variational Description of the Equations of Motion ..........................8–3

Table of Contents

§8.3. Calculus of Variations..............................................................................8–6 §8.4. Ordinary Maximum and Minimum Theory.........................................8–7

§8.4.1. Lagrangian Formalism and the Calculus of Variations.............8–8 §8.5. Generalized Coordinates..........................................................................8–10 §8.6. Hamiltonian Formalism...........................................................................8–11

§8.6.1. Canonical Coordinates and Poisson Brackets...............................8–14 §8.7. Standard Lagrangian of Classical Electrodynamics .........................8–16

§8.7.1. Time Independent Lagrangian ........................................................8–18 §8.7.2. Lagrangian Density ............................................................................8–19

§9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD .......................................9–1 §9.1. Field Energy Density................................................................................9–4

§10. A PREVIEW OF QUANTUM MECHANICS.........................................................10–1 §10.1. Domination of Modern Quantum Mechanics.......................................10–1 §10.2. Early Quantum Theory............................................................................10–2 §10.3. Experimental Necessity for the Quantum Theory of Radiation......10–3

§10.3.1. Black Body Radiation.........................................................................10–4 §10.4. States of a Mechanical System ...............................................................10–6 §10.5. Quantum Mechanics of Electromagnetic Fields .................................10–7 §10.6. Preliminaries to Quantizing the Radiation Field...............................10–8

§10.6.1. Vector Potential Expanded as a Fourier Series............................10–8 §10.6.2. Planck’s Conclusions Using the Vector Potential.........................10–10

§10.7. Radiation Field Expansion Using Canonical Variables ...................10–12 §10.8. Schrödinger’s Equation............................................................................10–13

§10.8.1. Development of Schrödinger’s Equation ........................................10–14 §10.9. Formulating Schrödinger’s Wave Equation........................................10–16 §10.10. Schrödinger’s Time Dependent Equation ............................................10–17

§10.10.1. The General Solution to Schrödinger’s Equation.............10–18 §10.10.2. Semi–Classical Theory of Radiation ...................................10–18

§11. GAUGE THEORY.................................................................................................11–1 §11.1. Classical Mechanics Example of a Gauge Invariance.......................11–2 §11.2. Electromagnetic Fields and Gauge Transformations........................11–3 §11.3. Lorentz and Coulomb Transformations ...............................................11–8 §11.4. Gauge Symmetries and Potential Fields..............................................11–9

§11.4.1. Gauge Invariance and the Lagrangian..........................................11–11 §11.4.2. Symmetry and Conservation ............................................................11–12

§11.5. Gauge Particles and the Conveyance of Force....................................11–12

§12. MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS......................12–1 §12.1. Vectors and Vector Spaces.......................................................................12–1

§12.1.1. Abstract Vector Algebra.....................................................................12–2

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§12.2. Linear Functionals....................................................................................12–4 §12.2.1. Linear Operators .................................................................................12–5

§12.3. Dirac Notation and Linear Operators ...................................................12–5 §12.3.1. Measurable Properties .......................................................................12–8 §12.3.2. Quantum Operators ............................................................................12–9 §12.3.3. Commutators and Poisson Brackets................................................12–10 §12.3.4. Commutators and the Electromagnetic Field ...............................12–12

§13. POSTULATES OF QUANTUM MECHANICS......................................................13–1 §13.1 Basic Theoretical Concepts .....................................................................13–1 §13.2 The Four Postulates of Quantum Mechanics (according to Bohr) 13–2

§13.2.1 Postulate 1 and Postulate 2...............................................................13–2 §13.2.2 Postulate 3 ............................................................................................13–3 §13.2.3 Postulate 4 ............................................................................................13–3 §13.2.4 Postulate 5 and Schrödinger's Equation ........................................13–5 §13.2.5 Lorentz Force Law from Schrödinger's Equation.........................13–7

§14. FOUNDATIONS OF QUANTUM FIELD THEORY..............................................14–1 §14.1 Problems with QFT ...................................................................................14–1 §14.2 Simple Approach to QFT..........................................................................14–2 §14.3 Mechanical Analogy .................................................................................14–3

§14.3.1 Canonical Coordinates of the String ...............................................14–7 §14.3.2 Quantizing the Mechanical System ................................................14–8

§14.4 Canonical Momentum of the String ......................................................14–9

§15. QUANTIZING THE CLASSICAL RADIATION FIELD.........................................15–1 §15.1 Quantizing the Schrödinger Equation .................................................15–2 §15.2 Quantizing the Radiation Field .............................................................15–3

§15.2.1 Field Commutation Modes ................................................................15–3 §15.2.2 Zero Point Energy ...............................................................................15–3

§16. GAUGE THEORY AND THE CREATION OF PHOTONS....................................16–1 §16.1 Annihilation and Creation Operators...................................................16–1 §16.2 Photons States............................................................................................16–2 §16.3 Photons as Radiated Field Excitations.................................................16–3

§16.3.1 Total Hamiltonian...............................................................................16–4 §16.3.2 Photon Polarization ............................................................................16–5

§17. VACUUM STATE FLUCTUATIONS....................................................................17–1 §17.1 Radiation Density of the Quantized Field............................................17–2 §17.2 Radiation Damping and Self Fields.......................................................17–4 §17.3 Open Questions about the QFT..............................................................17–7

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§18. BIBLIOGRAPHY ..................................................................................................18–1

I write to discover what I think. — Daniel J. Boorstin, Librarian of Congress

Copyright 2000, 2001 1–1

§1. THE FOUR FORCES OF NATURE

We see the effects of force all around us. The force of gravity, the electric and magnetic forces of natural and manmade objects, and the mechanical force of machines all have well known effects. In pre–twentieth century science, natural philosophers asked many of the same questions that are asked here — why does nature behave in the way it does? Although these questions have the tone of theological or philosophical inquiries, the study of these forces and their interaction with matter is generally the domain of physics [Alio87].

The development of the concept of a force marks the boundary between science and pre–science [Jamm62], [Agas68], [Cajo29]. In early history, objects were believed to have internal powers, which could account for their movements. The motion of the planets through the night sky was associated with gods, and supernatural powers. It was realized during the time of Galileo that the function of a force was not to produce the motion, but to produce a change in the motion [Whit58], [Koyr55], [Jamm62], [Hawk87], [Roge60]. This description of force was not significantly different from the previous occult force, since the origin of the force was not known. However, these forces could be measured which allowed quantitative order to be brought to nature.

One of the most significant scientific developments in the past several centuries was the concept of a continuous field of force [d’Abro39], [Adai87], [Hess61], [Sach73]. This discovery replaced action–at–a–distance with action conveyed through a field. The application of this concept by 19th century scientists lead to a new understanding of electricity and magnetism which strongly influenced 20th century physics [Beck74]. The special theory of relativity exploited the concept of a continuous field to describe the motion of objects, including electromagnetic waves, independent of any special reference frame. The second revolution in 20th century physics was quantum theory, which describes matter at the atomic level in the form of fields. With electromagnetism's fields of force, special relativity's fields of geometry and quantum theory's fields of probability, the notion of a field is capable of describing nearly all aspects of physical processes [Sach73], [Agas68].

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I do not know what I may appear to the world; but to myself I seem to have been only like a boy, playing on the sea–shore, and diverting myself, in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me

— Isaac Newton [Brew55]

§1.1 THE EVERYDAY FORCE OF GRAVITY

The most familiar force in everyday life is the gravitational force, which unifies the behavior of objects on the human scale of a few centimeters to the galactic scale of 2510 cm . This force holds objects to the earth, it keeps the planets in their orbits, it maintains the path of stars in the galaxy and it forms the glue the binds the galaxies together. The strength of the gravitational force is proportional to the product of an object’s mass and inversely proportional to the square of the distance between the objects. Gravity is the only force that acts in the same manner between all types of matter. Neutrons, protons, electrons, and the matter they form all attract each other according to the law of gravity. Since the same law applies to all objects, gravity can be considered the result of the geometrical properties of space itself [Tayl66], [Hawk87]. Einstein formulated the general theory of relativity on this basis. Unlike Newton’s inverse square law of gravity, the strength of the gravitational force in general relativity is not a simple inverse square relationship. [1] Although the force of gravity dominates the human experience, it is in fact the weakest force of nature.

1 The concept of the gravity in Newtonian mechanics implies that a test particle is

subject to an external force — the gravitational force. This force acts in a linear fashion on the test particle as it travels through the gravitational field. In the General Theory of Relativity, the presence of the test mass influences the behavior of the gravitational force, so that the force felt by the test particle is non–linear. In the Newtonian view of gravity, the force field is static and can be represented by a scalar potential, just as the electrostatic potential can be represented by the Coulomb potential. When the electromagnetic field is not static — it is dynamic — the addition of the vector field is required to represent the complete system. These scalar and vector portions of the electromagnetic field can be represented by a 4–vector potential. The consequence of this form of representation is that electromagnetic disturbances are propagated with the speed of light. In Maxwell’s representation, the potentials satisfy the wave equation, rather than the Poisson’s static potential equation. In General Relativity the Poisson equation ∇ = − πκρ2 4U describing the

static gravitational potential is replaced by ∗∇ − = πκρ200 4g , where ∗ρ is the density of

mass–energy, not just mass and 00g is the metric tensor describing the curvature of space–time. Space–time is curved as a result of the presence of matter [Fran79], [Misn75].

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The discovery of the gravitational force was made by Sir Isaac Newton (1642–1726) while attempting to explain Johannes Kepler's (1571–1630) three laws of planetary motion [Koyr55], [Holt56], [Step94]. The history of Newton’s discovery of the laws of gravity is surrounded in popular myth. The 17th century laws governing the motions of celestial objects were regarded quite differently from those governing the motions of bodies on earth. The study of the motion of a heavenly body, particularly the planets and the sun, was the primary subject taught in the university in the mid 1600’s. Students of natural philosophy at Cambridge in 1664 discussed these motions in detail. In 1665, the plague broke out in England and classes at Cambridge were suspended [Manu68], [Chri97], [Shre70]. The 23 year old Isaac Newton student was sent home in June to Woolsthrope of that year and did not return until March of 1666 [Manu68]. While pursuing his B. A. Degree in the Lent Term of 1665, Newton remained home to think about the question of planetary motion [Sedg39]. He was apparently inspired as he saw an apple fall to earth in an orchard. [2] It occurred to

2 Newton’s contribution to the science of physics is well documented. His formulation of mechanics and his ideas of absolute space and time were not seriously challenged until Albert Einstein developed the theory of special relativity nearly 250 years after Newton, in 1905. Newton also invented the fluxional calculus, conceived the idea of universal gravitation, discovered its law, and discovered the composition of white light [Resn60].

In a biography written by Newton’s friend Dr. William Stukeley in 1752, Memoirs of Sir Isaac Newton, Stukeley states that he was having tea with Newton in a garden under some apple trees, when Newton said that the setting was the same as when he got the idea of gravitation, earlier as he noticed an apple drawn to earth in his mother’s Woolsthrope garden [Asim82], [West80], [Fren88], [Stuk36], [Manu68], [Chan95].

It was occasioned by the fall of an apple, as he sat in a contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself? Why should it not go sideways or upwards, but constantly to the earths centre? Accordingly, the reason is that the earth draws it.

Another account of this incident is given by Newton himself through the words of his associate John Conduitt:

Whilst he was musing in a garden came into his thought that the power of gravity (which brought an apple from the tree to the ground) was not limited to a certain distance from the earth but that this power must extend much further then was usually thought. Why not as high as the moon said he to himself and if so that must influence her motion and perhaps retain her in her orbit, where upon he fell to calculating that would be the effect of that supposition. [West80]

The particular tree under which Newton was to have been siting has been identified as a yellow–green cooking apple in the front garden of Newton’s home in Woolsthrope. When the tree collapsed in the 18th century, a cutting was grafted to another tree in the botanical garden of Kew outside London.

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Newton that the same force that attracts the apple to the earth could also attract the moon to the earth. Newton postulated that the centripetal acceleration of the moon in its orbit and the downward acceleration of a body on the earth might have the same origin. The idea that celestial motions and terrestrial motions followed similar laws was a major break in the tradition of 17th century science [d’Arbo27], [d’Arbo39]

Newton postulated that a universal attractive force between two bodies could explain the motions of the moon around the earth as well as the motions of the planets [3]. Before the time of Galileo, most natural philosophers thought that some external influence or force was needed to keep a body moving. They thought that a body was in its natural state when it was at rest. In order for a body to move in a straight line at constant speed, they believed that some external agent had to continually propel it along — otherwise the body in motion would naturally stop [Resn60].

3 Newton wrote down his laws of motion in Philosophiae Naturalius Principia

Mathematica, between 1684 and 1687. In this text, Newton collected his previous incomplete studies in mechanics and mathematics. The writing of Principia arose from a discussion at the Royal Society in 1684 between astronomer Edmond Halley (1656–1742) the architect Sir Christopher Wren (1632–1723) and Newton’s archival Robert Hooke (1635–1703) [Manu68], [Rona69], [Chri97], [Hall32]. The discussion revolved around the conjecture by Wren that the inverse square law implies that elliptical orbits of the planets must be produced. Hooke claimed that he had a proof of this theory, but could not actually produce the mathematics. Halley went to ask Newton the same question. Newton claimed he could prove this conjecture, but he also did not have the mathematics to back up his claim. Using Kepler’s observations, Newton produced, in April of 1685, a nine page paper (in Latin) De Motu Corporum (On the motion of bodies in Orbit), which described the elliptical paths of the planets in terms of the Laws of Gravitation and the Laws of Motion [Manu68]. This paper laid the foundation for the mathematical description of the laws of classical mechanics described in Principia, first published in 1687.

Newton reasoned that the forces between bodies must be the consequences of a force between particles, which make up the bodies. 22 years after the Lent Term, Newton consolidated his ideas in Principia. Newton wrote the Principia in three parts, using the methods of Euclidean geometry to derive his results. The first part describes the motion of a body from the forces acting on it. The second part describes the forces encountered in nature and the third examines the solar system and the motion of planets under the force of gravity. All of these subjects are developed through axioms, lemmas and theorems in the same manner as a Greek mathematical exposition. The result is a text that is very difficult to read, even by today’s standards, because of the geometric language. The differential and integral calculus that was invented for describing motion was not included in Principia.

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§1.2 EARLY ASTRONOMY — THE HISTORY OF THEORY

Early astronomy provides a clear example of the growth and use of theory in the development of a deeper understanding of nature. Astronomy is almost as old as mankind. When early civilization ventured outside their known world, trade routes were formed. These routes required navigation aides in order to be reliably traveled. The compass, clock and calendar became essential components of modern civilizations. Astronomy provided all three.

The relative individualism of the history of science as opposed to general history, is also due to the fact that if it is not altogether easy to analyze and to estimate a man’s contributions in the field of science, at least it is a good deal easier than is any other field, except art.

— George Sarton [Sart31]

The earliest attempts to describe the solar system were made by the Greeks in the 4th century BC. Aristarchus of Samos (310–230 B.C.) proposed a heliocentric system [Heat13], [Clag55], [Cole60], [Neug52]. Archimedes (287–212 B.C.) assumed the earth moved in an orbit whose radius, when compared to the fixed stars, was the same ratio of the center of the earth to its surface. A detailed description of the conclusions of the Greek astronomers was published in the 2nd century by (Claudius Ptolemaeus) Ptolemy [4] and described a geocentric system in which the earth is stationary at the center of the universe. The sun, planets and all stars revolve around the earth in complex orbits. This theory had great influence on the philosophy and literature for fifteen centuries. Since the theory was computationally complex, it could not be used to quantitatively

4 The Egyptian astronomer Ptolemy (90–168 AD) recorded his astronomical observations

in The Almagest (Arabic for The Greatest). The exact birth, death and publication dates of Ptolemy are not reliably known. He lived during the reigns of emperors Trajan, Hadrian, Antonious Pius and Marcus Aurelius from around 100 AD to 178 AD. He worked in or near Alexandria Egypt. Ptolemy drew on the works of Hipparchus of Nicaea (180–125 BC) [Sart31], [Clag55], [Farr49], [Ging93], who was a well-respected Greek astronomer. Hipparchus’ observations led to the development of trigonometry using theorems of similar triangles. From these theorems, the concepts of sine, cosine and tangent were defined.

Ptolemy also wrote Geography, which summarizes all Greek knowledge on the subject of maps, including various methods of projecting the surface of the earth onto flat maps. Ptolemy’s book was lost during the Dark Ages and cartography became a lost science. Ptolemy remained one of the greatest astronomers until Copernicus, Tycho and Brahe [Farr49], [Whit58], [Adle60].

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account for the increasing number of accurate observations of the motions of the stars and planets. In 1514 Nicolus Copernicus (1473–1543) suggested that a simpler description of the motions of the planets could be developed, by placing the sun at the center of the universe, with the earth, planets orbiting this center [Rose71], [Kuhn56], [Kuhn57], [Ging93], [Armi57], [Banv76].

Copernicus agreed that for certain phenomena, which were used to justify the evidence for the stationary theory of the earth, this evidence would not be altered if the earth moved and the sun was stationary.

The centre of the earth is not the centre of the universe. We revolve around the sun like any other planet. The earth’s unmobility (is) due to an appearance [Rose59].

The controversy over the heliocentric theory prompted astronomers to gather more accurate data about the motions of celestial objects. The observations made by Tycho Brahe (1546–1601), [5] recorded in Astronomiae Instauratae Mechanica, were analyzed and interpreted by Kepler, who had been Brahe’s assistant [Holt56], [Banv81].

Using the precise observations of Tycho Brahe, including error measurements, Kepler found regularities in the motion of the planets and formulated his three laws of planetary motion. [6] Kepler’s laws reinforced

5 Tycho Brahe observed a supernova in 1572, which bears his name. Tycho’s visual

observations were made with great care and were sufficiently accurate to deduce the rate of decrease of the brightest supernova of the time. He was able to make these accurate and systematic measurements with the help of instruments constructed with funds provided by King Frederick II of Denmark. He made several advances in measuring celestial objects. He derived methods for measuring the flex in the instruments. He corrected for the effects of refraction when stars were observed at different elevations above the horizon. He included the error values for his observations. These techniques are recorded in Epistolarum Astronomicarum. This information was vital for the proper interpretation of Brahe’s observation by Kepler 20 years later.

6 Kepler assumed circular orbits, but the closest he could come to describing the plant’s motion had an error of 6–8 arc minutes. This error was outside the error band of Tycho Brahe’s observations. The 6–8 arc minutes is equivalent to the width of a wooden pencil when viewed from a distance of ten feet.

From these eight minutes, we will construct a new theory that will explain the motions of the planets – Kepler.

Kepler’s next attempt used ovoid orbits and an inverse square law of the force driving the planet’s motion. Kepler attempted to use this law to describe the velocity of the planets in their orbits. After some difficulty, he intuitively adopted the idea of equal areas swept out in equal time – the aeral law.

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the Copernican theory and showed the simplicity with which planetary motions could be described when the sun was placed at the center of the orbital system. These laws described the motions of the planets using empirical data, without any theoretical interpretation. However, Kepler had no concept of the force that caused the planets to move with regularity. It was Newton’s great triumph that the laws of motion, using gravity as the force, could be derived from Kepler’s laws of planetary motion. Newton could account for the motion of the planets in the solar system and for the motion of the bodies falling near the earth with the same concept. He unified, in one theory, the theory of terrestrial mechanics and celestial mechanics. [7]

Kepler’s first two laws were published in Astronomia Nova (The New Astronomy: Based

on Causes or Celestial Physics) (1609) [Kepl09] and the third in Harmonice Mundi (Harmony of the World) (1619) [Kepl16]. Kepler's three laws are: (i) each planet moves in an elliptical orbit, with the Sun at one focus of the ellipse; (ii) the focal radius from the Sun to a planet sweeps equal areas of space in equal intervals of time; (iii) the square of the sidereal periods of the planets are proportional to the cube of their mean distance to the Sun. This third law can be stated as =3 2A kT where T is the period of the planet and A is the semimajor axis of its elliptical orbit and k can be given in terms of Newton's gravitational constant [Emch84].

Kepler’s first law expresses the constancy of the observed orbits and the total angular momentum of the plant–sun system. This observation which was seen as...

... a marvelous manifestation of the harmony of Nature. [Banv81]

This observation revels itself today as a consequence of the laws of dynamics. In fact, Kepler’s Law is incorrect because it is not the angular momentum of the plant–sun system that remains constant, but the angular momentum of the entire solar system. The angular momentum vector for the entire system is perpendicular to the invariable plane of Laplace. Fortunately for Newton, Kepler’s error has negligible impact because of the weak interaction between the plants compared to the interaction between the Sun and the planets [Doug90].

7 Newton was the first to state that his work was the culmination of the work of others. In a letter to Robert Hooke...

If I have seen further (than you) it is by standing upon the shoulders of Giants.

Although the quote of Newton has been popularized into a comment regarding the substantial works of previous scientist, it is more complex. The reference to the shoulders of giants is taken from John of Salisbury’s The Metalogicon [Spey94], [Thor90]…

Bernard of Chatres used to compare us to (puny) dwarfs perched on the shoulders of giants. He pointed out that we see more and further than our predecessors, not because we have keener vision or greater height, but because we are lifted up and borne aloft on their gigantic stature [Sals55].

The quote can actually be taken as backhanded slap at Robert Hooke (1635–1703), who Newton carried on a life time rivalry. Newton used this quote in a letter responding to Hooke’s claim that Newton stole the hypothesis on light from Hooke’s Micrographia

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The force Newton postulated would be proportional to the product of the masses of the two bodies and inversely proportional to their separation. Newton then developed the laws of motion that govern the path taken by a body in the presence of this gravitational force [Cajo62], [Fren88], [West80]. [8]

The most well known of these laws is Newton's second law of motion, ma=F ,[9] which states that a force F produces an acceleration on a body proportional to the mass of body — given the same force, light bodies are accelerated faster than heavy bodies. [10] This law describes the acceleration, force and mass properties of material bodies [Wein61].

[Hook61]. Newton was familiar with Micrographia and claimed that Hooke took much of that work from Descartes who — claimed Newton — took his work from Marcantonio de Dominis and Ariotto [Hall62], [Manu68], [Koyr65].

8 There is evidence in Newton’s student notebooks that he had learned of Kepler’s first and third laws from, Astronomia Carolina written in 1661 by Street [Robi90].

9 Unlike computer languages, which we are familiar with, mathematical notation is read left to right. As the above sentence says, a force produces an acceleration on a mass. In Einstein's General Theory of Relativity, accelerations produce force, so that Newton's Second Law has reciprocity. In general though equations do not exhibit such behaviors. Although the point seems trivial, the mathematics of physics, unlike the mathematics of computing (some would argue this), is a language in which physical phenomenon are described in a self contained manner. The language of mathematics is capable of describing the behaviors of nature that can be visualized — in addition, mathematics is capable of describing unobservable behaviors as well. It is possible to invent a mathematical model for a process of nature that has no equivalent visualization. There are several models of nature that can not be visualized.

The behavior of objects whose size is small compared to molecules can be described by quantum mechanics. The laws of physics at the quantum level may have no equivalent visualization in the classical world [Polk85]. The situation of increasing abstraction was predicated by Joseph Lamor…

There has been of late a growing trend of opinion, promoted in part by general philosophical views in the direction that the theoretical constructions of physical science are largely factitious, that instead of presenting a valid image of the relation of things on which further progress can be based, they are still little better than a mirage [Lam05].

10 Newton's three laws of motion are formally given in Philosophiae Naturalis Principia Mathematica (Mathematical Principals of Natural Philosophy) [Cajo62], [Andr56], [Asim82], [Motz89], [Cohe78], [Heri65] as:

Lex I (in editions of 1687 and 1713) – Corpus omne perseverare in statu suo movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare.

Lex I (in edition of 1726) – Corpus omne perseverare in statu suo

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§1.3 THE FOUR FORCES OF NATURE

Although it is the electromagnetic force that is of interest here, three other forces exist in nature, the gravitational force, the nuclear or strong force and the weak force. These four forces are the source of all the variety in the universe [Neem86]. Without them attraction and repulsion of physical bodies would not be possible and interaction between matter would not take place. Bodies would simply pass through each other with no effect.

Gravity or the gravitational force was first identified by Isaac Newton in the 1680's. Although the gravitational force acts on all matter, its strength is the weakest of the four forces. As humans, we are conscious of

quiescendi vel movendi uniformiter in directum, nisi quantenus illud a viribus impressis cogitur statum suum mutare. (Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.)

A Body at rest remains at rest and a body in a state of uniform linear motion continues its uniform motion in a straight line unless acted on by an unbalanced force. This law is often called the law of inertia. This means that the state of motion in a straight line remains at rest of continues its uniform motion unless acted on by an unbalanced force. The presence of the unbalanced force is indicated by changes in the state of motion of a body.

Lex II – Mutationem motis proportionalem esse vi motrici impressae, et fieri secundum lineam qua vis illa imprimitur. (The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed).

An unbalanced force, F, applied to a body gives it an acceleration, a, in the direction of the force such that the magnitude of the force divided by the magnitude of the acceleration is a constant, m, independent of the applied force. This constant, m, is identified with the inertial mass of the body. The inertial mass is a derived rather than basic quantity. Newton's equations of motion establish a procedure for measuring this mass. This is done by applying a known force to a body and measuring its acceleration. The result of this measure is the mass of the body. There is an additional interpretation of the second law of motion. If a body is observed to be accelerating than a force must be acting on it, but if no force is known to be physically applied to the body, Newton concluded that this force must act–at–a–distance.

Lex III – Actioni contrariam semper et aequalem esse reactionem: sive corporum duorum actiones is se mutuo semper esse aequales et in partes contrarias dirigi. (To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.)

If a body exerts a force of any kind on another body, the latter exerts an exactly equal and opposite force on the former. This law introduces a symmetry that does not appear in the first two laws. It states that forces appear in equal and opposite pairs.

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the force of gravity only because of the immense mass of the earth and celestial objects.

The weak and strong forces are not detectable at human scales, not because of their relative strength but because of their short range. The weak force has a range of 1710 m− to 1810 m− . At distances small compared to the range of these forces, both the strong and the weak force obey the inverse square law the same as the gravitational and electromagnetic forces [Hugh91]. Although unfelt by humans, the weak force plays a critical role in the generation of energy in the sun and the building of heavy elements through nuclear synthesis [Kane93]. The weak force is also responsible for the instability of neutrons. Although neutrons are stable within the nucleus of an atom, under the influence of the weak force a neutron placed in isolation will split into a proton, an electron and an antielectron neutrino within fifteen minutes. [11] This instability is called Beta Decay.

In 1958 Robert E. Marshak (1916–1992) and E. C. G. Sudarshan (1931 – ) observed that the weak force appeared to involve an action between two currents similar to the attraction or repulsion between two current carrying wires [Neem86], [Mars92]. In the 1960's and 1970's a theory emerged which unified the weak force with the electromagnetic force — the electroweak force [Rent90], [Mars92].

The strong force which acts between protons and neutrons (nucleons) is effective only when the nucleons are within 1510 m− of each other. The strong force is responsible for the interactions between nucleons, nucleons and mesons and a number of other particles. The nucleus contains both protons and neutrons and the electrostatic repulsive force of protons must be overcome by an attractive force in order to maintain the stability of the nucleus. Since the 1930’s, some form of nuclear force has been postulated. In modern particle physics, it is believed the quarks are the particles that undergo strong nuclear interactions and are described by the theory of Quantum Chromodynamics (QCD).

A century after the discovery of the gravitational force, Charles A. Coulomb (1736–1806) measured the electrostatic force acting between two

11 One of the first experimental confirmations of the neutron’s instability was performed by Enrico Fermi (1901–1954). Using the atomic pile at the University of Chicago, Fermi placed an evacuated spherical container inside the reactor. After some time some of the fission neutrons while passing through the container would decay into a proton an electron and an antielectron neutrino. The electron and the proton would become trapped in the container and be combine to form hydrogen gas. The rate at which the gas formed could be used to estimate the neutron’s mean half–life of approximately 14 minutes [Gamo65].

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charged bodies. Like Newton's inverse square law for gravity, the electromagnetic force obeys an inverse square law — Coulomb's Law. Instead of being proportional to the masses of the bodies, the electric force is proportional to the product of the bodies’ electric charge. Since electric charges can be positive as well as negative, the electric force can attract as well as repel bodies.

In the late 19th century, the effects of magnetism were carefully measured and it was determined that magnetism was a force created by the current produced by the motion of electrically charged objects. The electromagnetic force was first thought to be two unrelated forces, electricity and magnetism. Experiments showed that they were connected and are a single force.

Although the electrostatic force acts only between charged bodies, the electromagnetic force can effect uncharged bodies as well. The neutral charged neutron has a non–zero magnetic moment and is influenced by magnetic fields. The photon, which has no charge or magnetic moment, is effected by the electromagnetic force during its absorption and remission by atoms.

§1.4 THE PARTICLE ZOO

The study of the universe can be described as the search for the basic constitutes of matter, the forces that effect this matter and the calculation of the motions of this matter given these forces [Kane92]. Starting with the Greeks and Chinese, there have been theories that describe the behavior of matter. The Greeks thought all matter was made up of four elements — air, fire, water and earth. This atomistic theory originated with the Greek philosopher Leucippus, the probable founder of the School of Abdera in Thrace, 5th century B.C. [12] This school of thought claims that both empty space and the matter composed of atoms that filled the space are real. The changing world was described in terms of the isolation of groups of atoms, which was in direct conflict with the views put forth by the teachings of the Eleatic School of Parmenides of Elea (515–450 B.C.), which stated that everything that had existed had always done so and could never change.

12 Little is known of the life of Leucippus. He was probably a contemporary of

Empedocles (490–435 B. C.) and Anaxagoras (499–428 B. C.) [Gres64] and possibly a pupil of Zeno of Elea (~462 B.C.). Leucippus assumed the existence of empty space as well as matter and held that all things are composed of atoms. Space is infinite in extent and atoms are infinite in number and are indivisible. The atoms are always engaged in activity and the worlds produced by them have various shapes and weights [Sedg39].

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The Chinese on the other hand thought there were five elements rather than four — metal, wood, water, fire and earth and named five planets accordingly.

Along with Newton’s work in celestial mechanics, he laid the groundwork for particle physics. Newton’s reasoning is considered traditional and theologically based in the times.

All these Things being considered, it seems probable to me that God in the beginning formed Matter in solid, massy, hard, impenetrable, moveable Particles, of such Sizes and Figures, and with other Properties, and in such Proportion to Space, as most conduced to the End for which He formed them; and that these primitive Particles being Solids are incomparably harder than any porous Bodies compounded of them; even so very hard, as never to wear or break in Pieces; no ordinary Power being able to divide what God himself made in the first Creation ... And therefore that Nature may be lasting, the Changes of corporeal Things are to be placed only in the various Separations and new Associations and Motions of these permanent Particles. [Cohe52]

With the beginning of chemistry in the early 17th century, John Dalton (1776–1844) proposed there was an elementary component within each element, which itself was unalterable, called an atom. [13] In the middle of the 19th century, Dmitry Ivanovich Mendeleyev (1834–1907) discovered that the chemical elements could be classified into a table that had a periodic structure.

In 1897 the electron was discovered by Joseph John (J. J.) Thompson (1856–1940) [Thom99] followed by the discovery of the nucleus of the atom by Ernest Rutherford (1871–1937) in 1911 [Ruth11]. These discoveries resulted in a model of the atom based on the planetary like motion of electrons orbiting the nucleus. The nucleus of an element can have the same numbers of electrons but have a different mass and still have identical chemical properties. These elements are called isotopes. The study of chemical isotopes suggested that it is the number electrons in the element that is responsible for its chemical properties. With the discovery of the neutron by Sir James Chadwick (1891–1974) in 1932, the behavior of

13 In the Principia, page 6, Newton laid the foundation of the atomic theory ...

Because the hardness of the whole arises from the hardness of the parts, we ... justly infer the hardness of the individual particles not only from the bodies we fell but of all others.

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isotopes was explained [Chad32]. Since the electrically neutral neutron resides in the nucleus it has no effect on the chemistry of the element, but changes its atomic weight. With the additional discovery of the proton, the description of the nucleons making up the nucleus of the atom was complete. Isotopes are now understood to be elements with different numbers of neutrons, but the same numbers of proton and electrons.

With the detection of cosmic rays in the 1930’s, other particles were discovered to exist. Using accelerators, still other constituents of matter were produced through the collisions between particles. The existence of these particles lead to the discovery of the nuclear force and the classification of particles that are subject to the nuclear force — hadrons. Protons and neutrons are hadrons that are held together in the nucleus of the atom by the nuclear force. Electrons are not hadrons since they are held in the atom by the electromagnetic force. By 1939 the fundamental constituents of matter were composed of the proton (p), the neutron (n), the electron (e) and the neutrino (ν ), plus their anti–particles [Mars93]. After World War II, the number of particles exploded. Using accelerators hundreds of particles were created adding to the complexity of the underlying structure of nature.

This situation can be simplified if there is some order given to the vast zoo of particles. The first approach is to classify a particle by how it behaves in the presence of an identical particle. This can be done by describing the statistics of its interaction between large numbers of identical particles. Two types of particles exist using this description — fermions and bosons. If a number of identical fermions are placed in a confined area, they will statistically tend to avoid each other. If a number of identical bosons are placed in a confined area together, they will statistically tend to stay together [Kim91].

A second method of classifying particles is by describing their interaction with the forces of nature. By sorting through the remnants of the particle collisions, it was discovered that there are two nuclear forces at work, the weak nuclear force and the strong nuclear force. Particles subject to the weak nuclear force and the electromagnetic force are a class of fermions called leptons. These leptons are — the electron (e), muon (µ ), tau (τ ) and the neutrino (ν ). The particles that are subject to the strong nuclear force remained hadrons. The hadrons can be further classified into mesons and baryons. [Clos87], [Clos86], [Dodd84], [Frau74], [Schw92].

As early as 1964, Murray Gell–Mann (1929– ) and George Zweig (1937– ) independently produced a theory that would explain the growing complexity of the hadrons and their interaction. Their original theory

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described a universe made up of three types of elementary particles: (i) quarks, which come in two flavors, up and down, (ii) the electron and (iii) electron neutrino. In this theory, forces including gravity are carried by other particles — gauge bosons [Bloo82].

In 1970, there was no theory capable of describing the strong force. A nuclear force had been postulated in the 1930’s since the nucleus contains several protons that must be held together while the electrical force attempts to pull them part. It is the quarks that participate in the strong force [Ishi82], [Clos79]. Quarks carry color charge and combine to make color neutral hadrons just as electrically charged electrons combine with charged nuclei to form electrically neutral atoms [Chew64].

The material of the universe can be described as being made up of leptons and quarks, which are held together by the force carrying bosons. The force carriers are the photon for the electromagnetic force, gluons for the strong force and the 0, andW W Z+ − for the weak force [Garv93].

The theory of quarks and their interaction with each other and other matter is called Quantum Chromodynamics (QCD). The simplifying theory of quarks quickly became complex as more behaviors of hadron interaction were discovered at higher collision energies.

The original up and down quarks were joined by four other quarks named charm, strange, top, and bottom [Namb76]. These six quarks and their related leptons can be classified into three generations. The first generation makes up the matter we see in everyday life. The constituents of the second and third generation are unstable at normal energies and are only produced in accelerators — or during the formation of the universe.

The six quarks can be arranged into three groups or doublets u c td s b

. The top rows of quarks have charge 2 3 and the bottom row

have charge 1 3− . The six leptons can also be arranged in three doublets

e

eµ τν ν ν

µ τ where e is an electron, µ is a muon, τ is a tau all of

which have charge –1, while each particles' neutrino has no charge [Namb76], [Neem86], [Okun85]. [14]

14 Neutrinos are massless (or nearly massless) particles with no charge. The neutrino

was invented by Wolfgang Pauli (1900–1958) in 1930 to account for the missing energy created during Beta decay [Rein79]. Beta decay was discovered in 1896 when it was observed that certain atoms decay into other atoms [Frit83]. Early theories of Beta decay predicted

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that the neutron in the nucleus of an atom would be changed into a proton and a free electron [Lipk62]. It was also predicted that the products of the decay would conserve charge, energy and momentum with a fixed value. In an experiment performed in 1927 it was found that the free electron produced by the Beta decay had a continuous spectrum of energy values, contrary to the theory [Sutt92], [Brow78].

Pauli's new particle was needed to carry off the momentum and energy, preserving the conservation laws that were violated by the earlier naive models [Sutt92]. This particle was named the neutrino after Enrico Fermi developed the theory of Beta decay and was quoted as saying...

It is a little neutron, it is a neutrino [Rein79].

Pauli’s original particle was named the neutron since today’s chargeless particle called the neutron had yet to be discovered. Pauli’s neutron name was not copyrighted since it only appeared in private correspondence and never in print. In 1932 James Chadwick (1891–1974) presented evidence of a neutral charge particle with nearly the same mass as the proton he called the neutron. When Enrico Fermi (1901–1954) reported Chadwick’s discovery, a member of the audience asked if Chadwick’s neutron was the same as Pauli’s neutron, Fermi answered...

No, the neutrons of Chadwick are large and heavy, Pauli’s neutrons are small and light, they have to be called Neutrinos [Gamo65], [Ferm54], [Segr70].

The neutrino has an extremely low interaction rate with other forms of matter. In a cubic centimeter of water there are approximately × 227 10 free protons available in the nuclei of hydrogen. The protons in the nuclei of oxygen are bound and unavailable for any interaction. A neutrino passing through this cubic centimeter of water has one chance in

4410 of being captured by any one of the 2210 protons. The result is one chance in 2210 of any proton capturing the neutrino — very low odds. Converting this probability to a human scale it would require 22 310 cm of water to capture a single neutrino. This length is 1000 light years or 63,000 times the distance between the sun and the earth [Sutt92].

Free neutrinos were first observed in 1956 by Fred Reines (1918 – ) and Cylde L. Cowan (1919 – ) using a liquid scintillator placed in a neutrino beam generated by a nuclear reactor [Cowa56], [Rein56], [Rein5]. Their first proposal was to place the scintillator 40 m from ground zero during the test of the first atomic bomb. After 100 days of operation over a period of a year, on June 14th, 1956 Reines and Cowan captured the poltergeist particle [Rein79], [Rein79a], [Rein94], [Krop94].

This discussion of neutrinos may seem far removed from the goal of the book, but it does have several connections. The speculation of the existence of the neutrino by Pauli and its subsequent theoretical prediction by Fermi lead to the theory of Beta decay. Fermi's theory was built on a quantum field theory in which particles need not preexist but can be created from a vacuum [Bern89]. No theorist was saying the neutrinos preexist inside the nucleus and are ejected during Beta decay. They are rather created during the Beta decay process, then ejected [Brow78]. The concept of the creation and subsequent annihilation of particles will be used later in the quantum field description of the electromagnetic field.

The second connection is between quantum field theory and observational astronomy. On the night of February 23, 1987 a star named Sanduleak (SK) –69° 202, cataloged by Nicholas Sanduleak in 1969, located in the region of the Tarantula Nebula, on the edge of the Large Magellanic Cloud became the first supernova to occur in our own galaxy in four

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Why matter is composed of leptons and quarks and why these leptons and quarks should be arranged in families with specific masses is not known. The search for the answer to this question is the quest of the current generation of physicist [Wein93], [Lede93]. Using the quark model, material objects can be built from these particles. Protons, neutrons, pions, etc. are built from quarks. Since these hadrons are constructed from quarks, they are not considered elementary [Robe79]. [15]

§1.5 FUNDAMENTAL FORCES IN QUANTUM CHROMODYNAMICS

In Quantum Chromodynamics, the quarks that compose hadrons are bound together by gluons. The residual force of the gluon, when seen outside a hadron becomes the nuclear force that binds hadrons into stable nuclei. The electrically stable nuclei and the only electrically stable lepton — the electron — are bound into atoms by the electromagnetic force. The residual electromagnetic force outside the atom binds atoms into molecules. Since these molecules form the basis of life, the study of particle physics can be considered of primary interest to mankind [Geor81], [Wein93], [Lede93], [Geog80], [Frit83], [Barr91].

centuries. Other than the observation of this very rare event SK 69° 202, a.k.a. SN 1987A was important for what was not seen by the astronomers. A burst of approximately × 581 10 neutrinos, lasting nearly 6 seconds were emitted from SN 1987A. Nearly 30 million billion of the neutrinos then passed through a detector located 2000 feet deep in a salt mine in Painesville, Ohio. Out of these particles × 1530 10 neutrinos, 8 interactions occurred. Nearly three hours later the visible photons from SN 1987A arrived at the telescopes in the Southern Hemisphere. The energy necessary to produce × 581 10 neutrinos is approximately

× 583 10 ergs sec or × 201 10 times the total energy production of the sun.

15 Neutrons are composed of 2 down–quarks and an up–quark whose charges are summed produced a neutral particle ( ) ( )( ) ( )− − = =1 3 1 3 2 3 0 0 0 while a proton is

composed of two up–quarks and one down–quark whose charge is ( ) ( )( )− =2 3 2 3 1 3 1 .

Other hadrons are composed of three different quarks and are called baryons, while other hadrons are composed of a quark and an anti–quark and are called mesons.

There is symmetry between quarks and leptons in that leptons have integral units of charge while the electrical units of quarks are multiples of 1/3. This factor of 3 is actually accompanied and compensated by another factor 3; each quark comes in three invented labels called color, which is simply a quantum number for the behavior of quarks, not an actual color as we know green or red.

All of these behaviors are described through a theory based on group symmetries called SU(3) or the Eightfold Way [Gell64], [Dodd84], [Clos83]. In this theory the nucleons belong to a multiplet of eight and the pions and kaons, which are mesons with quark contents

, , ,su sd us ds , belong to a separate multiplet which also has a multiplicity of eight.

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In addition to quarks and bosons there is one more particle needed to complete the theory in a consistent manner — the Higgs Boson (Peter Ware Higgs (1929– )) [Velt86]. The theory of the electroweak interaction and the large masses of the 0, andW W Z+ − particles requires one electrically neutral Higgs Boson [Clin82], [Clin74]. The field produced by the Higgs Boson is a background field pervading all space, ever present, even in the vacuum state [Guid91]. The presence of this field produces an energy density in the vacuum which would curve space–time through the gravitational interaction. At this time, there is no experimental evidence for the Higgs Boson, but the search continues [Roln94].

The electromagnetic and gravitational forces have long range, the weak and nuclear forces have short ranges, all four forces obey an inverse square law. Why are their ranges different? Why do all these forces obey the inverse square law? What generates these four forces? How are the forces conveyed?

Although the four forces of nature all appear to follow the inverse square law, the force binding quarks together — the chromostatic force — behaves differently. The electrostatic force described by ( ) 2 4V e= − πr r is replaced by ( )V = κr r , when r is very large. This chromostatic force behaves Coulombic when r is small, but the potential increases linearly for large r. The result is that this force permanently confines the quarks inside their host [Adle81].

A theory of the electromagnetic force must not only explain observable effects of electromagnetic fields, but also must explain the source of the forces and the mechanism that conveys these forces. The search for the answer to these questions is similar to the mathematically undecidable question in Gödel's Theorem (Kurt Gödel (1906–1978)) [Gode62], [Hofs79], [Nage58], [Penr89b] — that in order to describe sufficiently one set of axioms, an external (super) set of axioms (theories) is needed, which in turn requires another external set of axioms. [16] Gödel summarized this

16 Gödel's Theorem appears as Proposition VI in his 1931 paper, "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". It states: "To every ω –consistent recursive class κ of formulae there correspond recursive class–signs r, such that neither ~Gen r nor Neg(~Gen r) belongs to κ( )Flg (where v is a free variable of r).

In layman's terms this says: All consistent axiomatic formulations of number theory include undecidable propositions — arithmetic is not completely formalizable. Gödel observed that a statement about number theory could be about a statement of number theory (possibly even itself), if only the numbers could somehow stand for statements. Gödel's work was part of a long attempt to define what proofs are. Proofs are

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dilemma in the development of the understanding of nature, using the tools of mathematics as…

The human mind is incapable of formulating (or mechanizing) all its mathematical intuitions, i.e. if it has succeeded in formulating some of them, this very fact yields new intuitive knowledge, e.g. the consistency of this formalism. This fact may be called the incompletability of mathematics. On the other hand, on the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discoverable) a theorem–proving machine which in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor even be proved to yield only correct theorems… [Gode51].

demonstrations within fixed systems of propositions. Gödel was saying that the system of Principia Mathematica [Whit27] is incomplete — there are true statements of number theory that its methods of proof are too weak to demonstrate. (The Principia Mathematica is a monumental work consisting of 4 volumes that attempted to build the foundation of mathematics upon a paradox–free set of logical axioms.)

The concept that mathematics is nothing but symbols in some formal mathematical system is the definition of formalism in which mathematics becomes a meaningless game. Gödel dealt formalism a devastating blow with his theorem and restored meaning to the symbols. In the world of physics it is the meaning of the symbols that provides the means for describing nature through mathematics.

In the early 1920's the logician Alfred Tarski (1901 – 1983) took Gödel's argument further to show that logical systems are also semantically incomplete as well [Tars56]. He showed that if a mathematical system is consistent then the notion of truth is not definable. The result of this discovery is that logical and mathematical systems are logically incomplete in that there is no formal system in which the truth of all mathematical statements could be decided or in which all mathematical concepts could be defined [Barr92]. The totality of mathematics cannot be brought to complete order on the basis of any system of axioms. Just as Heisenberg had done for the physical sciences, Gödel ended the search for certainty in mathematics [Asim82b].

In the proof of Gödel’s theorem there are two important notions that must be dealt the simultaneously: (i) the notation that mathematics is simply the manipulation of symbols and (ii) the concept that a mathematical proof can be substituted for the concept of the truth. Together these concepts provide for the translation of a verbally stated logical paradox into an arithmetic statement. One example of such a paradox is the Liar’s Paradox: This sentence is false.

All of this background would be of no interest if not for the logical paradoxes that will be encounted when quantum mechanics is applied to the measurement and behavior of photons as they interact with matter [Bohm51], [Bohm62], [Stew92], [Cast96].

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In the realm of classical physics, these questions are unanswerable and perhaps meaningless — the force is just there. In the quantum world however new answers may be found. [17]

§1.6 QUANTUM FIELD THEORY

In order to address the questions raised in classical physics, a new theory has been developed — Quantum Field Theory [Gros93], [Guid91], [Hari72], [Itzy80], [Kaku93], [Mand84], [Brow90], [Chai84], [Chen83], [Quig83], [Roma69], [Visc69]. When quantum mechanics and special relativity are combined, the resulting description of nature is based on the interaction of quantized fields and their associated force carrying particles — gauge bosons. Like classical field theories, quantum field theory describes a force created when one particle acts on another particle, after an appropriate delay due to the finite propagation speed of light. When quantum theory is used, the energy carriers of the force can only assume discrete values. It is these quanta of the field energy that are identified with the particles that transmit force. In quantum field theory, the interaction of elementary particles is interpreted as the exchange of force carrying particles among the material particles.

It will be shown later that these interactions obey specific symmetry rules and the force of the interaction is proportional to a charge of some kind. In such theories, the interaction of objects takes place locally in the form of the creation and annihilation of particles. Forces are transmitted by the propagation of particles known as exchange particles. These exchange particles have properties of mass, spin and charge, like the material particles. The four forces of nature differ because the exchange particles differ. In quantum field theory, the electromagnetic force is conveyed by an exchange particle — the photon.

17 Although it may be meaningless to ask the question how is the force conveyed using

the vocabulary of classical mechanics, quantum mechanics produces a similar set of meaningless questions. Heisenberg's uncertainty principle restricts the description of the words position and velocity to any accuracy exceeding the uncertainty relation [Heis30]. Heisenberg cautioned that

... one should be particularly careful to remember that the human language permits the construction of sentences which do not involve any consequences and which therefore have no content ...

The use of the words reality often leads to a picture of a physical process that can neither be proved nor disproved. The description of the physical process of the electromagnetic force will become increasingly abstract as this book progresses. The conceptualization of modern physical theory will be the most difficult hurdle to overcome.

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According to the quantum theory of Maxwell's electrodynamics — quantum electrodynamics (QED) — electromagnetic forces between two charged particles are generated by the transmission of a massless gauge boson — the photon — between them. [18] This photon brings with it a command from one particle to another. The receiving particle obeys the command of the arriving gauge particle with the result interpreted as the conveyance of the force. In this theory, the photon passes a force message to the receiving particle, rather than imparting some physical force to the receiving particle. In the classical electrodynamics, during the transmission of radio waves, external energy is available to communicate commands. Such energy is not always available in the quantum world of charged particles. According to the uncertainty principle of quantum mechanics, this energy can be borrowed for a short time to enable the command to be carried by the exchange particle, thus allowing force to be conveyed while obeying the conservation of energy.

§1.7 PRELIMINARIES TO MODERN PHYSICS

Modern physics is so strongly based on quantum models of the microworld it is difficult to bridge the gap between classical electrodynamics and quantum theories of Maxwell's equations and their interaction with matter. A course of instruction at a university covers classical electromagnetic theory, developing an understanding of electrostatics, magentics and Maxwell's equations. This knowledge is then used in the formulation of electromagnetic radiation and its practical applications, usually in antenna theory, wave guides and electro–optics. In parallel, the student develops an understanding of quantum mechanics. This material describes the duality of particles and waves, which is the basis of the descriptions of atomic and subatomic phenomena. What is missing from these parallel courses of study is the description of the electromagnetic field as a Quantum Field Theory. Although the development of the quantum description of Maxwell's radiation field presented prior to proceeding with the Quantum Field descriptions of subatomic particles, the details of the

18 The massless nature of the photon is a consequence of the gauge (phase) symmetry

of the electromagnetic field. The photon’s masslessness also guarantees that the electric charge is conserved since the symmetry responsible for the masslessness is a result of the invariance of Maxwell’s equations to arbitrary phase changes in the quantum fields associated with the electron and the photon. The arbitrary phase changes in the electron field can be compensated for by the redefinition of the photon field which leaves the form and structure of the equations of motion for the electromagnetic interactions unchanged [West93].

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Quantum Radiation Field of Maxwell's equations is not fully developed at the undergraduate level.

There will be an attempt to bring some understanding to the phenomenon of the electromagnetic force, through the various theories beginning with Classical Electrostatics, Electrodynamics, Quantum Mechanics and concluding with Quantum Field Theory. Because of the compressed form of this book, mathematical expressions are brief and usually given in non–rigorous form and in some cases simply stated without supporting derivations. Whenever possible the derivation or expanded material is given in an endnote.

The electromagnetic force is well understood from the view of classical physics. Its characteristics include: an infinite range, allowing macroscopic phenomena to be observed and a reasonably strong force, allowing microscopic phenomena to be observed. It is the interaction of the electromagnetic force with microscopic matter that leads to the formulation of Quantum Field Theory (QFT) — the goal of this thesis.

The path taken from classical electrodynamics to quantum field theory will often take a diversion to cover background material needed to illuminate the primary subject. This approach is necessary, since the reader may require an additional understanding of the mathematical developments. [19] The unification of the concepts found in the literature and texts as well the diverse notation has been a significant effort. I apologize for any over simplifications that may have entered the text. Finally, none of the material presented here should be considered original, but is a compendium of ideas found in the literature given in the bibliography. Direct quotations have been kept to a minimum, but there is material which is best conveyed verbatim from the source. In such cases, references to the original text are given within the sentence. All other references provide the reader with a rich set of source materials to continue the search for understanding of the primary forces of nature.

19 Much of the difficulty in learning physics comes from understanding the background

material, including the mathematical notation.

In any branch of science the terminology becomes so cumbersome in the process of its progress that it is very difficult to put it in a simple way for a reader who encounters all these complicated notations for the first time [Gamo66], pp. 68.

A second problem is the notation used in the description of electromagnetic phenomena can be somewhat cryptic when compared to ordinary notation of calculus and differential equations. When the notation is beyond the norm, footnotes will be used to provide additional information on the subject matter.

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§1.8 UNIFYING PRINCIPALS OF NATURE

One of the great achievements of 20th century physics has been the attempted unification of the four forces of nature. Although Einstein first proposed the grand unified theory, it wasn’t until the 1970’s that any serious progress was made. The first unification took place at the end of the 19th century, when Maxwell formulated the theory of electro–magnetism. In modern terms – Quantum Electrodynamics (QED) – all phenomena can be understood in terms of the force–carrying particles exchanged between electrically charged matter and the photon [West93].

In the last 20 years, the four forces of nature, electromagnetic, weak, strong and gravitational are being described by a small number of unifying principals [West93]. Four of these principals will be developed further in this book.

The first is the Principal of Relativity, which restricts the kinematic description of the motion of particles and fields, both classical and quantum. This principal states that the laws of physics are independent of spatial location and the passage of time, while being observed in a uniformly moving reference frame.

The second is the Principal of Stationary Action which describes the motion of both classical and quantum mechanical systems in terms of the minimization of action.

The third is the Gauge Principal, which describes the rules governing the interaction of fields of force with material particles. This principal is formulated as a classical description of nature, but it has inherent quantum mechanical properties. It may also be formulated as a relativity principal which allows internal and external properties of a physical system to be described independent of the observers reference frame.

The fourth principal is the Quantum Principal, which states that all physical systems in nature are inherently quantizable. This principal is founded on extensive experimental experience.

Using these four simple principals, it is surprising that the basic properties of various forms of matter and the interaction between matter can be described in detail. One question raised in the late 20th century is whether there is a Theory of Everything [Davi88], [Wein77], [Wein77], [Wein93], [Lede93]. A theory in which the forces and matter of nature are described with a very small number of principals, which would reduce the current gap between the mathematical model of the world and physical reality.

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The cornerstone of all physical sciences is experiment. The understanding of the basic forces of nature is advanced through the interplay of physical experiment and theoretical ideas. The largest impediment in the formulation of the Theory of Everything is the increasing energies needed to probe deeper into nature. With this increased energy, comes increased cost. At the energy level required for complete unification of natures forces — the Planck energy 191 10 GeV≈ × — the direct observation of this unification may be beyond our ability to fund the experiments [Doug90].

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I wish we could derive the rest of the phenomena of nature by the same kind of reasoning as for mechanical principals. For I am induced by many reasons to suspect that they may all depend on certain forces.

— Preface to the 1st Edition (1686) of Newton’s Principia

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§2. CLASSICAL FIELD THEORY

Newtonian mechanics and the theory of gravity dominated much of the physical sciences into the middle of the 19th century. When applied to astronomy, the mathematics of Newton's equations of motion yielded dramatic results. Problems such as the rise and fall of tides, the flow of rivers, the orbits of projectiles and the motion of machinery were well understood. Compared to the ease of these mechanical problems, the discovery of light and the study of the structure of matter were significantly more difficult. Much of the work on electromagnetism and physical chemistry took place by trial and error. Unlike Newtonian mechanics, these branches of science had no mathematical basis from which to develop. [1]

In the physical sciences, the word field describes a continuous distribution of some type of condition, which pervades a continuum [d’Abr39]. The nature and magnitude of this condition can take on many forms. If the condition can be described by a single valued function for each point in space, then a scalar field is said to exist. A temperature distribution in a volume of gas is an example of a scalar field. In many cases, the condition at each point in space has a direction as well as a magnitude. In this case, a vector field is said to exist. A field of velocities of a fluid in motion is an example of a vector field. The distribution of stress in an elastic medium can be described by a tensor field, as can the gravitational field using a 4–dimensional space–time coordinate system [Sope75].

Before the development of modern electrodynamics, all field theories

1 The description of nature using the language of mathematics was introduced by

Pythogorus. The Greeks before him envisioned the world in terms of matter. Pythogorus envisioned the world through a mathematical description of form. This form can be described by the fact that matter exists under definite structured conditions and develops or moves according to definite laws, which can be described mathematically.

Mathematics is the way of describing the relationships of these movements. The discovering (by the Greeks) that nature moves according to the laws of mathematics is profound insight into the basic order of the universe. Although Kant has stated that mathematics is merely a …category of our thinking, every phenomenon of Nature can be placed in some form of mathematical language [Kant87], [Kant88]. Mathematics is the vocabulary by which the human mind preserves the intrinsic order of nature. All of this classical order will be challenged in later chapters by the mathematics of quantum theory. For now however, chaos does not rule the realm of electrodynamics and classical field theory.

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represented conditions of space, usually mechanical categories such as force, velocity or stress. The aether of the electromagnetic field was first described using the notation of a mechanical field theory. After the withdrawal of the aether, the field theory describing electromagnetics still referred to the aether as merely another name for empty space. It was a field of space, which became an active agent instead of a passive void. [2]

§2.1. ELECTRODYNAMICS

The force controlling the interaction of particles, whether charged or neutral, macro or micro in size can be described with the concept of a field of force. Instead of saying that one particle acts on another through direct contact or through some physical medium, as was believed in Newtonian physics, it can be said that the particle creates a field and a certain force resulting from the field, unique to the particle class, then acts on every other particle of the same or similar class, located in this field. In classical physics, the concept of a field is merely a mathematical description of the physical phenomenon—the interaction of particles. [3]

Here and elsewhere, we shall not obtain the best insights into things until we actually see them growing from the beginning.

— Aristotle

§2.2. ELECTROSTATICS AND EARLY EXPERIMENTS

When a piece of amber is rubbed, it attracts small pieces of material. This discovery is attributed to Thales of Miletus (640–548 B. C.) [Jean25]. [4] A second discovery by Titus Lucretius Carus (~ 99/94–55/51

2 The development of field theories took place in the early 19th century only after the development of the theory of partial differential equations [d’Abr39]. Even after the mathematics was available, field theories were not immediately constructed. The mechanical models of stress in elastic media, with their mechanical magnitudes were the basis of much of the work. Only when Maxwell formulated the non–mechanical description of the electromagnetic field did field theories take hold. Maxwell’s formulation though depended on the experimental and theoretical results of the pioneers of electrical research, who awaited Maxwell to make the necessary connections before the theory can be turned into practice.

3 The use of the term particle does not imply subatomic particles, but rather a body of matter that is physically very small, so as to approach a mathematical point.

4 Thales was considered one of the Seven Sages of Ancient Greece. He was the father of Greek, and consequently of European philosophy and science. His speculations embraced a wide range of topics relating to political and celestial matters.

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B.C.) and described in De Rerum Natura (On the Nature of the Universe) [Lucr52] was that a mineral ore — lodestone — possessed the ability to attract iron. [5] Plato (428–348/7 B.C.) refers to the attributes of amber in his dialog Timaues [Bury29]. By the Middle Ages, the properties of a compressed form of coal called jet had been described by Venerable Bede (673–735). [6] In the 13th century, Pierre de Maricourt demonstrated the existence of two poles in a magnet by tracing the direction of a needle, which was laid onto a neutral magnetized material. His publication described the first observation connected with one of the modern laws of electromagnetics — Gauss’s Law for the absence of magnetic charges.

The first use of the word electricity has been attributed to William Gilbert (1544–1603) in 1600. [7] He used electricity to characterize a

Thales studied astronomy in Egypt which allow him to construct accurate tables forecasting the flooding of the River Nile. He first became widely known by anticipating an eclipse of the sun in May of 585 B.C., which coincided with the final battle in the war between the Lydians and the Persians. He used tables drawn by the Babylonian astronomers, but did not succeed in forecasting the exact day (May 28th) or the hour of the event.

He believed that certain substances, like lodestone (magnetic rock) and the resin amber, possessed psyche (a soul). Many centuries lapsed before Thales’ soul was identified as static electricity and magnetism. William Gilbert (1544–1603) who had read about the unexplained observation of Thales, also became interested in the intangible property, and decided to call it electricity, from the classical Greek word for amber, which is electron.

5 Little is known of Lucretius since he lived by the motto of the Epicureans Live in Obscurity [Lucr65]. The only contemporary he mentions in his writings was a work of Memmius, who was a politician and praetor in the 58 B.C. De Rerum Natura, is a didactic poem on the subject of Nature, creation, and the universe. The basic concepts developed in the poem are that nothing can be created from nothing, nothing can be reduced, the elemental form of matter is tiny particles which are invisible and indivisible. These particles were called seeds, first bodies, and first–beginnings. In modern physics of course, these particles are called atoms. Lucretius also conjectured that there is also empty space or void. All things in the universe consist of a mixture of particles and this void.

6 Bede was an English monk who also studied the tides, calculated the dates of Easter for centuries to come and wrote one of the world’s great works of history, The Ecclesiastical History of English. In this book Bede describes the material jet as..

... like amber, when it is warmed by friction, it clings to whatever is applied to it [Bede55].

The cause of this attraction is not well founded, since Bede confuses friction and the warmth produced by friction.

7 Gilbert was a London physician and president of the Royal College of Surgeons and court physician to Elizabeth I and James I. Gilbert studied the effects of lodestone (magnetite) and introduced the term electric, in his 1600 Latin text De magnate,

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quantity that many substances share with amber when they are rubbed, including glass, sulfur, wax and certain gems. These observations of electrical attraction led to the idea that electricity was not an intrinsic property of the material, but rather a substance unto itself, which was produced or transferred when the material was rubbed.

Stephen Gray (1670–1736) showed in 1729 that static electricity can be moved between bodies by some substances, among them metals, which are now called conductors [Gray31]. Early laboratory experiments by Benjamin Franklin (1706–90) [Fran51] and Charles–François de Cisternay Du Fay (1698–1739) [Dfay33] established that electrically charged bodies both attract and repel each other, resulting from the presence of negative or positive charges. [8]

Du Fay concluded that there are...

...two electricities, very different from each other; one of these I call vitreous electricity; the other resinous electricity [Dfay33].

magneticisque corporibus, et de magno telluse; physiologia nova, plurimis & argumentis, & experimentis demonstrata (On the Magnet: Magnetic Bodies also, and On the Great Magnet the Earth; A New Physiology Demonstrated by Many Arguments and Experiments) after the Greek word elektron (ηλεκτρου) for amber. This book was an attempt to explain the nature of the lodestone and to account for the five movements connected with magnetic phenomena [Gill77]. Gilbert's work is one of the oldest publications on the theory of magnetism. An English translation of Gilbert's work On the Magnet, [Pric58] is a facsimile edition of a previous translation [Thom00], which itself is a replica of Gilbert's original Latin edition published in London in 1600.

In his book, Gilbert stated that many substances besides amber could be electrified when rubbed and they would attract light objects. Although Gilbert denied the existence of electrical repulsion, several other researchers observed repulsion later in the 17th century. Niccolò Cabeo (1596–1650) has often been credited with first observing electrical repulsion, but he regarded it as a mere mechanical rebounding of the attracted objects from amber [Home92b]. Gilbert made the distinction between the behavior of dissimilar electrified objects and the behavior of the properties of the lodestone [Heil79]. This separation of electricity from magnetism would be rejoined two centuries later by Øersted in 1813.

Gilbert went on to speculate that magnetism was responsible for holding the planets in their place around the Sun. His improper explanation of orbital mechanics did lay the groundwork for the concept of action–at–a–distance which paved the way for the future concept of universal gravity in the 1680's [Benn80].

8 It has been known since early times that a piece of amber, when rubbed with fur, will acquire the power to attract ... feathers, straws, sticks and other small things [Bari68]. The amber acquires a negative charge when rubbed by the fur, while glass rubbed with silk acquires a positive charge. The exact mechanism involved in transferring a charge (positive or negative) from the surface of one material to another is still not well understood [Wein90], [Moor73].

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The vitreous electricity, from the Latin vitreus for glassy, is produced when glass or crystal is rubbed with silk. The resinous electricity is produced in amber or copal, when they are rubbed with fur. Both of these types of electricity were observed to attract ordinary matter. Vitreous electricity was assumed at attract resinous electricity, but materials containing vitreous electricity were assumed to repel each other and likewise for materials containing resinous electricity.

The strength of this attraction or repulsion is given by an inverse square law. The natural philosophers of the 18th century were disposed to the idea of an inverse square law for electrostatics, following the success of Newton's inverse square law for gravity. It was Charles Augustin de Coulomb's (1736–1806) careful experiments in 1785, using a very sensitive torsion balance, which gave direct quantitative verification of the inverse square law, know as Coulomb's Law [Heav50], [Elli66], [Whit52], [Shan59]. Coulomb stated that

... the repulsive forces between two small spheres charged with the same kind of electricity is in inverse ratio to the square of the distance between the center of the two spheres... (and) the law of inverse square was found to hold also the case of attraction.

The inverse square law was found to hold accurately for various charges and separations. [9] This law states that the force between two point charges, 1q and 2q — whose dimensions are small compared to their separation — exerted on one another has the direction of the line joining the charges and is inversely proportional to the square of their separation

9 The inverse force law for electrostatic charges was first found by Joseph Priestly

(1733–1804), who also was the discoverer of oxygen in 1767 [Segr84]. He formulated the inverse square law in a unique manner, which was more convincing that any direct measurement that had been performed at the time. By the same reasoning, Cavendish derived a similar law [Crow62]. After Coulomb’s Law had been established, the science of electrostatics became mathematical rather than subjective. The most important problem faced by the scientist of the time was,

Given the total quantity of electricity on conducting bodies, calculate the distribution of charges on them under the action of their mutual influence and also the forces due to these charges [Born24].

The solution to this problem, usually called the theory of potentials, does not represent the true theory of contiguous action since the differential equations (or difference equations used by Coulomb) describe the charge density as a function of position. They do not describe the transmission of the electrostatic force as a function time. Therefore, they still represent an instantaneous action at a distance.

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distance rr , [10] as given by,

10 The idea that electric forces act like gravity at a distance was first conceived by

Æpinus in 1795. He did not succeed in formulating the correct law for the dependence of the electric action at a distance, but he was able to explain the phenomenon of electrostatic induction. The original concept of the inverse square law was discovered through experiment by Henry Cavendish (1731–1810) in 1772. Although Coulomb's method was a direct measurement of the electrostatic force as a function of distance, Cavendish measured the force exerted by a large metallic sphere on a test particle placed inside and outside the sphere. Cavendish made use of the analogy between his experimental apparatus and Newton's description of gravitational forces.

Newton stated that...

... a test particle placed anywhere within the sphere will experience no net force exactly when the two–body central force between point charges is exactly proportional to 1 2r , while a net force toward or away from the

center of the large sphere would be observed if ( ) 21F r ε+≈r with ε ≠ 0.

In Cavendish's laboratory notes, edited by Maxwell and published in 1879, Maxwell concluded ...

... that the electric attraction and repulsion is inversely as the square of the distances, or to speak more properly, that the theory will not agree with experiment on the supposition that it varies according to any other law.

Cavendish put an upper bound on ε of ε ≤ 1 60 . Experiments of higher precision

involving different sized objects have been performed over the years. Recent results have given error limits on the inverse square relationship of ε = ± × −( . . )2 7 3 1 10 16 for 1 2r +ε [Will71]. The unit of electric charge is the coulomb (C) defined as the electric charge that passes a given point in a wire in one second that is carried by one Ampère of current. The constant f in Eq. (2.1) has the measured value 8 987 10 9 2 2. × −Nm C [Roll54].

The quantitative work of Coulomb and Cavendish on electrostatics as well as Joseph Black (1728–1799) and Antoine Laurent Lavoiser (1743–1859) and Pierre Simon de Laplace (1749–1827) on heat conduction, and Johann Tobias Meyer (1723–1762), Johann Heinrich Lambert (1728–1777) on magnetism established the process of precise experimentation and the use of mathematical models as the scientific methodology of the nineteenth century [Harm82].

Lavoiser can be considered the Isaac Newton of chemistry (or Newton could be considered the Lavoiser of physics). In his 1787 work Methode de Nomenclature Chimque, he restructured chemical theory by organizing and renaming the chemicals of the time. Prior to Lavoiser’s work chemicals were named with colorful but cryptic terms. Ethiop was changed to iron oxide, orpiment to arsenic sulfide. Using prefixes like ox and sulf and suffixes like ide and ous he cataloged compounds. Lavoiser’s major contribution was his theory of gases and combustion. He determined that combustion was a chemical process and gases could be compressed which lead to the realization that an element could exist in three states: solid, liquid and vapor.

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1 22 ,

q qf=FF

rr (2.1)

where the constant f, analogous to Newton's gravitational constant G, has a value which depends on the system of units used to measure the electric charge. [11]

If more charges are added to the expression, the description of FF becomes more complex. Experimentally is can be shown that these new charges have no effect on the interaction of the original two charges.

The effect felt by any charge from other charges in the vicinity can be described by restating Coulomb's Law as,

13

0

14

qr

=πε

E rE r , (2.2)

where 0ε is the permitivity or capacitivity of free space surrounding the charge and EE is the electric field or electric intensity surrounding the charge 1q — independent of any charges. [12] When another charge 2q is

11 The factor f is independent not only of the condition of the bodies but of their

position in space as well. The quantity f has the dimensions of ( ) ( )2 2force length charge× .

Although the force FF is what is measured in the expression, it is useful to introduce the concept of an electric field due to some array of charges. At this point, the electric field can be defined as the force per unit charge acting from a given point in space. This force is a vector function of position, denoted by EE . It is assumed that the charge is located at a point. The discreteness of the electric charge means that this mathematical limit is impossible to realize physically. In order to escape from this dilemma, the Dirac delta function can be employed. In one dimension the delta function δ( )x a− is a mathematically improper

function having the properties δ( )x a− = 0 for x a≠ and ( ) 1x a dxδ − =∫ if the region of

integration includes a and zero otherwise, and ( ) ( ) ( )f x x a dx f aδ − =∫ . A rigorous account

of the seemingly trivial function is given in [Brac65], [Titc48], [Halp52].

12 Faraday made a fundamental discovery that the capacitance of a parallel plate capacitor changes when different substances are placed between the plates. A measure of the capacitance is given by the permitivity. If air is the material between the plates, the normalized value of the permitivity is 1.0005. For a vacuum, this value is 1.0. For all substances other than a vacuum, the permitivity between the parallel plates increases in value. Through a series of experiments it was determined that the ratio between the vacuum ε0 permitivity and the permitivity of various materials ε could be considered a

constant, ε ε0 , where ε0128 85 10= × −. Farad meter (in MKS units).

The material placed between the plates of the capacitor is called a dielectric — which is usually a good insulator. The increase in the normalized permitivity and the resulting increase in the capacitance is the result of the electrical behavior of the dielectric. The molecules of some dielectrics are non–polar since the average electrical center of their

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introduced, the electric force it experiences which is generated by the first charge 1q is given by 2q=F EF E .

§2.3. ELECTROMAGNETIC INTERACTIONS

The description of the interaction of electrical matter was first codified by William Watson (1715–1787) and Benjamin Franklin [Fran51], [Cohe41], [Home92]. Franklin stated that there was only one kind of electricity, consisting of extremely subtle particles, which were identified with Du Fay’s vitreous electricity. Franklin proposed that electrical matter differed from ordinary matter in that electrical matter is strongly attracted to ordinary matter and repelled by other electrical matter. Ordinary matter was like a sponge capable of absorbing electrical fluid. When ordinary matter was squeezed the electrical matter was forced out and became negatively charged. When excess electrical matter was absorbed by ordinary matter it became positively charged [Home92]. Franklin also introduce the concept of conservation of charge, in that electricity is never created or destroyed, only transferred [Fran51]

Franklin's theory of one type of electricity, rather than Du Fay’s two types, was able to describe the mutual repulsion of charged particles as well as the attraction of materials to a charged body. This description accounted for the observed repulsion between materials carrying the vitreous electricity and for the attraction between materials carrying the resinous electricity and the vitreous electricity. What his theory could not describe was the repulsion of two resinous charged bodies. The problem with Franklin’s single type of electricity was addressed by Franz Ulrich Theodor Æpinus (1724–1802) in 1759, followed by Coulomb's explanation

negative and positive charge is coincident. The molecules of other dielectrics are polar since each molecule acts as a small electric dipole, even in the absence of an external electric field. For non–polar dielectrics the average center of the electrons in the dielectric molecule shifts relative to the positive nucleus containing the molecules protons. For polar dielectrics, the individual molecules are aligned with the external electric field. In both cases, a positive charge appears on the surface of the dielectric nearest the negative plate of the capacitor, while a negative charge appears on the surface of the dielectric nearest the positive plate of the capacitor. If the dielectric material is homogeneous, its interior remains neutral and the material becomes polarized. The effect of the dielectric material is to increase the amount of charge stored in the capacitor. The measured effect is an increase in the normalized permittivity of the capacitor [Roja71]. At a macro level the dielectric constant ε ε0 is a measured constant and is used in the empirical description of Coulomb's Law.

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in 1788. [13]

Æpinus proposed that...

... in the absence of a counterbalancing quantity of electricity, ordinary matter repels itself [Aepi59].

The repulsion between materials that carried resinous electricity was explained as the repulsion between materials that had been stripped of their normal amount of electricity. This explanation allowed Franklin’s theory of one type of electricity to account for all the observations made by Du Fay and others in the 18th century. It was not until the discovery of the electron that the full explanation of the forces experienced by electrically charged materials would be available. [14]

§2.4. UNIFYING ELECTRICITY AND MAGNETISM

One problem encountered in the description of electromagnetic force is how to unify differences between electricity and magnetism. Experiment shows that the interaction between bar magnets can be interpreted in terms of the interaction between the poles of the magnets. Early explanations of magnetism were based in two magnetic fluids, the boreal or north and the austral the south. Magnetic particles of the opposite kind attract each other and magnetic fluids of similar kind repeal each other. There were attempts by Coulomb to measure the circulation of the north and south magnetic fluids, but as was common in these early attempts to

13 Æpinus described his electrical theory in Tentamem theoriae electricitatis et

magnetisimi published in St. Petersburg in 1759 [Wein90]. His theory of electricity differs from Franklin's theory published ten years earlier in two ways. Franklin proposed that electrified bodies were surrounded by an electrical atmosphere where Æpinus' theory did not require such a medium. Franklin had also proposed there were two kinds of interparticulate force — a mutual repulsion force between particles of the electrical fluid and an attractive force between particles of the fluid and ordinary matter. Æpinus additionally proposed that particles of ordinary mater mutually repelled each other [Home92a].

14 Under normal circumstances, electricity is carried by electrons, which Franklin proposed as the electricity of only one type. However, in Franklin’s theory these particles of extreme subtileness carried a positive charge. In fact, the particles carry a charge of the type proposed by Du Fay as resinous, not vitreous electricity. Modern physics follows Franklin by calling vitreous electricity positive and resinous electricity negative, which forever creates the situation where the common carriers of electricity carry a negative electrical charge.

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explain nature, they failed. [15]

Each pole, north and south exerts, an inverse square force on the other pole similar to Coulomb's force law between electric charges. It is useful at this point, to picture the poles of a magnet as accumulations of magnetic

15 The explanation of magnetism is complicated by the existence of three types of magnetic phenomena — ferromagnetism, paramagentism and diamagentism. The common magnetic attraction of ferrous metals, such as iron filings, to permanent magnets is sufficiently complex to require a quantum mechanical explanation. Substances other than iron, nickel, cobalt and some exotic metals below a temperature of 16° C exhibit magnetic effects. These effects however are many orders of magnitude less than those found in ferromagnetic materials. These small effects can be attractive or repulsive. If a material is placed in a strong external magnetic field and is repelled from the south pole of the field, then it is said to be diamagnetic. If the material is attracted to the south pole of the external field is it said to be paramagnetic.

The qualitative explanation of diamagentism and paramagentism is straightforward. In diamagnetic materials, the atoms have no permanent net angular magnetic moment — in which the electron spins and their orbital motions balance so that any one atom has an average magnetic moment of zero. When an external magnetic field is applied to a diamagnetic material, a current is generated by induction inside the each atom. According to Lenz's law, these currents are in the opposite direction to the applied magnetic field.

In a paramagnetic material the individual atomic magnetic moments are aligned with the external field rather than opposing the field, resulting in an enhancement of the applied field. Paramagnetic forces are small when compared to the mechanical forces caused by the motions of the individual atoms due to the thermal agitation of the material. Because of these thermal excitations, many paramagnetic materials increase their effect at lower temperatures. In any material with atomic magnetic moments, there is a diamagnetic as well as a paramagnetic effect with paramagentism dominating diamagentism [Jack75].

In ferromagnetic materials the net effect of the externally induced magnetic moments are orders of magnitude greater than in diamagnetic and paramagnetic materials. The large magnetic moment in ferromagnetic materials come from the magnetic moment of the electrons in the inner shell of the atom.

When examined at a deeper level each of these qualitative explanations are entirely wrong [Feyn64]. It is not possible to explain the magnetic effects of material without using quantum mechanics. A description of the interaction of the spinning electrons which produce the magnetic moments of the classical magnetism, requires that the interaction generate a strong tendency to align their spin in the same direction. This behavior can be described by Pauli's exclusion principal. Werner Heisenberg proposed a theory in 1928 that ferromagnetism was due to the exchange interaction of the electrons of the incomplete inner shells of iron, nickel, cobalt, etc.

What is remarkable is that the experimentalists of the 18th century were able to describe in sufficient detail the macro–level phenomena of magnetism. It was not until a clear understanding of solid state physics came about that the mechanism of magnetism could be explained in a manner consistent with the experimental evidence. Although these magnetic effects can be measured and theories constructed describing the interaction between the magnetic field and the various magnetic materials, the underlying reason why magnetism exists is still a mystery of nature.

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charge. There are references to magnetic polarizations in many texts, since they are in some ways analogous to the polarization charges found in electrically charged materials [Pais69], [Roja71], [Jack75].

Isolated electrical charges can be studied in the laboratory, while isolated magnetic charges or magnetic monopoles have not yet been observed. [16] In the development of Maxwell's equations, it is useful to

16 The quantization of the electric charge is one of the most striking features of atomic physics and there is no explanation for its existence, other than one proposed by P. A. M Dirac. In 1931 Dirac described a theory in which magnetic charges were the consequence of quantizing the electric charge [Dira31], [Krag90], [Amal68], Barr91], [Cabr83], [Carr82a], [Crai82], [Davi88], [Dira48], [Good87], [Lind94], [Yang77]. Isolated magnetic charges or magnetic monopoles were predicted, although there is currently no experimental evidence for them. The discovery of magnetic monopoles would be of great importance and allow the generalization of Maxwell's equations [Efin69], [Lapi69]. Dirac's argument was that the existence of magnetic monopoles could explain the discrete nature of the electric charge.

Theories of matter, which allow magnetic monopoles, are called dual. In the present theory of electromagnetism if the electric and magnetic quantities are exchanged in Maxwell’s equations, a different theory is produced. However, if magnetic charges are assumed to exist, then the exchange of the electric and magnetic terms in the equations results in the symmetric theory [Mule96]. Such a theory is called dual since the two theories are nearly identical. Duality makes elementary and composite particles interchangeable.

The current theory of the formation of the universe predicts the existence of the magnetic monopole [Lind87], [Cabr83], [Carr82]. The monopole would have a mass 1 1016× times greater than the proton. These monopoles would have been formed in the very early stages of the big bang and should be as abundant as protons [Carr82a]. The result would be a universe with a density approaching 1 10 29 3× − g cm which is nearly 1 1015× times denser than actual observed — a troublesome result [Lind94]. The current vehicle for the explanation of charge quantization is the Standard Model of particle physics. When several anomalies have been removed from this theory the reason for charge quantization will most likely become clear [Mars93].

There are theoretical issues with admitting magnetic monopoles to the classical electromagnetic theory. If such a monopole were to exist, the curl equation for the magnetic field would be given as, ∇ × = ′BB ρ , where ′ρ is the magnetic current density, such that

′ = ′jj vρ . However when Maxwell’s equation are written in their potential form, where

BB AA= ∇ × and EE AA+ = −∇φ∂ ∂t , it would not be possible for monopoles to exist since the

identity ( ) 0∇ ⋅ ∇ × =A results in the monopole versions of the Maxwell curl equations

being inconsistent with the normal Maxwell equations. Although Dirac invented the monopole, it was not his aim to do so. The monopole appeared as a result of his calculations.

It often happens in scientific research that when one is looking for one thing, one is led to discover something else that one wasn’t expecting. This is what happened to me with the monopole concept. I was not searching for anything like monopoles at the time [Dira78].

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maintain the symmetry between electricity and magnetism by representing both phenomenon with poles — positive and negative electric charge and north and south magnetic poles. This representation provides a natural extension to the lines of force description of electric and magnetic fields.

The concept of magnetic charge must be used with caution though. The magnetic fields of a bar magnet can be attributed to the electric currents flowing in the magnetic material, either from spinning electrons or form the motion of electrons in the atom [Feyn64] (I II I 36–2).

§2.4.1. LINES OF FORCE

The breakthrough in the theory of electromagnetics came when Hans Christian Øersted (1777–1851) observed direct evidence for electromagnetism. [17] Through experiments with a galvanic cell, he discovered there were electrical forces beyond the static forces previously observed. In 1813, Øersted proposed that electrical force could be transformed into magnetic forces. Through seven years of laboratory experiments, he reached his goal in 1820. Øersted believed...

... that magnetic effects are produced by the same powers as electrical ... also that heat and light are produced by the same powers which might be only two different forms of one primordial power [Dibn62].

During the winter of 1819–20 Øersted prepared a lecture on the principals of electricity and magnetism [Shan59]. His usual demonstration was to pass a current generated by a galvanic cell through a thin platina wire and observe the glow produced by the heating of the wire. During the lecture, Øersted placed the wire over a compass. The motion of the compass

17 Øersted was born to a village apothecary in Rudkoebing on the Baltic island of

Langeland in Denmark. Øersted became an ardent follower of Emanual Kant's (1724–1804) Naturphilosophie which was articulated in General Natural History and Theory of the Heaven, published in 1755 and The Critique of Pure Reason, published in 1781 and again as a second edition in 1787 [Kant81], [Kant88], [Appl95], [Frie92]. This philosophy describes the universe as

... a Divine work of art held together by a few simple forces. The basic principle of Øersted’s philosophy was that all phenomena are produced by the same force and the science was not merely the discovery of nature; that is the scientist did not just record empirical facts and sum them up in mathematical formulas. Rather, the human mind imposed patterns upon perceptions; and the patterns were scientific. [Oers80].

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needle due to the magnetic field produced by the flowing current, was so small that it was first attributed to the confusion surrounding the demonstration equipment. [18] In July of 1820, Øersted resumed his experiments with a stronger galvanic device. The magnetic effect was still small because of the small wires used — limiting the current flowing through the wires. Øersted repeated the experiment with larger diameter wire and discovered that the compass needle was strongly effected by the current flowing in the wire and concluded …

... that the magnetic effect of the electrical current has a circular motion round it [Will66].

Other scientists had attempted the same experiment, but had placed the compass needle at right angles to the wire, with no observed effect. They had expected that the magnetic effect should act in the direction of the current. Using this premise, the needle should have been moved to a position parallel to the conducting wire. What Øersted observed however was that the compass needle moved in a direction perpendicular to the wire. [19]

Although Øersted anticipated some form of magnetic action from the conduction of electricity the resulting circular force was both unanticipated and inexplicable [Will71]. Øersted’s discovery came as a surprise to the scientific community of the early 1800's. The concept of the helical force surrounding the currently carrying wire was a hopelessly confusing idea [Will66], [Plaa94]. The Naturphilosophie of the time demanded that all forces be central in nature — a circular force could not be explained. In 1821, the problem appeared to be solved when Ampère proposed the magnetism was simply electricity in motion. He observed

18 Although it has been popular to write that Øersted’s discovery was an accident of

scientific experimentation, there is ample evidence to show this was not the case. The exact description of Øersted’s experiments were made by Øersted himself in a article on thermo–electricity in the Edinburgh Encyclopedia [Will71]. A complete survey of this myth is given in [Stau53] and [Stau57].

19 Although Øersted would have observed the compass needle move in a perpendicular direction, the needle could have done so only through a small angle before the earth's magnetic field balanced the effect of the magnetic field induced by the current carrying wire [Will83]. The result was that the needle did not end up pointing at right angles to the wire which has been the popular science description of the experiment. In Øersted’s original paper he states…

If the distance of the joining wire from the magnetic needle does not exceed ¾ of an inch, the declination of the needle makes an angle of about 45º.

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through an experiment in which two current carrying wires attracted each other when the current flowed in the same direction and repelled each other when the current flowed in opposite directions. Ampère made a revolutionary conclusion that since magnetism is electricity in motion it should be possible to reproduce all the effects of a permanent magnet by various arrangements of current carrying wires. A wire wound in a helix could be made to behave as if it were a bar magnet, with a north and south pole.

After the publication of Øersted’s paper on July 21, 1820, [Øers20] attempts were made to explain the motion of the compass needle. Two theories were put forward, one from André–Marie Ampère (1775–1836) [20] and a second by Johann Josef van Prechtl (1778–1854). Although Amprère's theory won out in the end, Prechtl's theory was given some consideration by Faraday in [Fara65]. At this time the various theories of electromagnetism were based on Coulomb's assertion that the interaction between electricity and magnetism was not possible [Will71]. Coulomb's influence was so strong that Ampère was an ardent believer that...

... electric and magnetic phenomena are due to "two different fluids" which act independently of each other. [21]

Ampère was in the audience on September 4, 1820, when Dominique François Jean Arago (1786–1853) reported that Øersted had discovered that electricity created magnetic effects surrounding a current carrying wire. Despite what Coulomb had claimed decades before — that electricity and magnetism were independent — Ampère refined Øersted’s experiments by neutralizing the effect of the earth's magnetic field.

By placing magnets in suitable locations, the influence of terrestrial magnetism was canceled and the effect of the current induced magnetic field more noticeable [Will83]. He also observed that the compass needle

20 Ampère was the son of a well–to–do merchant, who through a self education process

laid the foundations for the science of electromagnetics. Ampère educated himself by reading the family library. He survived the French revolution to become a science teacher, first in Lyons then in Bourg. He later took a post at the Ecole Polytechnique, and in 1808 became the inspector general of the university system in Paris. Beginning in 1824, he taught physics at the College de France and philosophy at the Faculte des Lettres. Although successful in his professional career, Ampère’s personal live was a tragedy. His father was executed by guillotine during the Revolution and after his first wife’s early death, his second marriage was deemed a catastrophe.

21 Quoted in Arago's éloge of Ampère in Domonique François Jean Arago (1786–1853), Œuvres Complètes [Will71], Ref: 16, pp.185.

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pointed one direction above the wire and an opposite direction below the wire. This lead Ampère to conclude the magnetic force formed a …

… circle in space, concentric about the wire.[Will83]

Ampère's approach to the problem followed Coulomb's logic — but the results advanced the state of the art. Starting with Coulomb's theory that …

... rejected electric and magnetic interaction because these fluids were essentially dissimilar and like only acted on like fluids...

Ampère restated the experimental evidence from a different point of view [Will71], [Will62]. Ampère postulated that since like (only) acts on like the action of the electric current on the compass needle (a magnetic interaction known at the time),

... electricity, then, must be the cause of magnetism and the action of the electric current upon the magnetic needle was ... not an interaction between two dissimilar entities ... but rather the action of electricity upon itself [Will71].

Ampère then performed his now famous experiment of passing a current through two parallel wires an observing the attraction and repulsion of each wire to and from the other. [22] These observations were documented in Ampère's memoirs giving birth to the science of electrodynamics.

The fundamental question posed by Ampère and Faraday was whether the magnetic field could be produced by a moving current or whether it

22 The details of this experiment are reported in Ampère's memoirs which were collected together by the Société française de physique, Mémoires sur l'életrodynamique published in 2 volumes in Paris during 1887–8 [Will71]. The standard high school textbook description of Ampère's experiment shows two parallel wires with a current flowing in the wires. An attractive or repulsive force is then measured between the two wire. This explanation skips over a few of the details of the process. The direction of the current in each wire — relative to the current's direction in the other wire — produces different effects. Since the magnetic field produced by the following current follows the right rule and is circular, currents flowing in similar directions cause the wires to be attracted. Current flowing in opposite directions therefore cause the wires to be repelled. Ampère's experiment was repeated by Faraday during the writing of a survey paper in 1821 and 1822 titled, "Historical Sketch of Electromagnetism," for the Annals of Philosophy 1818 , pp. 195–200, 274–290, 1919 , pp. 107–121 [Good85], [Will85] and an article "On some new Electro–Magnetical Motions, and on the Theory of Magnetism," in the Quarterly Journal of Science, 1212 , pp. 74–96. During this period Faraday came to grips [Will71] with the theory of electric and magnetic action and confirmed that the magnetic rotation or circular force of the magnetic field was a verifiable phenomena.

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also resulted from the interaction of a moving electric charge and a metallic conductor in which the charge traveled. If the later proved correct than magnetic fields would be produced when electric currents moved through a conductor. This problem was stated by Henry Rowland (1848–1901) [23] in a letter to Helmholtz...

The question I first wish to take up is that of whether it is the mere motion of something through space which produces the magnetic effect of an electric current, or whether those effects are due to some change in the conducting body which, by affecting some medium around the body, produces the magnetic effect [Mill72], [Mill76], [Wear76].

§2.4.2. BEGINNINGS OF FIELD THEORY

A next problem in the description of the electromagnetic force is how to represent the quantitative properties of the electric and magnetic fields. The electric field has the properties of a vector field — direction and magnitude or intensity. The field can be imagined to be seen as lines of force occupying space.

Although there is much debate the origin of field theory [Ners85], this concept can be traced to Michael Faraday (1791–1867) and James Clerk Maxwell (1831–1879) [Ners85], [Will65], [Agas75], [Berk74]. Maxwell interpreted Faraday's work as a replacement of the concept of...

... action–at–a–distance with a continuous action. Faraday, ... saw lines of force transversing all space where the mathematicians saw centers of force attracting at a distance... [Maxw65], [Hess61], [Agas71].

The intensity of the electric and magnetic fields can be defined as the number of lines of force passing through a unit area which is at right angles to the direction of the lines of force.

This visualization of the field intensity and lines of force has limited use in today’s physics, since the lines of force do not represent any real

23 Henry Rowland was a graduate of Rensselaer Polytechnic Institute in 1870. In 1873 Rowland submitted a paper title “On Magnetic Permeability and the Maximum of Magnetism of Iron, Steel and Nickel,” Philosophical Magazine, 4646 , 1873, pp. 140–159, to the American Journal of Science, on his studies of magnetic permeability. When this paper was rejected he sent it to James Clerk Maxwell who forwarded it to the Philosophical Magazine. Rowland’s primary contribution to American science was through his construction of defraction gratings. These gratings advanced the science of spectroscopy in the late 1800’s [Rein64].

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physical phenomenon in electromagnetic theory — however they can be useful in illustrating the underlying mathematics. Faraday used this concept to communicate his discoveries in a way that could be visualized. Even though the lines of force as were used by Faraday may implied a physical mode of the transmission of action, they were never meant to represent any physical lines flowing in space.

Faraday also understood the limitations of this concept when he wrote ...

The term "lines of magnetic force" is intended to express simply the direction of the force in any given place, and any physical idea or notion of the manner in which the force may be exerted... [Maxw65], [Hess61], [Agas71], [Pari69]. [24]

§2.4.3. REMOVAL OF ACTION AT A DISTANCE

One of the great laws of classical physics is Isaac Newton's (1642–1726) law of gravitation. This law embodies the concept of action–at–a–distance in which masses exert gravitational force by virtue of their position in

24 Throughout the sections dealing with classical electrodynamics, Michael Faraday's

influence can be found. Faraday has been described as the Cinderella of science because of his rise from a childhood as the son of a blacksmith in the slums of London to the pinnacle of scientific achievement in the mid 19th century [Wein90], [Whit37], [Will66].

Several biographies have described Faraday's life and scientific accomplishments with [Will71] being one of the better known, so that the details need not be repeated here. In Experimental Researches in Electricity published in three volumes and a partial fourth, Faraday figuratively and in some cases literally revolutionized theoretical and experimental physics in the mid 17th century. The results of his efforts are seen all around us, the electric motor, the electric generator, the origins of modern field theory and the basis for the description of most electromagnetic phenomena of the modern world.

Faraday did not start his life as a scientist. As a young man Faraday worked as a booksellers apprentice. While binding books Faraday became engrossed in reading the books particularly books about science. He wrote later that...

... there were two that especially helped me, Encyclopedia Britanica, from which I gained my first notions of electricity and Mrs. Jane Marcet’s Conversations on Chemistry which gave me my foundations in that science.

Faraday began to attend public lectures in science, many which were given by Sir Humphrey Davy (1790 – 1868). In 1812 Faraday completed his apprenticeship, but did not continue as journeyman bookbinder. Instead, he wrote Sir Davy seeking employment, enclosing notes he had written based on Davy's lectures. Faraday provided the insight necessary to move the theory of magnetism from one based on imponderable fluids to one based on the beginnings of field theory.

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space, with the intervening space playing no active role in the conveyance of the gravitational force. In Newton's theory, the gravitational force was conveyed instantaneously to the remote body. It is the instantaneous transmission of force that is the basis of action–at–a–distance. Although Einstein's theories of relativity were to change this concept by the introduction of a finite propagation velocity, both Newtonian and modern field theories provide a mechanism for forces to be felt at a remote location.

Through laboratory experiments, it is observed that the presence of an electric charge produces an electrostatic field, which extends outward through space surrounding the charge. The presence of this field is inferred from the action it has on other electric charges placed in the field. In this manner, action–at–a–distance between charges becomes action–by–contact, where the action is between the remote particle and the field generated by the source charged particle. The basis of electromagnetic theory is that the field does have properties that are associated with matter. The electromagnetic field can posses energy and momentum. The field becomes a dynamical concept and not merely a mathematical concept described by coordinate and time variables [Will66].

A field can now be defined as the representation of the way in which some distributed quantity, the electric or magnetic field behaves. The introduction of the field concept moves the discussion away from the electric and magnetic charges to the behavior of the space surrounding the charges.

§2.5. SPECIAL RELATIVITY AND ELECTROMAGNETIC FIELDS

When the special theory of relativity was introduced in the early 1900's by Albert Einstein (1879–1955), the action–by–contact between the electromagnetic field and charged particles was further altered by the finite propagation velocity of the fields force [Clar71], [Bers73], [Fren79], [Hoff83], [Holt82], [Eins55]. [25] A change in the position of one of the

25 The theory of relativity is attributed to Albert Einstein and the birth of modern

physics. In fact Newtonian physics also contains a relativity principal. In Newton's principal of relativity the speed of the observer was taken for granted, were in Einstein's theory the motion of the observer was explicitly constructed to be consistent with the experimental evidence that the speed of light appears the same not matter at what speed the observer was traveling.

In Einstein's theory there is no meaning to the statement that two separate events occur simultaneously. In order to correct the problem of instantaneous action–at–a–distance Einstein reformulated the concept of force as a field.

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charged particles influences other charged particles only after the lapse of a time interval, t c∆ ⋅ , where 103 10 cm/secc ≅ × . [26] In classical mechanics, the field is merely a mode of description of the physical phenomenon — the interaction of particles [Eisn61]. In the theory of relativity, because of the finite velocity of propagation of the interactions, the forces acting on a particle at a given time are not determined by the positions of the particles at the time a force is felt by a specific particle. A change in position of one particle influences other particles only after a lapse of time. Because of the propagation delay, resulting from the effects of special relativity, the field itself assumes a reality. Therefore, particle interaction is described as an interaction between one particle's action and its generated field and the subsequent interaction of the field with a second particle [Land71].

The two postulates of Einstein’s Special Relativity are: (i) The Principal of Relativity, which states that the local laws of physics have the same form in frames of reference that are related by a uniform relative velocity and (ii) The invariance or frame independence of the speed of light, which states that the propagation speed of information or energy has a finite upper bound [Eisn55], [Doug92]. [27] These principals restrict the form of

26 Einstein introduced the Special Theory of Relativity in 1905. This theory overturned the Newtonian interpretation of space–time, in which the knowledge of an event propagates at an infinite velocity. He postulated that the laws of physics should be completely objective — that they should be identical for all observers. They should not depend in any way on how the observer is moving relative to an observed object. This can be stated more precisely as: there exists a triply infinite set of equivalent Euclidean reference frames moving with constant velocities in rectilinear paths relative to one another in which all physical phenomena occur in an identical manner. This means that the laws of physics should appear to be identical in an inertial reference frame (non accelerating reference frame) and that all inertial reference frames are equivalent. Einstein also postulated that the speed of light was a universal constant in all inertial reference frames, representing the maximum speed at which the occurrence of an event could be signaled. It is important to remember that it is the laws of physics that are invariant between reference frames. Physical values such as force, electric and magnetic fields and momentum will differ between reference frames.

The Special Theory of Relativity can be restated as:

The laws of physics cannot provide a way to distinguish one inertial reference frame from another [Tayl63].

27 Einstein’s original paper [Eisn05] submitted in June of 1905, made use of many of the ad hoc assumptions made by his immediate predecessors. This paper placed on a simple conceptual basis, the form of Special Relativity in two postulates. The paper makes no references to the existing literature in particular the work of Lorentz and Poincaré [Lore04]. A balanced description of the contributions of all these authors to the field of Special Relativity is given by [Holt60], [Gold84] and [Gold67]. A more controversial account is given in [Whit60] who states:

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the laws of physics and are supported by a vast amount of experimental evidence [Fren68], [Haug87], [Macd80], [Mari82], [Merm84], [Schw84], [Terl68], [Will81], [Will86], [Will87].

The first postulate, when applied to the mechanics of material particles and electromagnetic fields was not a unique feature of the Theory of Special Relativity. This same concept can be used applied in Galileo — Newtonian mechanics. It is the second postulate that revolutionized physics. By limiting the speed of light, the propagation of information and energy, and their associated forces, Einstein redefined the meaning of simultaneity.

This can be illustrated by the following example. Suppose two charged particles, 1q and 2q are separated by a distance d. If particle 1q is rapidly moved then returned to rest, it will emit a pulse of electromagnetic radiation which travels at velocity c and will be felt by particle 2q at a time

( )t d c= . If the system of two particles is observed some time after 1q has

come to rest but before the time ( )t d c> , both particles will be observed at

rest and would constitute an isolated system whose kinetic energy is zero. At time, ( )t d c= the second particle 2q could be observed to move. The

energy and momentum of the isolated system would appear to change although there were no external forces acting on it, violating the conservation of energy and momentum. This situation can be reconciled in field theory by observing that there are other forms of energy at work in the system. The other physical entity at work, which contains energy and momentum, is the electromagnetic field [Eyge72]. The energy and momentum transmitted to particle 2q at time ( )t d c= is contained in the

electromagnetic field. [28]

... Einstein published a paper, which set forth the relativity theory of Poincaré and Lorentz, with some amplifications and which attracted much attention [Whit60], pp. 60.

28 Throughout this book there has been an attempt to present material that is understandable, with a minimum of mathematical background. By using examples such as this it is hoped that the reader will grasp the concepts of Field Theory. However, in reality many of these examples have serious flaws, whose correction requires mathematics, which will be developed later. This example is such a case.

When the observer makes a measurement of the total kinetic energy of the system, consisting of the two particles, the measurement must be made through some form of energy measuring device. This device will depend on the finite propagation velocity of the field conveying the kinetic energy measurements. If the total kinetic energy of the system could be measured without delay for both particles, then the kinetic energy discrepancy

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The basis of electromagnetism as a field theory states that charge and currents produced at each point in a field have reality of their own. The field can contain and propagate energy and momentum , which acts on other charges, embedded in the field. The discussion of the interaction between charged particles is now directed to the interaction between one charged particle with the field it generates, and the subsequent interaction of the generated field with a remote charged particle [Heit54].

These two sets of equations describe the consequences of electromagnetic field theory. Maxwell's equations that describe the fields produced by a given set of charges and the Lorentz force equation that describes how a given field acts on these charges.

§2.6. LIGHT — PARTICLE OR WAVE

The nature of light, its propagation though space, its material composition and the formation and absorption by material objects has long been the source of questioning and mystery. Starting with the Biblical creation myth...

… on the first day God created light.

To the modern creation theory based on the Big Bang [Wein77], light has always been at the origin of scientific questioning. The Greeks [29] based their theory of light on geometric optics and the representation of perspective, the properties of light to reflect and refract, have been part of our culture.

In the 17th century, an era…

… of change … so radical that classical optics was destroyed and disappeared for good. Today a book on optics written earlier than the seventeenth century would be incomprehensible to a majority of people. [Ronc70].

It was the 17th century instrument, the telescope, that revolutionized the optical world and the methods used to study it. In 1676, Ole Rømer

would be observed, as described in the example. Since such a measurement is not possible, the real system kinetic energy can not be determined simultaneously for both particles and communicated to a common point, thus the example becomes invalid. Although this example serves to illustrate the concepts behind the force field and its conveyance of energy and momentum, the description of this concept is beyond the mathematics of this monograph.

29 Aristotle unsuccessfully attempted to interpret the rainbow as a sign of the covenant [Dale78].

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(1644–1710) pointed his telescope at Jupiter’s innermost moon, Io. He observed that this satellite shows a variation in its motion around Jupiter [Magi65]. Using this instrument, Rømer deduced that the speed of light was 214,300 kilometers per second — about 2 3 ' s of the modern value.

In 1666, at the age of 23, Isaac Newton began his experiments with sunlight passing through a triangular glass prism. Newton showed …

… Colours are not “Qualifications of Light” derived from refractions or reflections of natural bodies (as ‘tis generally believed) but “Original and connale properties” … [white light] is not similar to homogenial but consists of Difforn rays, some of which are more Refrangible than other … [it is] a confused aggregate of rays induced with all sorts of colours as they were promiscuously darted from the various parts of luminous bodies. [Newt72]

Newton recorded his feelings in Opticks [Shap84], [Newt31]. One interesting conjecture in Opticks, which pertains to the subject mater here, is summarized in Newton’s 29th Query:

Are not the Rays of light very mall bodies emitted from shining substances? … Nothing more is required for producing all the variety of Colours, and degrees of Refragibility, than that the Rays of light be bodies of different Sizes, the least of which may make violet the weakest and darkest of the Colours, and be more easily diverted by refracting Surfaces from the right Course; and the rest as they are bigger and bigger may make the stringer and more lucid Colours, blue, green, yellow and red, and be more and more difficulty diverted.

In 1690 an alternative to Newton’s description of light as small bullets was put forth by Christian Huyghen’s (1629–1693) [Huyg90]. Huyghen’s theory proposed that …

… Light spreads as sound does, by spherical surface waves. I call them waves from their resemblance to those which are seen to be found in water when a stone is thrown into it. [30]

Not until 1925 did Newton’s description of light begin to fail and the corpuscular theory of Newton and the wave theory of Huyghen’s merge into the quantum theory of light.

Underlying this revolution in the description of light, the concept of the

30 Huyghens postulated that the waves of light traveled through a medium, just as the wave on the surface of the water.

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aether was also undone. The notion of the all–prevasive ether (aether) goes back to Plato.

Nature’s abhorrence of a vacuum was a sufficient reason for imagining an all surrounding aether … aether’s were invented for plants to swim in, to constitute electric atmospheres and magnetic effluvia [Niven].

§2.7. OVERVIEW OF THE WAVE EQUATION

The next chapters present Maxwell’s equations and their solutions in the form of the electromagnetic wave equation. The mathematical description if waves is at the heart of many fields in physics. The foundations of the theory of waves can be found in almost any good physics text. This section will provide a brief review of the equations of a traveling wave and prepare the reader for the description of Maxwell’s wave equation.

The development of the mathematics of the wave equation can be traced to the Greek’s. A vibrating string moves much too fast for the human eye to see the actual shape of the string. The Greek’s however showed that the pitch at which a string vibrates depends on the position of the nodes [Stew95]. These nodes are locations along the string where the motion of the string is stationary. At the string’s fundamental frequency, only the end points of the string are stationary. If the center of the string is held stationary, when it is plucked it will vibrate at one octave higher. If a point 1 3 the length is held stationary, the string will vibrate in two modes. The waves of the vibrating string are stationary waves that do not travel along the string, but only move up and down. These waves are transverse waves.

In 1715, Brook Taylor (1635–1731) published a paper Methodus incrementorum in which he described the theorem on power series expansions. Taylor also wrote on the theory of the vibrations of a string, in terms of its length, tension and mass. In 1743, Jean le Rond d’Alembert (1717–1783) published Traité de Dynamique, in which he developed Newton’s dynamics using the concept of energy rather than force. He later described the standing waves in violin strings as composed of any shaped wave.

In 1748, Leonhard Euler (1707–1783) described in Introductio in Analysin Infinitorum the wave equation of a vibrating string using a differential equation. Euler’s solution to the wave equation is based on the superposition of two arbitrary shaped waves, each traveling in opposite

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directions along the vibrating string. When these two waveforms are combined, Euler claimed that the motion of the string could be described if the ends of the string are fixed.

Daniel Bernoulli (1700–1782) solved the wave equation in a different manner. In Bernoulli’s method, the motion of the string could be described as the superposition of an infinite number of sinusoidal waves. The superposition principal was used by Fourier to describe the series expansion of a periodic function in terms of sine’s and cosine’s.

The general form the wave equation is given by,

2

22 2

10

uu

v t∂

∇ − =∂

, (2.3)

where u is the disturbance that is being propagated and v is the velocity of the propagation. In general v f= λ , where f is the frequency of the propagated disturbance and λ is the wavelength of the propagated disturbance. Using these definitions, 2= π λkk and 2 fω = π . The relationships between the spatial oscillation rate (kk) and the temporal oscillation rate (ω ) is called the dispersion.

Figure 2.1 — A plane wave front traveling in the x–direction. The wave

λ

kk

x

y

z

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fronts spaced λ distance apart with a constant phase ( )t kxω − + φ .

By choosing a specific coordinate axis for the propagation direction a propagation vector, kk , can be used to describe the motion of the wave through space. This propagation vector, kk , is the number of cycles of the wave in a unit, whose normalized length is k = kk . The propagation

vector, kk , is perpendicular to the wave front.

Since the use of the plane wave description may entail combining waves, which have different coordinate systems, the selection of a single coordinate system may not be possible. Two other vectors are also needed for this arbitrary coordinate system description. The direction of the wave fronts — in a single dimension — will be given by xx .

Figure 2.2 — A Section of a plane wave front traveling along the x’ axis. The coordinates x’, y’, and z’ are an arbitrary angle relative to the original coordinates x, y and z. The radius vector rr joins the origin with a point PP on the wave front. The propagation vector kk is parallel with the coordinate vector x’.

x

y

z

x'

PP

kk

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This direction can in fact be any of the three dimensional coordinates, but xx is a common one. The second vector that connects the origin of the propagating plane wave with the wave front is rr , and is usually called the radius vector. The propagation vector, kk , lies along the wave front’s direction vector xx . The radius vector can be defined as = + +r x y zr x y z , since it represents a point on a given wave front as measured from the origin. In this definition the vectors xx , yy , and zz are the vectors along each Cartesian coordinate system axis. [31] The wave front is then described by,

( ) Constantt k ′ω − + ϕ =xx , (2.4)

where ϕ is a phase factor.

Note that ′xx is a different coordinate that the xx coordinate of the Cartesian coordinate system centered on the origin. This plane wave can be described in a new coordinate system (unprimed) by the plane wave,

( ) Constantt k ′ ω − ⋅ + ϕ = r x (2.5)

Using some vector algebra gives,

( ) ( )k k′ ′⋅ = ⋅r x x rr x x r . (2.6)

The product k ′xx is a vector lying in the same direction as ′xx , whose magnitude is k v= ω . This is the final definition of the propagation vector kk .

The disturbance traveling in the ′xx direction can now be described by,

( )0 cosu u t x= ω − + ϕkk . (2.7)

The wave equation given in Eq. (2.3) does not state how many dimensions are in coordinate system of the disturbance. It will be assumed that the wave is propagating in three dimensions. However, in many of the examples given later, the wave equation will be simplified to one dimension. Without any loss in its descriptive power.

In principal, there are two kinds of waves, (i) longitudinal waves in which k v u and (ii) transversal waves in which k v u⊥ . It will be the

transversal waves that will be of interest in the study of the propagating electromagnetic field.

31 In most texts this expression is given by ( ) 0∇ ⋅ ∇ × =A , where x, y and z are unit

vectors in the vector space of the coordinate system.

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The phase velocity of the propagating disturbance is given by,

phasevkω

= . (2.8)

The group velocity is given by,

1phasegroup phase phase

dvd k dnv v f v

dk dk ndkω = = + = −

, (2.9)

where n is the refractive index of the medium through which the disturbance is propagating. Information transferred by the disturbance by modulating it does so at the group velocity. If phasev does not depend on ω ,

then phase groupv v= .

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I have also a paper afloat, containing an electromagnetic theory of light, which till I am convinced to the contrary, I hold to be great guns.

— James Clerk Maxwell

in a letter to his cousin Charles Cay [Ever70]

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§3. MAXWELL'S EQUATIONS

Electric and magnetic fields effect matter on a range of scales from the atomic to the cosmic. Although these fields are only visible to humans in a narrow band of frequencies, they are responsible for the majority of chemical and biological effects that make up life as we know it. Electric fields are produced by electric charge and magnetic fields are produced by motion of electric charge. Electric fields can also be produced by a changing magnetic field and magnetic fields can be produced by a changing electric field. These two fields are coupled into one field, the electromagnetic field described by Maxwell’s equations.

Maxwell’s codification of the field equations transcends their physical description. Einstein put this achievement in perspective in the commemoration of the 100th birthday of James Clerk Maxwell:

We may say that, before Maxwell, physical reality in so far as it was to represent the process of nature, was thought of as consisting in material particles, whose variations consist only in movements governed by partial differential equations. Since Maxwell’s time, physical reality has been thought of as represented by continuous fields, governed by partial differential equations, and not capable of any mechanical interpretation. This change in the conception of reality is the most profound and most fruitful that physics has experienced since the time of Newton… [Eins31]

Since the time of Rene Descartes (1596–1642), natural philosophers have speculated how the electric force is transmitted through space [Whit51], [Cott92]. [1] Karl Friedrich Gauss (1777–1855) wrote Wilhelm

1 René Descartes' is one of several 17th century natural philosophers who considered

the question of action–at–a–distance including Galileo, Newton and Huygens. Although many of Descartes' explanations did not survive, he laid the foundation for later theories that did. In practice Descartes' method was a mixture of rationalism and empiricism, but his theoretical approach tended to be entirely rationalist when he claimed that when the first principles or simple natures of philosophy are clearly and distinctly perceived, all other truths are deduced from them [Hess61].

Since Descartes saw verification of the simple natures by comparing them with experience as a mere formality, there was no process by which his theories could be tested. He did use observation and experimentation to arrive at his simple natures through the deductive process put forward by Aristotle and Newton. He stated that his chief innovation to physics, when compared to Aristotelian science was the ...

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Weber (1804–90) [2] in 1845 to remark that...

...he had proposed to himself to supplement the known forces which act between electrical charges by other forces, such as would cause electric actions to be propagated between charges at finite velocity.

In a paper by Gauss published posthumously in 1867, the concept of electric action propagating through space is presented. Maxwell stated that...

...this paper first introduced into mathematical science that idea of electric action carried by means of a continuous medium... [Maxw65], [Buch85].

The concept of a field was introduced by Faraday [Fara65], [Agas71] and extensively developed by Maxwell [Maxw65], [Ever74], [Larm37], [Agas68]. [3] In Maxwell's theory electric charge is regarded as the source

...explanatory principles .. that he clearly and distinctly perceived. [Hess61].

Descartes’ scientific method was laid out in 1637 in his Discours de la Methode Pour bien conduire fa raison & cherches la veritè dans kes sciences (Discourse on the Method of Properly Guiding the Reason in the Search of Truth in Science). His method consists of:

(a) accepting only what is clear in one’s own mind as to exclude any doubt,

(b) splitting large difficulties into smaller ones,

(c) arguing from the simple to the complex,

(d) checking when one is done [Davi86a].

Appended to Discours are three essays on which Descartes gives examples of discoveries may by the use of his methods. The first appendix title Optics presents the laws of the refraction light, which had earlier been discovered by Willebrord Snell. The second appendix discusses meteorology. The third appendix presents analytical geometry.

2 In 1855 Weber and Kohlrausch determined the limiting velocity in Weber’s electromagnetic theory to be c km s= 439 450, [Webe56], [Webe93], [Doug90], [Jung86].

3 James Clerk Maxwell was the son of a Scottish nobleman who encouraged Clerk's curiosity by taking him to see the manufacturing plants being developed in the 1830's and 1840's. During high school Maxwell became interested in mathematics and at age 15 published a paper in the Proceedings of the Royal Society of Edinburgh titled "On the Description of Oval Curves." [Macd64]. In 1857 he wrote to Faraday commenting on some of his ideas and Faraday replied with encouragement. In 1873 Maxwell published his treatise Electricity and Magnetism [Maxw65] in which he unified electricity and magnetism through four equations, predicted electromagnetic wave propagation, calculated the velocity of light and conjectured that light was an electromagnetic wave phenomena [Ever74].

Maxwell's equations were experimentally confirmed by Heinrich Hertz in two papers

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of the field and the electric and magnetic fields are fields of force, represented as vector fields. Laboratory experiments have shown that electric currents produce magnetic fields and vice versa. Electricity and magnetism are not distinct forces but are in fact a single electromagnetic force. The first steps to unify the theory of electromagnetism were taken in the late 1800's [Buch85]. Maxwell set out to write down a set of equations, which when taken with the strengths of the electric and magnetic fields and the strengths and arrangements of the sources of these fields, would result in the equations of motion for a test charge placed in the field.

Maxwell succeeded in showing that the force a charge exerts on another charge and the energies of each charge could be expressed not only in terms of the magnitudes of the charges and their positions in space, but in terms of a stress energy tensor that was defined throughout the medium, even if that medium was free space. The need for a medium to convey the electromagnetic field was fundamental to Maxwell's concept of waves. Like velocity waves in hydrodynamics or the stress and strain field in elasticity, these fields were not considered to exist by themselves but were somehow considered to be vibrations of an underlying luminiferous aether, whose properties were akin to a elastic solid [Larm37], [Whit60], [Buch85]. The electromagnetic waves were secondary. The aether could exist without the waves, but the waves could not exist without the aether. [4]

Years after the discovery by Maxwell of the wave equations, an experiment in 1886 by Albert Abraham Michelson (1852–1931) and Edward Williams Morely (1838–1923), showed that the aether did not exist. [5] As a result, the electromagnetic field can be viewed as an entity of

published in 1881, "On Electromagnetic Waves in Air" and "On Electric Radiation."

4 The spelling aether with the initial diphthong is an arcane reversion that even Michelson and Maxwell did not use. However modern texts on the history of electromagnetism [Whit60], [Swen72] make use of this spelling. The word aether is related to the Greek αιθηρ which means upper air or sky and relates to the refined fire of the empyrean [Shan64].

5 The existence of the aether was accepted by many scientist of the time as a logical conclusion of Maxwell's theories, since some form of media would be necessary to carry the electromagnetic waves. One consequence of the aether would be that the earth's motion through a motionless aether produce a drag effect resulting in measurable differences in the speed of light depending on the direction it is traveling relative to the earth. This idea was put to a stringent test by Michelson and Morely in 1881 [Mich81], [Livi88]. They used the earth as a moving reference frame and compared the round trip speed of light along the line of the earth's motion with the speed of light perpendicular to the line of the earth's motion. They found no evidence for such a drag effect and no evidence for relative motion

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its own, as real as physical matter. The essence of electromagnetic field theory is that the field does have properties that usually as associated with matter. The four equations, in Rationalized units, describing the electromagnetic theory developed by Maxwell are given below. [6]

§3.1. MAXWELL'S 1ST EQUATION — COULOMB'S LAW

The first Maxwell equation deals with static electric fields. Given a collection of charges, surrounded by closed surface, the number of lines of force passing through a unit area, A, on the surface is given by,

1

n

iiA

N d q=

= ⋅ = ∑∫E A . (3.1)

between the aether and the earth [Lore95]. This was one of the most important negative experiments performed and resulted in the 1907 Nobel prize being awarded to Michelson [Bagg92]. Hendrich Antoon Lorentz (1853–1928) proposed an explanation to this negative experiment by introducing the Lorentz contraction of the light as it travels parallel to the earth's motion, which canceled of the expected change in the interference patterns proposed by Michelson and Morley [Swen88]. Although Lorentz's explanation of the null experiment proved to be incorrect, portions of his concept were later included in Einstein's 1905 theory of relativity. (This connection between Lorentz, Michelson and Einstein is somewhat controversial [Buch88], [Swen88].) Einstein showed that both Michelson's and Lorentz's concepts could be reconciled by using newer ideas. Several modern experiments have been performed to confirm the negative of Michelson–Morley's observations [Pano66], [Jase64].

6 The rationalized units is used in this monograph as one of the five systems of units found in electromagnetic field theory. The rationalized units notation provides the simplest form in which to describe the workings of electromagnetic field theory, since no actual calculations are necessary in the approach taken here [Brid31]. The desirable features of a system of units in any field of study are clarity and convenience. By choosing the rationalized units clarity is provided by the absence of terms dealing with permeability and permitivity. One additional simplification has also taken place in this paper, which makes the notation convenient, but erroneously describes the electromagnetic field in a practical sense. The notation for the magnetic field used here is BB , when in fact it is HH , where HH BB MM= −1 0µ , where MM is the magnetization term and µ 0 is the field permitivity. The

electric field EE and the electric displacement DD are simplified to EE . This results in a technically incorrect formulation of Maxwell's equations for the propagation of electromagnetic waves in an isotropic medium (air), but provides a simplified notation for the development of the quantum mechanical formulation found later in the monograph. This form of Maxwell's equations, in which the properties of matter are not considered, is properly called the microscopic Maxwell equations in contrast to the original macroscopic equations.

The final adjustment to the notation is to set 1c= =h resulting in natural units. In the MKS system of units this allows the constants µ 0 and ε0 to be assigned values of 1. The factor 4π can then be moved from the field equations to the force equations.

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The expression in Eq. (3.1) is referred to as Gauss's Law. Charge is always given in discrete units of e, but it is reasonable to assume when dealing with macroscopic phenomena that charge can be described by a continuous distribution function, the charge density, ( )ρ r . Gauss's Law can then be rewritten as,

A V

d dV⋅ = ρ∫ ∫E A . (3.2)

The left side of Gauss's Law can be rewritten again using the divergence theorem of vector calculus which states,

0

1limV

A

dAV→

∇⋅ = ⋅

∫E E , (3.3)

which is called Gauss's Theorem (different from Gauss's Law). The divergence theorem (Gauss's Theorem) is used to express a surface integral as a volume integral of the divergence of the electric field ∇ ⋅E , and is valid for any volume V. The divergence of a vector field measures the net amount of charge entering or leaving a small volume. For the electric field, the divergence is a scalar value representing the number of lines of force passing through the surface surrounding a charge. If the surrounding surface does not contain any charge, then the number of lines entering the volume is equal to the number of lines leaving the volume resulting in 0

AdA⋅ =∫ E . If the surrounding surface contains charge, then

Gauss's Theorem produces a nonzero result.

Equating the expression for the number of lines passing through the surface of the total charge within the surface gives,

V V

dV dV∇ ⋅ = ρ∫ ∫E . (3.4)

Differentiating both sides with respect to the volume surrounding the charge leads to Maxwell's first equation or Coulomb's Law,

(I) ∇ ⋅ = ρE (3.5)

which states that,

Flux of EE through a closed surface = charge inside the closed surface. [7]

7 Experiment shows that in the simultaneous action of several charges, 1 2 3, ,e e e … on

a test charge their field contributions, as well as their force actions, behave according to

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All of electrostatics stems from the quantitative statement of Coulomb's Law. Coulomb showed experimentally that the total force produced on a small charged body by a number of other small charged bodies placed around it was the vector sum of the individual two–body forces. [8]

§3.2. MAXWELL'S 2ND EQUATION — ABSENCE OF MAGNETIC MONOPOLES

The second Maxwell equation deals with static magnetic fields. The magnetic intensity BB corresponds to the electrical intensity EE even though BB is usually called the magnetic induction. Laboratory experiments in the late 19th century showed that the properties attributed to electrical charges can also be applied to magnetic poles. The number of lines of magnetic force passing through a surface can be written as,

A V

N dA dV= ⋅ = ∇ ⋅∫ ∫B B . (3.6)

If a surface is placed around a distribution of magnets, with each magnetic pole composed of a north and south pole, the number of lines of force going outward through the surface from the north poles is equal to the number of lines of force coming inwards to the south poles, such that,

0V

N dV= ∇⋅ =∫ B , (3.7)

the vector law of addition; 4j

n je V

E ds e∈

= ∑∫Ñ π , where the summation is applied to the charge

inside the volume V. Given a large number of charges, within the volume element, using the ratio of volume, dv, to a total charge ρdv , and applying the electric flux theorem

( )n iu d s u v= ∇ ⋅∫Ñ , gives the differential form of the electric flux theorem as ∇ ⋅ =EE ρ which

is Coulomb's Law, Poisson's generalization of the inverse square law and Maxwell's first equation as well as Gauss' theorem.

8 The concept of flux was first used by Maxwell in his analogy between the flow of heat and the flow of electricity. Maxwell borrowed this idea from William Thomson (1824–1907), later Lord Kelvin. Thomson had shown that the equations describing static electricity are of the same form as those describing the flow of heat [Larm37]. This analogy of electrical flux with heat flux can be extended to the concept of a streamline. Maxwell suggested that Faraday's lines of force were similar to streamlines in the flow pattern of a fluid.

Thompson was considered the most eminent experimental physicist in Britain. Thompson was selected to head the Cavendish (Henry Cavendish (1731–1810)) laboratory at Cambridge University. However he wished to in his native Glasgow and instead the Cavendish Professorship went to another Scotsman James Clerk Maxwell, who at age 39 was living in retirement on his estate at Glenair [Wein90].

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which gives Maxwell's second equation as,

( II ) 0∇ ⋅ =B , (3.8)

which states that the,

Magnetic flux BB through a closed surface = zero,

which implies the absence of magnetic monopoles, which in turn implies that the lines of magnetic flux produced by the electric field do not start or stop.

§3.3. AMPÈRE'S LAW FOR STEADY STATE FIELDS

Maxwell's third and fourth equations involve the description of time varying electric and magnetic fields. If a continuous distribution of charge is placed in motion, resulting in the production of current, the current density can be described in a manner similar to the electric and magnetic intensity of the previous two equations.

The current density jj per unit area A can be used to define a current,

Ai dA= ⋅∫ j , which can be rewritten as,

S A

dS dA⋅ = ⋅∫ ∫B j . (3.9)

The curl of a vector field is defined as,

0

1limA

S

dSA→

∇× = ⋅

∫B B , (3.10)

where dS is an element of the path surrounding the unit area A. If follows that,

S A

dS dA⋅ = ∇ × ⋅∫ ∫B B . (3.11)

This result is called Stokes Theorem and can be rewritten as,

A A

dA dA∇× ⋅ = ⋅∫ ∫B j . (3.12)

Differentiating both sides of Eq. (3.12) gives,

∇× =B j , (3.13)

which is known as Ampère's Law and restates that currents are generated by the motion of charge.

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The problem with Eq. (3.13) is that although the magnetic field and the current is generates are linked by Ampère's Law, it contains no reference to how the current varies with time, since Ampère's Law was derived from static fields.

If a closed surface is place around an electrical circuit, the net current flowing through the surface is defined as,

AdA⋅∫ j . This integral will have a

positive value when the net flow of current of outward through the surface. This flow of current then must be balanced by an equivalent reduction of the electrical charge within the enclosing volume. The total charge within the volume is defined as

VdVρ ⋅∫ and the rate of loss of the this over time

is ( )V

t dV∂ ∂ ρ ⋅∫ .

The loss of this charge produces a current defined as,

A VdA dV

t∂

⋅ = − ρ∂∫ ∫j

. (3.14)

Using the definition for the divergence of a vector field gives,

( )V A

dV dA∇ ⋅ = ⋅∫ ∫j j , (3.15)

which when substituted into Eq. (3.14) gives,

( )V V

dV dVt

∂∇ ⋅ = − ρ

∂∫ ∫j . (3.16)

Differentiating both sides of Eq. (3.16) gives,

t

∂ρ∇⋅ = −

∂j , (3.17)

which is called the continuity equations and will be discussed in detail later.

Returning to Ampère's Law and taking the divergence of both sides gives,

( )∇⋅ ∇× = ∇ ⋅B j . (3.18)

The left hand side of this equation is of the form ( )⋅ ×A A B where A A and BB are vectors. The term ( )×A B is a vector a right angles to both AA and BB . The scalar product of AA with ( )×A B must be zero, since the two vectors are at right angles, resulting in ( ) 0∇ ⋅ ∇× =B . This corresponds to

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the physical situation of the curl of BB follows the lines of magnetic force for which the divergence is zero.

The result of this development is that Ampère's Law implies 0∇ ⋅ =j . Using Eq. (3.17), Ampere's equation then implies 0t∂ρ ∂ = . This results in the confirmation that Ampère's Law holds for steady currents where the rate of flow of charge is constant.

§3.4. MAXWELL'S 3RD EQUATION — AMPERE 'S LAW

In Maxwell's time the magnetostatic equation, ∇× =B j (Eq. (3.13)) was extended to include the concept of a displacement current. In modern physics, this extension would be considered a postulate. With the advantage of historical hindsight the divergence of Eq. (3.15) is zero — since the divergence of any curl is zero.

When the divergence of a vector filed is zero, it means that the lines of force for that field close in on themselves. In the case of the magnetic field, current flows is closed loops when static fields are present, but not with time varying fields. Maxwell understood this process through the concept of Faraday's polarization current. Faraday discovered this phenomenon of polarization — the separation of charges in a dielectric placed in an electric field. Since charges in motion are defined as a current, the process of polarization implies the existence of a polarization current. Maxwell developed these ideas in a series of papers during 1861–1862 [Whit51], [Maxw62].

In Maxwell’s paper [Maxw62] the concept of an elastic media played an important part in t he development of the displacement current paradigm. Maxwell describes how energy is stored in insulators, by displacing the electric particles from their equilibrium position by the action of the electric field. The movement of these electric charges produces a current. Since the media was assumed to be elastic, it was possible to calculate the velocity of the electric charges as they traveled through the media. The displacement current would then be proportional to the electric field strength. This calculation involved two constants α andβ , which were unknown to Maxwell. These constants were later associated with the permitivity and permeability of free space.

Using the steady state condition of Ampere’s Law ∇× =B j , this equation can be altered for electromagnetic fields that vary with time. Taking the divergence of both sides of Ampere’s Law, ( )∇⋅ ∇× = ∇ ⋅B j gives

0∇ ⋅ =j , since the divergence of the curl of a vector fields vanishes

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identically. It can be shown that in general, t∇ ⋅ = − ∂ρ ∂j (the continuity equations developed in §3.8) the term ∇ ⋅j vanishes only in the case that the charge density is static. Therefore, Ampere’s Law given as ∇× =B j is insufficient for time dependent – charge density varying – fields. Maxwell made several attempts to modify Ampere’s Law. In 1861 Maxwell noted his ideas in a letter to Sir William Thomson (Lord Kelvin), but did not develop the idea fully until 1865. Maxwell made the substitution

t→ + ∂ ∂j j E so that Ampere’s Law becomes t∇× = + ∂ ∂B j E . The continuity law can be recovered from this equation by taking the divergence of both sides, so that ( ) ( )t∇⋅ ∇× = ∇ ⋅ +∇⋅ ∂ ∂B j E to give

( )0 t= ∇ ⋅ + ∇ ⋅ ∂ ∂j E . By interchanging the space and time derivatives of EE ,

gives, ( ) 0t∇⋅ + ∂ ∂ ∇ ⋅ =j E . Using Coulomb’s Law, ∇ ⋅ = ρE , the continuity law can be restored as 0t∇ ⋅ +∂ρ ∂ =j . This provides for the conservation of charge where the original Ampere’s Law did not [Nive66].

Since polarization currents exist in ordinary matter — capacitors and dielectrics — Maxwell assumed that similar current could exist in the ether. In order to salvage Maxwell's original magnetostatic equation a displacement current was introduced, such that the magnetostatic equation is given as,

D∇× = +B j j , (3.19)

where Dj is the displacement current, which implies,

D∇ ⋅ = − ∇ ⋅j j . (3.20)

Since t∇ ⋅ = − ∂ρ ∂j , it follows D t∇ ⋅ = − ∂ρ ∂j . Using Maxwell's first equation, ∇ ⋅ = ρE and differentiating both sides gives,

( ) Dt t t∂ ∂ ∂ρ

∇ ⋅ = ∇ ⋅ = = ∇ ⋅∂ ∂ ∂

EE j , (3.21)

which can be rewritten as,

0Dt∂ ∇ ⋅ − = ∂ E j , (3.22)

which can be reduced to,

Dt∂

=∂E j . (3.23)

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Substituting this result into Eq. (3.19) gives Maxwell's equation as,

( III ) ,t

∂∇× = +

∂EB j

(3.24)

which states the,

(Line integral of BB around a loop) = Current thorough the loop

t

∂+

∂ (the flux EE through the loop);

which is Ampère's Law (1775–1836) adjusted for time varying fields. [9]

Maxwell was inspired by the equivalence of the displacement current and ordinary currents in order to generalize Ampère's Law. The addition of this displacement current allowed Maxwell to predict the existence of electromagnetic waves [Eyge72].

9 The four equations given above are referred to as Maxwell's Free Space equations,

but it must be remembered that these equations are actually assumptions of the Maxwell theory. Equations I, II, and III have been derived from experiments performed in steady state situations where fields and currents are not changing with time. Maxwell was largely responsible for putting these laws in the form of differential equations and contemplated the possibility that they were all valid even in time-dependent situations and realized that Ampère's Law (III) was inconsistent with the continuity equation (Eq.(19)). Maxwell saw that if Ampère's Law is modified by the addition of a time derivative (which leaves it unaltered in a steady state) to become ∇ × − =BB EE jj∂ ∂t , then there is a set of equations which are mathematically consistent even for ρ( , )rr t and j r( , )t varying with time. In Maxwell's time the first magnetostatic equation (III) was given by ∇ × =BB jj . Taking the divergence of this equation, since the divergence of the curl is always zero, gives ∇ ⋅ =j 0 . The fact that the divergence of a vector field is zero means that the field lines close on themselves. Current does flow in closed loops when static fields are present, but not when the fields vary with time. However, Maxwell's conception of this process somewhat different then modern formulations. Faraday had discovered and Maxwell knew about the phenomenon of polarization currents, or the separation of charges in a dielectric when placed in an electric field. Charges must move to separate, and since charges in motion constitute a current, this motion was referred to as the polarization current. Since polarization currents were know to exist in ordinary matter, it was natural for Maxwell to assume that similar currents could exist in the ether (in Maxwell's terms) that filled empty space [Whit60], [Buch85]. Maxwell then salvaged what appears to have been considered the basic equation for current flow in loops. From ∇ ⋅ =EE ρ, the time rate of change on the charge density is given by ∂ρ ∂ ∂ ∂t t= ∇ ⋅( )EE . Placing this result into the continuity equation ∇ ⋅ + =jj ∂ρ ∂t 0 gives ∇ ⋅ + =( )jj EE∂ ∂t 0 . The second term, ∂ ∂EE t, has the proper dimensions to be considered a current density and is called Maxwell's Displacement Current. The physical interpretation of this additional term is the description of the displacement current which produces a circulation in the magnetic field [Jack75].

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§3.5. MAXWELL'S 4TH EQUATION — FARADAY'S LAW OF INDUCTION

Michael Faraday found that when he magnetic flux passing through a conducting circuit was changed, a current was observed to flow in the circuit. The current was induced in such a way that it opposed the original flow of current in the circuit. This behavior is called Lenz's Law and states that the change of flux through a loop induces an electromotive force and associated current, which opposes the original flux. The induced current is a secondary effect whose value depends on the resistance of the circuit. This change in the magnetic flux results in an electromotive force EE , which produces the observed current,

dNdt

= −E , (3.25)

where N is the density of the magnetic lines of force.

The left hand side of Eq. (3.25) can now be converted into a form compatible with Maxwell's equations by defining the electromotive force as the force acting on a collection of electric charges,

S

dS= ⋅∫E E . (3.26)

Using the previous definition for the curl of a vector, the expression for the electromotive force can be rewritten as,

S A

dS dA⋅ = ∇ × ⋅∫ ∫E E , (3.27)

where dA is the unit are enclosed by the circuit. The right hand side of Eq. (3.27) can be rewritten by inserting the definition of the number of lines of magnetic force,

AN dA= ∇× ⋅∫ B and differentiating both sides to

give,

A

dN dAdt t

∂= ⋅

∂∫B

. (3.28)

Combing both sides of the rewritten Eq. (3.27) gives,

A A

dA dAt

∂∇× ⋅ = − ⋅

∂∫ ∫BE . (3.29)

Differentiating both sides gives, the fourth Maxwell equation is,

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( IV ) 0t

∂∇× + =

∂BE

(3.30)

which states the,

(Line integral of EE around a loop) t

∂+

∂(the flux BB through the loop) =

0;

which is Faraday's Law of induction. [10] In 1831 Faraday made the first quantitative observations of time–dependent electric and magnetic fields. He observed that a transient current is induced in a circuit if the steady current flowing in an adjacent circuit is turned off or on; or the adjacent circuit with a steady current flowing is moved relative to the first circuit; or a permanent magnet is moved into or out of the circuit. faraday interpreted the transient current flow as being caused by a changing magnetic flux linked by the two circuits. The changing flux induces an electric field around the circuit, the integral of which is the electromotive force. According to Ohm's Law, this electromotive force causes a current to flow in a circuit [Agas71]. [11]

Faraday's Law is a kind of relativity effect with respect to the motion of magnetic fields [Eins09a]. If the circuit in a magnetic field is moved with velocity v, the charges surround this circuit will be acted upon by a forcev ×B . If the circuit is held stationary and the magnets are moved, the same effect should occur. But since the charges are at rest, they must be acted on by the electric force qE generated by the circuit. The equality of these two motions is expressed in Faraday's Law integral form,

10 Equations (III) and (IV) are commonly referred to as the Curl equations, leaving

Equations (I) and (II) to be referred to as the Divergence equations.

11 Faraday found experimentally that a nonconservative electric field accompanies varying magnetic fields. Contrary to the description in many texts, Faraday's law of induction is not the consequence of the law of conservation of energy applied to the overall energy balance of current's in magnetic fields. Given a circuit of resistance RR carrying current jj resulting in a electromotive force E, the magnetic flux Φm surrounding this

circuit is m dSΦ = ⋅∫B , where the surface of integration is bounded by the circuit. When

the current changes with time, there are experimental observations that show j E tmRR − = − ∂ ∂Φ . This means that the current in the circuit is different than the current predicted by Ohm's law, by an amount equal to the negative time rate of change of the magnetic flux through the circuit. This expression is an experimentally derived law, not a law deduced from any theoretical understanding available to Faraday [Pans55].

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⋅ = − ⋅∫ ∫ÑF C

d df dsdt

B E and Ampère's Law ⋅ = ⋅∫ ∫ÑF C

df dsC B , where

= + d dtC j B . For a closed surface Faraday's Law and Ampère's Law

become ⋅ =∫Ñ constant in timedfB and ⋅ =∫Ñ 0dfC . The second of these

equations indicates that the electric current lines are closed. Because BB is finite and the surrounding space is homogeneous, the same must hold for magnetic lines in agreement with ⋅ =∫Ñ 0dfB . This relation will be

formalized in a following section, by developing the concept of continuity.

§3.6. NEWTON–LORENTZ FORCE EQUATION

The description of the interaction between charged particles by the intermediary electromagnetic field originated with Faraday and Maxwell. The field is produced by the charged particles — whose existence is assumed — and is measured by the acceleration it produces when acting on other charged particles. [12]

One approach to the development of this force equation is through the experimental observation that when a wire carrying a current, i, is place in a magnetic field a force dF is exerted on a short length of wire d s , such that,

( )d i d= ×F s B . (3.31)

This force is a right angles to the current flowing in the wire and is at right angles to the magnetic field BB , since the force is the result of a vector product.

In the early nineteenth century experiments were performed to observe what happens when a current flows through a conductor. [13] Two

12 The interaction of the electromagnetic field generated by a charged particle on the charged particle itself is one of the intractable problems of electrodynamics. An accelerating particle produces a field itself which changes the external field in which the particle is moving, which in turn will effect the motion of the particle. The theory of interacting particles and the fields they generate is intricate and involved, both conceptually and mathematically. The solution to this problem is provided by the relativistic formulation of Maxwell's equations and Quantum Electrodynamics.

13 The law describing the behavior of the magnetic field produced by the current is referred to as the Biot–Savart Law, but the assignment of this name is still open to question. Following Øersted’s announcement of the effects of a current on a permanent magnet in 1820, Ampère announced the observation of similar forces produced by a current or other currents flowing in a nearby wire. Biot and Savart presented the first quantitative

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French scientists, Jean–Baptiste Biot (1774–1862) and Félix Savart (1791–1841) found a relationship between the current flowing in a wire and the magnetic field produced by the flowing current. [14] The magnetic field BB due a current i flowing along a segment d s is given by,

( )034id dr

µ= ×

πB s r . (3.32)

This equation is known as the Biot–Savart law and describes the inverse square relationship between current and magnetic field intensity, with the magnetic field vector pointing at right angles to the current flow. Both the Biot–Savart equation and the equation for the force due to a flowing current involve the flow of current through a segment of wire, ids . Since this current is the result of the movement of charge q along with segment of wire, the current is nowi dq dt= . If the charge is moving with

velocity v then d vdt=s , which results in ( )id dq dt vdt dqv= =s .

The Biot–Savart can be rewritten is the form,

( )034

d dqvr

µ= ×

πB r , (3.33)

and integrating over the charge volume,

( )034q vr

µ= ×

πB r . (3.34)

The magnetic force equation, Eq. (3.31) can be rewritten in a similar manner to give,

( )d dq v= ×F B , (3.35)

and after integrating gives,

statement for the special case of a current flowing in a straight wire. Ampère later formulated a general description of this effect for currents flowing in arbitrary paths [Whit60], [Mott22].

14 In 1816 Biot published a work on the experimental methods in physics Traité de Physique expérimental et mathématique (Treatise on experimental and mathematical physics). This work established the importance of precise experimentation, stressing the need to improve the accuracy and precision of measurements by introducing new procedures and instruments to laboratory work. Biot described a mathematical and experimental method in which quantification of the results became the paradigm of science [Harm82].

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( )q v= ×F B . (3.36)

Inserting Eq. (3.34) into Eq. (3.36) gives an expression for the magnetic force as,

( )2

034magentic

q v vr

µ= × × π

F B . (3.37)

Comparing this to the electric force ,

2

304electric

qr

=πε

F r , (3.38)

allows the magnitudes of the two forces (ignoring the vector nature of the electric and magnetic quantities) to be compared,

2

20 0 2

magnetic

electric

vvc

= ε µ =FF

, (3.39)

where 0 01 cε µ = has the dimensions of a velocity and its magnitude is the speed of light is a vacuum.

Using the previous definition of the electric force, electric q=F E and adding this to the magnetic force gives, an expression for the force applied to a charged particle in the presence of the electric and magnetic fields can be written as the Newton–Lorentz equation. Experimentally it is determined that a particle carry charge q, and moving in a vacuum with a velocity v, experiences a force FF which is given by,

( V ) ( ).magnetictotal electric q v= + = + ×F F F E B (3.40)

Although this force equation was derived for currents flowing in a wire, it is also valid of charged particles moving in electric and magnetic fields in free space. [15]

15 The discovery of the Lorentz force ( )F e v= + ×E B was made in a 1895 paper

[Lore85] written by Lorentz, which lays the foundation for the Special Theory of Relativity. It was in this paper that Lorentz proved the concept of corresponding events [Pais86] in which the transformation of spatial and temporal coordinates between moving reference frames was first described mathematically. The Lorentz transformations were given in 1899 to a factor of ε . In 1904 this factor was fixed to unity (1), but Lorentz made an error in transforming the velocity components in the inhomogeneous Maxwell equations. Because of this Lorentz did not obtain a covariant solution to Maxwell’s equation which was needed to move forward with special relativity.

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§3.7. COUPLING STRENGTH OF THE ELECTROMAGNETIC FIELD

The electromagnetic field results in the discovery that fields and forces are tightly connected. It appears that the mass of particles is directly connected from the energies of the fields from which the particles serve as the source — the mass of the electron appears to be largely the mass corresponding to the energy of the electromagnetic field generated by its charge. From this view, a particle can be seen as a source or singular point in the field. The field of force can then be considered to consist of particles, such as the electromagnetic field can be described as a spectrum of photons emitted and absorbed by charged sources [Adai87].

In the previous section an expression of the force felt by a charged particle traveling in an electromagnetic field was given by Eq. (3.40). The relative strength of the electromagnetic force, weak force, strong force and gravitational force becomes a dimensional analysis problem — what units can be used to define force when the forces involved have very different characteristics. [16] In many texts the electromagnetic coupling constant is simply stated as a fact, without derivation nor a description of its importance to the understanding of the electromagnetic force. This section provides a diversion in order to describe the dimensional analysis approach to coupling constants and their importance to the understanding of all forces of nature.

There are several methods for approaching the electromagnetic field coupling strength problem. One approach involves the development of the axiomatic description of the underlying quantum field theory. Such a description is provided in [Feyn62]. In Feynman’s description the properties of the electromagnetic field are extended are made consistent with the tenants of quantum mechanics. Although these techniques have produced accurate results, there are still many problems to be solved. In this section a dimensional analysis approach will be used to develop the coupling constants required for a consistent description of the forces of

16 The foundations of dimensional analysis were laid out in Fourier’s Theorie

Analytique de la Chaleus, published in Paris in 1822. This work contains the formulation of Fourier Analysis as well. Fourier realized that every physical quantity...

... has one dimension proper to itself, and that the terms of one and the same exponent of dimension. We have introduced this consideration ... to verify the analysis ... it is the equivalent of the fundamental lemmas which the Greeks have left us without proof [West88].

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nature. [17]

For an electron traveling in an electromagnetic field its coupling strength can be compared to the corresponding centripetal force of an orbiting electron, such that,

2202

emvr r

= , (3.41)

where v is the constant speed of the electron and,

2

20

04ee =πε

, (3.42)

is the force obtained from the potential energy,

2

0

( )4eV r

r=

πε, (3.43)

where 0ε is the permitivity of free the vacuum. In a simple atom such as hydrogen, this potential is interchangeable with the Coulomb force.

The dimensions of 20ε are energy times length. The fundamental

constants 20ε , h and m are now components of the dimensional analysis. In

this approach A will be defined as an arbitrary physical quantity and dimdim A will denote its dimension. The dimensions of mass, length, and time are denoted by M, L, and T. The dimension A is then given by,

A M L Tα β γ=dim , (3.44)

where , ,andα β γ are definition exponents [West88].

17 The coupling force of electromagnetism produces the macro effects of everyday life.

If this coupling strength were very different, the world around us would be equally very different. If the electromagnetic force were much strong than the strong nuclear force, than the electrostatic repulsion between protons would overcome the nuclear attractive force and the nucleus containing more than one proton would break up. The world would then be composed of only hydrogen and hydrogen isotopes and possibly large hydrogen molecules. With a weak electromagnetic force, electrons would be free to form a plasma and light would be constantly emitted and absorbed by the charged particles of the plasma. Any light traveling through this plasma would be scattered and distorted.

If the electromagnetic force were very strong then it would be impossible to observe the world through our senses. With a very strong electromagnetic force light traveling through space would scatter from other light preventing the image of an object from reaching our eyes or sensors undisturbed.

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In this analysis the expression,

h0x y ze m , (3.45)

will be defined as a length with x, y, and z as unknowns to be solved through dimensional analysis. For the fundamental constants m, 0e and h ,

m M=dim , (3.46)

−=312 2 1

0e M L Tdim , (3.47)

−=h 2 1M L Tdim . (3.48)

Since Eq. (3.45) is a length, the unknown exponents are determined by,

( ) ( ) − −= =h 2 3 2 20

x xx y z x y z x ze m M L T M M L T Ldim (3.49)

Solving for the three unknowns can be done using the following,

0,

2 0,3 2 2 1,

x zx y zx z

+ =+ + =

+ = (3.50)

whose solutions are,

2,1,

2.

xyz

= −= −=

(3.51)

The characteristic length 0a is now given by,

=h2

0 20

ame

, (3.52)

which for the hydrogen atom, a single electron orbiting in an electromagnetic field of the nucleus, is the Bohr radius. The characteristic energy of the electron can be determined in a similar manner to give,

= h40

0 2

meE , (3.53)

When values are inserted into Eq. (3.52) and Eq. (3.53),

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−= = ×h2

100 2

0

0.525 10a mme

, (3.54)

= =h40

0 2 27.2meE eV . (3.55)

If the speed of light is added to the dimensional analysis, the rest energy of the orbiting electron can be derived as,

= 2E mc . (3.56)

The ratio between the characteristic energy and the rest energy is then,

= α202

Emc

, (3.57)

which is a dimensionless constant called the coupling constant or fine structure constant,

α = ≅h20

electromagnetic1

137.0359895(61)ec

. (3.58)

This constant relates the strength of the electromagnetic field in dimensionless units as the relative strength of the appropriate force between two protons at a separation distance of 1310 cm− [West88]. The significance of the constant is that it allows the coupling between the electromagnetic field and charged particles to be treated as a small perturbation that could be approximated in a series expansion [Davi79], [Hugh91], [Kaku93].

The coupling constants for weak, strong and gravitational interactions vary tremendously. The weak coupling constant is approximately:

2 2 5protonweak 2 8 1.02 10g m −α ≅ ≅ × , (3.59)

where the strong force's interaction is approximately,

2strong 4 1sg cα = π ≅h . (3.60)

The gravitational force is significantly weaker with a value of

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2 39 2Newton proton5.9 10N pG m m−α ≅ ≅ × . [18] (3.61)

With these developments, the description of the electromagnetic force is largely complete. Maxwell’s equations can now be combined with the coupling constant to describe the forces between charged particles. When quantum mechanics is added to produce Quantum Electrodynamics

§3.8. CONTINUITY EQUATIONS

In order for Maxwell's equations to properly describe electromagnetic process an expression for the conservation of charge must be developed. This will be done through the Divergence Theorem or Gauss's Theorem [Byro69]. Consider a stationary volume V, with a surface area S, containing an electric charge with density ρ which is moving with velocity v. The total charge contained in the volume V is given by,

V

dVρ∫ , (3.62)

18 At normal energy levels the coupling constants for the electromagnetic, weak and

strong forces are given above. It is believed that at energy levels found during the first instants of the creation of the universe — The Big Bang — the coupling constants for these three forces were the same value and there existed a single universal force. When the universe cooled the three forces condensed and the coupling constants become what we measure them to be today [Kaku93].

Using the coupling constants provided above, the relative force strengths can be described, using dimensional analysis [Isha89]. The gravitational force between two objects whose masses are m1 and m2 which are a distance r apart is F Gm m r= 1 2

2 where the

gravitational constant is G Nm kg= × − −6 67 10 11 2 2. . The gravitational constant, G, which

sets the scale of gravitational forces, the velocity of light, c, which is assumed to be constant throughout the universe and the electrical charge, e, which is also assumed to be constant throughout the universe — form the fundamental units of physics. These fundamental constants can be used to form dimensionless ratios or ratios with only the dimension of length.

The Fine Structure constant can be given again as, ≈h2 1 137e c . The ratio of the

gravitational force to the electromagnetic force in a hydrogen atom is Gm m e1 22 4010≈ − ,

which rules out any gravitational influence at the microphysics level. The gravitational

constant, h and c form a dimensionless constant ( )3 3310 cmG c −≈h , which is called the

Planck scale. At this distance, the quantum effects of gravity become important. However the energy required to observe this effect is ≈ 1019Gev , which is four (4) orders of magnitude beyond which the laws of physics are current understood to be exact — which is the limit of the so–called Theory of Everything [Clos83].

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so that the time rate of change of this total charge is now,

V

dVt

∂ρ

∂ ∫ . (3.63)

The total charge leaving the volume V per unit time is given by,

( )V V

dV v dVt

∂ρ = ∇⋅ ρ

∂ ∫ ∫ , (3.64)

where the divergence theorem of Eq. (3.3) is used for the right side of this expression. The conservation of charge requires that,

( ) 0V V

dV v dVt

∂ρ + ∇⋅ ρ =

∂ ∫ ∫ , (3.65)

or,

( ) 0V

v dVt

∂ρ + ∇ ⋅ ρ = ∂ ∫ . (3.66)

For this expression to hold for an arbitrary volume V, it must be true that,

( ) 0vt

∂ρ+ ∇ ⋅ ρ =

∂. (3.67)

When a charge ρ moves with velocity v a current is generated such that v= ρj , which produces an equation of continuity or conservation of charge as,

( VI ) 0t

∂ρ+ ∇ ⋅ =

∂j . (3.68)

This concept allows ρ and j to be considered as sources on the electromagnetic field so that Eq. ( I ) – ( IV ) determine the fields produced by a given system of charges and currents. This is the radiation view where ρ and jj specify the properties of the antenna, further developed in later sections.

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§3.9. SUMMARY OF MAXWELL’S EQUATIONS

Maxwell’s equations, using the natural units notation in which

0 0 1c = µ = ε = , [19] can now be summarized as:

( )

0, (a)

, (b)

0, (c)

, (d)

, (e)

0 (f).

t

tF q v

t

∇ ⋅ =

∂∇× = −

∂∇ ⋅ =

∂∇× =

∂= + ×

∂ρ∇ ⋅ + =

∂

EBE

BEB

E B

j

(3.69)

These equations describe the propagation of electromagnetic waves through free space, in the absence of the source of the radiated energy. It will be these equations that will form the basis of the ultimate goal of this text, the explanation of the conveyance of the electromagnetic force from the antenna of a radio transmitter to the electrons in the metallic antenna of the receiver.

19 One of the consequences of using natural units = = ε = µ =h 0 0c 1 removes the

need to develop the value of c from Maxwell’s equations. In a generic wave equation

( )( )2 2 2 2 21t v x∂ ξ ∂ = ∂ ξ ∂ , whose general solution is ( ) ( )1 2f x vt f x vtξ = − + + for any

arbitrary functions of f1 and f2 . In general form the propagation velocity 1 2v is given by

ω2 2k , where ω is the wave frequency and k is the wave number [Lind56].

In 1857 Wilhelm Weber (1804–1890) and Rudolph Kohlrausch (1809–1858) measured this propagation constant which appears in Maxwell’s equation, when a combined system of electrostatic and electromagnetic units are used (esu and msu). They determined the value to be approximately 3 1010× cm sec . The importance of this measurement, which

was very close to observed speed of light, was first noted by Gustav Kirchoff (1824–1887). In 1864 Maxwell used this information in his revised electromagnetic theory to assert that electromagnetic waves and light waves are equivalent.

In 1887 Heinrich Hertz (1857–1894) succeeded in generating electromagnetic waves which possessed all the properties of light waves – interference, defraction, and reflection. Joseph Henry observed electrical oscillations and perhaps even propagation of electromagnetic waves as early as 1842 [Hert93].

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In general these equations are difficult to solve, since they require special techniques to deal with the boundary conditions and the coupling between the various terms. In the Cartesian coordinate system, it is laborious to expand the vector algebra and apply the differential operators to the individual terms. This level of detail will be avoided since it is handled so well in other texts [Bala89], [Baru64], [Beke77], [Cott91], [Elli93], [Eyge72], [Jack75], [John88], [Lorr70], [Mari65], [O’rah65], [Pano55], [Ramo84], [Roja71], [Stra41], [Thom85]. The approach taken in later chapters will be to solve the equations for a source free radiation field, propagating through free space. The first four Maxwell equations actually represent eight equations — two scalar (divergence) equations, three Cartesian components of the Electric Field’s curl equation and three Cartesian components of the Magnetic Field’s curl equation. The solution to these eight equations will be six wave equations, three each for the electric and magnetic fields — the electromagnetic wave equations.

Maxwell’s treatment of Electrical Science was differentiated from that of other writers by his insistence on Faraday’s conception of electric and magnetic energy as residing in the medium...in this view, the forces acting on electrified...bodies did not form the whole system of forces...but served only to reveal the presence of a vastly more intricate system of forces, which acted through the ether which other material bodies were supposed to be surrounded.

— J. H. Jeans [Jean25]

Copyright 2000, 2001 4–1

§4. SOLUTIONS TO MAXWELL'S EQUATIONS

The equations given in Eq. (3.69) are almost perfectly symmetric in EE and BB . The solution to these four equations will be done using two different approaches. The first will develop the wave equation by using vector algebra. The second approach will make use of the vector and scalar potentials. Both methods result in Maxwell’s wave equations, however the vector potential solution will prepare the way for the quantum mechanical description of the propagating electromagnetic wave.

§4.1. VECTOR ALGEBRA SOLUTION TO MAXWELL’S EQUATIONS

If the two Divergence equations are set aside for a moment, the two Curl equations describe coupled electric and magnetic fields since they both contain EE and BB . This approach to the solution will eliminate the duplicate terms and produce the wave equation directly.

Taking the Curl of Eq. (3.69b) gives,

( ) ( )

t∂

∇× ∇× = − ∇×∂

E B.

(4.1)

inserting Eq. (3.69d) into Eq. (4.1) gives,

( )2

2t t t∂ ∂ ∂ ∇× ∇× ≡ − = − ∂ ∂ ∂

E EE . (4.2)

Since ( ) ( ) ( )∇× ∇× = ∇ ∇⋅ − ∇⋅∇E E E , the first term of Eq. (4.2) is zero, because Gauss’s law states 0∇ ⋅ =E and the second term is the definition of the Laplacian operating on EE . In Cartesian coordinates,

2 2 2 2

22 2 2 2x y z t

∂ ∂ ∂ ∂∇ ≡ + + =

∂ ∂ ∂ ∂E E E EE (4.3)

The same manipulation can take place for Eq. (3.69c) giving,

2

22t

∂∇ =

∂BB (4.4)

The results of Eq. (4.3) and Eq. (4.4) are Maxwell’s wave equations propagating through free space.

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§4.2. VECTOR POTENTIAL SOLUTION TO MAXWELL’S EQUATIONS

Maxwell’s equations can be solved as they stand in simple situations, but it is often convenient to introduce potentials, obtaining a smaller number of second–order equations while satisfying some of the Maxwell equations with an identity. [1]

Since 0∇ ⋅ =B (Eq. ( II )) still holds, BB can be defined in terms of a vector potential, such that,

= ∇ ×B A (4.5)

Faraday's Law Eq. ( IV ) can be written as,

0t

∂ ∇× + = ∂ AE . (4.6)

This means that the quantity with a vanishing curl in Eq. (4.6) can be written as the gradient of some scalar function,

t

∂+ =−∇φ

∂AE , (4.7)

or,

t

∂= −∇φ−

∂AE . (4.8)

The definitions of BB and EE in terms of the potentials AA and φ will be

determined by the homogeneous equations Eq. (3.8) and Eq. (3.30) [2] The

1 The solutions to Maxwell's equations using the scalar and vector potential is the

modern approach and is given in hindsight as the logical approach to the problem. As is usual with such revisionist matters the logic of this approach was not obvious to the investigators of Maxwell's time, but the developed over a period of intense effort, resulting in the hindsight of today [Buch85].

2 The origins of the potential solution to Maxwell's equations is obscured by history. The earliest accounts of partitioning any well–behaved vector field into its irrotational (curl free) and solenoidial components can be found in [Helm58]. In general, vector fields are determined by the knowledge of their curl and divergence.

In 1846 — the year after he had taken his degree as second wrangler at Cambridge, J. J. Thompson (1856–1940) investigated the analogy between electric phenomena and mechanical elasticity. He examined the equations of equilibrium of an incompressible elastic solid which is under strain. He showed that the distribution of the vector which represents the displacement could be compared to the distribution of the electric force in a electrostatic system. The elastic displacement could be identified with a vector AA , defined in

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inhomogeneous equations, Eq. (3.5) and Eq. (3.24), can be written in terms of the potentials as,

( )2

t∂

∇ φ + ∇ ⋅ = −ρ∂

A , (4.9)

and,

2

22t t

∂ ∂φ ∇ − − ∇ ∇⋅ + = − ∂ ∂ AA A j . (4.10)

The four Maxwell equations are now reduced to two coupled inhomogeneous differential equations, Eq. (4.9) and Eq. (4.10) The uncoupling of these equations can be accomplished by exploiting the arbitrariness involved in the definition of the potentials [Brom70]. Since the BB field is defined through Eq. (68) in terms of AA , the vector potential is arbitrary to the extent that the gradient of some scalar function χ can be

added to the vector potential. [3] The magnetic field BB is left unchanged by the transformation,

′→ = +∇χA A A . (4.11)

In order that the electric field be unchanged as well, the scalar potential must be simultaneously transformed as,

t

∂χ′φ → φ = φ −∂

. (4.12)

The transformations in equation Eq. (4.11) and Eq. (4.12) are called gauge transformations and the invariance a gauge invariance.

The solution to Maxwell's equations using the scalar potential field φ and vector potential field AA , which in turn relate to the electric and magnetic fields by Eq. (3.5) and Eq. (3.24) [Eyge72] can be further

terms of the magnetic induction BB , by the familiar ∇ × =AA BB . The vector AA is equivalent to the vector potential which had been mentioned in the memoirs of Weber and Kerchief on the induction of currents; but Thompson arrived at this independently. Although Thompson laid the groundwork, it was J. C. Maxwell who provided the solution to the electromagnetic propagation question through his wave equation formulation.

3 The vector AA is not completely determined by the magnetic field BB , since for any scalar function χ , ∇ × ∇χ = 0 the gradient of an arbitrary function χ can be added to the

vector field AA . For the scalar potential φ , the time derivative of the arbitrary scalar function χ is subtracted in order to maintain the electric field's invariance.

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simplified using several techniques. [4] The solutions provided below are given without the formalism of their derivations which can be found in [Feyn64], [Cowa68] and [Land71]. One approach to the solution of Maxwell's equations is the use of Green's Theorem, developed in detail in the next sections. The approach is taken with the intention of developing a radiation oriented set of equations that can be used to solve problems in the radiation zone of the electromagnetic field.

By using the vector identity, 2( )∇×∇× = ∇ ∇ ⋅ − ∇A A A , and substituting Eq. (4.1) and Eq. (4.4) into Maxwell's equations, results in the corresponding scalar and vector field equations,

( )

22

2

22

2

, ( )

0 0.,( )

,( )

( )

at t t

b

t t t c

dt t t

∂ φ ∂ ∂φ ∇ ⋅ = ρ ⇒ ∇ φ − + ∇⋅ + =−ρ ∂ ∂ ∂ ∇ ⋅ = ⇒ ∇⋅ ∇× =∂∂ ∂φ ∇× = + ⇒ ∇ − − ∇ ∇ ⋅ + = − ∂ ∂ ∂ ∂ ∂ ∂ ∇× = − ⇒ ∇× − −∇φ = − ∇× ∂ ∂ ∂

E A

B A

AEB j A A j

B AE A

(4.13)

§4.3. INTEGRAL FORM OF MAXWELL'S FIELD EQUATIONS

The freedom of defining an arbitrary scalar and vector potential implied by Eq. (4.11) and Eq. (4.12) means that a set of potentials can be chosen such that,

4 The introduction of the scalar and vector potential fields is motivated by the search for solutions to the EE and BB fields whose form is appropriate for the traveling waves development in the next section. The vector and scalar potentials in Eq. (4.7)) and Eq. (4.8) are not unique. The simplification obtained by the introduction of AA and φ must be paid for by the fact that φ and A A are not unique for a given E E and BB field. Starting with a given choice of AA and φ , the same fields may be obtained from the alternative potentials, AA AA AA→ ′ = + ∇χ and φ φ φ ∂χ ∂→ ′ = − t where χ is an arbitrary scalar function that does not effect the individual field components. Since it is the fields which are the observable quantities there is no physical basis for choosing AA or ′AA . The field calculated from ′AA is the same as the one calculated from AA . Therefore only those quantities invariant under the gauge transformation will have direct experimental significance. The transformation is called a gauge transformation and is further developed in the section on Gauge Theory; the Maxwell equations are said to be invariant with respect to gauge transformations. The useful gauge transformations found in electrodynamics are: the Lorentz Gauge: ∂ ∂AA xi = 0 , the Radiation Gauge: ∇⋅ =AA 0 and the Coulomb Gauge: ∇ ⋅ = =AA 0 0, φ . The concept of gauge invariance and its relationship to the electromagnetic potential is developed further in later sections.

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0,t

∂φ∇ ⋅ + =

∂A (4.14)

that defines the Lorentz condition which allows an arbitrary specification of the vector and scalar potentials. [5] The Lorentz condition can now be used to uncouple the pair of equations Eq. (4.9) and Eq. (4.10) and leave two inhomogeneous equations [Borm70] (which later will be developed as the wave equations) one for φ and one for AA ,

22

2

22

2

,

.

t

t

∂ φ∇ φ − =−ρ ∂

∂ ∇ − = − ∂ AA j

. (4.15)

§4.3.1. Green's Function and the Potential Solution

The problem[6] of finding the solution to the potential equations AA and φ in terms of currents and charges may be approached using Fourier analysis and its differential equation solution technique, Green's Functions. Using Green's functions [Gree71], [Boch01], [Butk68], [Mors53], [Arfk85], and the Lorentz gauge is an approach that allows the introduction of gauge invariance — which will later be important in the quantum mechanical description of the radiation field. [7] Using Eq. (4.13)

5 The Lorentz condition is not as arbitrary as some texts state, in that it leads to a

symmetry between the vector and scalar potentials that allows these potentials to satisfy the same wave equation. In addition the Lorentz condition also provides a relativistic covariant relation between the scalar and vector potentials [Pano66].

6 This section is one the diversions necessary to explain what is usually glossed over in many text books. The solution to Maxwell's equations using the scalar and vector potential makes use of the Green's function method. In this method partial differential equations may be solved in a straight forward manner, without explicit consideration of the boundary conditions.

7 This theorem was first presented in 1828 by Georg Green (1793–1841) in his “Essay on the Application of Mathematics to Electricity and Magnetism” [Maxw65]. In the general theory of boundary value problems, an important role is played by a mathematical theorem called Green's Theorem and by certain integral expressions for the potential that are derived from it. Consider a volume V bounded by a closed surface S, let nn be an outwardly pointing normal to the surface. Let ′r be a position vector in a coordinate system with arbitrary origin. Given two arbitrary functions ψ( )′rr and χ( )′rr that are appropriately

continuous in V and form the function ( ) ( ) ( )′ ′ ′= ψ ∇χA r r r . Then the divergence of the

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(a) and Eq. (4.13)(b), a Green's function will be used to construct a general solution of these potential equations in integral form. A Green function ( , ; , )G t t′ ′r r , where ′r and t′ are passive parameters that satisfy,

2

22 2

1 4 ( ) ( )GG t tc t

∂ ′ ′∇ − = − πδ − δ −∂

r r , (4.16)

will be used as the basis of the solution of Maxwell's potential equations.

Beginning with the point source charge at the origin 0t′ ′= =r , 0 ( , ) ( , ; ,0)G t G t=r r 0 such that 0G satisfies the Laplacian,

2

2 00 2 2

1 4 ( ) ( )GG tc t

∂∇ − = − πδ δ

∂r . (4.17)

Since the source is a point charge, the solution is dependent only on r = r .

Using the Laplacian in spherical coordinates, but with the angular coordinate, φ , equal to zero, Eq. (4.17) becomes,

2

2 002 2 2

1 1 4 ( ) ( )GGr tr r r c t

∂∂∂ − = − πδ δ ∂ ∂ ∂ r . (4.18)

The solution to this equation at a distance from the origin is,

0( / )( , ) f t r cG t

r±

=r , (4.19)

where f is an arbitrary function of time and space.

Integrating Eq. (4.18) over a small volume V∆ containing the origin gives,

2

2 002 2 2

1 1 4 ( )V V

Gr dv G dv tr r r c t∆ ∆

∂∂∂ − = − πδ ∂ ∂ ∂ ∫ ∫ (4.20)

function is given by ∇ ⋅ = ∇ψ ⋅ ∇χ +AA ψ∇ χ2. The divergence theorem

V S

dv ds′∇ ⋅ = ⋅∫ ∫A A

yields 2( )V S

dv ds′∇ ψ ⋅ ∇ χ + ψ ∇ χ = ψ∇χ⋅∫ ∫ . Writing a similar equation with the roles of ψ and

χ interchanged and subtracting the two equations and using ( ) ( )ds n ds′ ′ψ∇χ ⋅ = ψ ∂χ ∂

gives ( ) ( )[ ]2 2( )V S

dv n n ds′ ′ψ∇ χ + χ ∇ ψ = ψ ∂χ ∂ − χ ∂ψ ∂∫ ∫ . This relation between the surface

and volume integrals is Green's Theorem.

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With 0G of Eq. (4.19), the integrand of the first integral in Eq. (4.20) is singular resulting in an improper integral.

The integral in Eq. (4.20) can be properly redefined by recognizing the integrand is 2

0 0G G∇ = ∇ ⋅ ∇ . By the divergence theorem the volume integral of the divergence can be converted to a surface integral over the small bounding surface 0S∆ → . [8]

With the conversion from a volume integral to a surface integral, the δ function condition in Eq. (4.20) becomes,

2

0 02 2

1( ) 4 ( ).S V

G ds G dv tc t∆ ∆

∂∇ ⋅ − = − πδ

∂∫ ∫ (4.21)

Substituting the explicit form Eq. (4.19) into the equation and using,

0 2

1f fGr c r

′ ∇ = ± ⋅

r , (4.22)

where the prime denotes differentiation with respect to the argument of the function and 0r is the radius of the small spherical volume about the origin, gives,

[ ] [ ]0 0 2

02 200

14 4 ( )

f t r c f t r cr dv t

cr c rr′ ± ± ′′δ

− π − = − πδ

∫∓ . (4.23)

Letting 0 0r → results in ( ) ( )f t t= δ , giving,

0

( )( , )

t t cG r t

′ ′δ − ± −=

′−r r

r r, (4.24)

and,

( )

( , ; , )t t c

G t t′ ′δ − ± −

′ ′ =′−

r rr r

r r. (4.25)

8 The divergence theorem states that for any well–behaved vector field ( )xA defined

within a volume V surrounded by the closed surface S the relation V

A nda Adv⋅ = ∇ ⋅∫ ∫Ñ

holds between the volume integral of the outwardly directed normal component of AA . This relation can be used as the definition of the divergence of a vector field [Stra41], pg. 4.

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§4.3.2. Field Potential Solutions

This Green function allows the integration of Eq. (4.13)(a) and Eq. (4.13)(b) so that the solution to φ and AA is,

( , )

( , ) ,t c

t dv′ ′ρ − −

φ =′−∫

r r rr

r r (4.26)

( , )

( , ) .t c

t dv′ ′− −

=′−∫

j r r rA r

r r (4.27)

The solutions provided by equations Eq. (4.26) and Eq. (4.27) are particular integrals of the inhomogeneous equations Eq. (4.13)(a) and Eq. (4.13)(b). For the purpose of adapting to given initial and boundary conditions, integrals of the homogeneous potential equations can be added to produce the wave equations for AA and φ . [9]

In Eq. (4.26) and Eq. (4.27), t c′− −r r denotes, that for ( )tφ , the value of the charge density ρ at time t c′− −r r should be used. That is, for each element of charge dvρ , the equation states that the contribution to the potential is the same form as in the static charge density equation,

14

dVr

φ = ρπ ∫ , except that the finite propagation time for the charge effect

must be accounted for. For computing the total contribution to the potential φ at a point x at time t, the values of charge density from points distance ′r away at an earlier time t c′− −r r , since for a given element it is that effect which reaches x at time t. A similar interpretation applies to the computation of AA from currents in Eq. (4.27). Because of this retardation effect, the potentials ( )tφ and AA (t) are called the retarded

potentials. [10]

9 The solutions to the inhomogeneous Maxwell equations using Green's functions is

based on the existence of the Fourier transforms of the vector and scalar potential functions. These solutions do not in principal apply to monochromatic radiation sources. Using the Fourier transform, a monochromatic source (radiating at a single frequency) would radiate over an infinite time. This situation can be delete with if a limiting process is used starting with a finite duration time domain pulse.

10 The physical content of equations Eq. (4.22) and Eq. (4.23) is not identical with that of Eq. (4.11) and Eq. (4.11) While in the differential form the sign of the time is in no way distinguished, i.e. the equations are not altered by an exchange of past with future, the integral forms make an essential distinction between past and future. Mathematically,

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Maxwell's equations ( I ) – ( V ) provide several relations between the vector potential AA and the scalar potential φ . By virtue of Maxwell's Equation ( I ),

0,∇ ⋅ =A [11] (4.28)

and,

2 .∇ φ = −ρ [12] (4.29)

As the electromagnetic wave propagates through sdpace, energy is transferred form the source (transmitter antenna) to the destination (receiving antenna). This energy is subject to the Laws of Conservation of Energy given by,

V S

dUdV S da

dt= − ⋅∫ ∫ (4.30)

The energy density U is given by the instantaneous value of the electric and magnetic fields,

(VI) 2 31 joules m ,2

U =E E (4.31)

and,

( VII ) 2 31 joules m .2

U =B B (4.32)

Eq. (4.22) and Eq. (4.23) would also be possible in which values of jj and ρ at the source point are chosen for later time, t c+ rr , giving the advanced potentials. Such solutions would, however be contradictory to the concepts of causality, since charges and currents are considered to by the sources of potentials, since the electromagnetic field does not proceed the charges and currents which cause it [Pano66].

A theory of radiation involving the advanced potential was put forward by J. A. Wheeler (1911– ) and R. P. Feynman (1918–1988) [Whee45] in which a covariant action–at–a–distance theory of electrodynamics can be formulated in terms of the symmetrical combination of the retarded and advanced potentials [Pano66].

11This is the result of the choice of the Lorentz Gauge.

12This is Poisson's equation and it is the description of an irrotational vector field derived from its sources. The solution may be found using Green's theorem [Gree71] in which φ is equal to the desired potential function and ψ is set to 1 r giving,

( ) ( ) ( ) ( ) ( ) 2 21 1 1 1r N r N ds r r dV∂φ ∂ − φ ∂ ∂ = ∇φ−φ∇∫ ∫Ñ .

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where the total energy density is given by,

( )2 212

U = +E B (4.33)

Eq. (4.31) and Eq. (4.32) represent the electrical and magnetic energy densities of the microscopic electromagnetic field. These energies reside in the field itself in a localized volume element.

This volume element contains the total energy of,

( )2 2 312 V

d x+∫ E B . (4.34)

This result form the basis of the Poynting Theorem developed in §5.2 and the Lagrangian and Hamiltonian approach to quantizing the electromagnetic field.

Finally Maxwell's Equation ( III ) and ( I ) imply that charge is conserved through an equation of continuity,

( VIII ) 0.t

∂ρ+ ∇ ⋅ =

∂j (4.35)

§4.4. TRAVELING WAVES

The four Maxwell equations, the continuity equation and the energy density equation, represent the sum of all knowledge regarding classical electrodynamics in the early twentieth century. From these equations all macro–world physics can be derived, since they describe the interaction of electromagnetism, including light, and matter in non–quantum mechanical and non–relativistic terms.

Life is a wave, which in no two consecutive moments of its existence is composed of the same particles.

— John Tyndall

The mathematical form of Maxwell's equations (( I ) – ( IV )) leads to the discovery of wave–like motion of the electric and magnetic fields. [13] The

13 Heinrich Rudolf Hertz (1857–94) devised an experimental test of Maxwell's traveling wave theory [Suss64], [D’ago75], [Hert95], [Long83], [Buch94]. He constructed a spark–gap generator which was used as a transmitter and a loop of wire as a receiver. The spark produced by the transmitter would produce a similar spark between a gap in the receiving loop. Using a zinc plate Hertz showed the standing waves were present. By moving the

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equations also indicate that the speed of the waves should be a constant, related to the Permittivity and Permeability of free space. When Maxwell calculated the value of this constant, he found it ...

… so nearly that of light, that it seems we have strong reason to conclude that light itself ... is an electromagnetic disturbance in the form of waves propagating through the electromagnetic field according to the electromagnetic laws [Zaja74].

The existence of such waves was known theoretically prior to the engineering skills necessary to construct equipment capable utilizing them — a feat of theoretical physics not recently repeated. [14]

To this point the equations describing the electric and magnetic fields have assumed that the waves are propagating in free space, free of any sources of the electromagnetic radiation, e.g. 0ρ = =j . The solutions to Maxwell's equations ((I) – (IV)) describe electromagnetic plane waves that are transverse to the direction of propagation, given by the vector k, such that the electric and magnetic field vectors are mutually orthogonal with k. [15]

receiving antenna, the intensity of the received spark would vary. He confirmed these electric waves would pass through wooden doors, be reflected like light and were polarized. Faraday's lines of force as well as Maxwell's electric waves were confirmed leading to the electromagnetic devices of today.

14 The electromagnetic waves described by Maxwell's equations are classified into several types, although they are all part of a continuous spectrum whose wave lengths range from 102m for radio waves to 10 2− m for microwaves, 10 10− m for x–rays through 10

15−m for cosmic rays. Maxwell predicted the existence of electromagnetic waves on

theoretical grounds when he derived the wave equation.

The velocity of these waves depends n certain electrical constants — the permittivity and permeability of the propagating medium. The measurement of these constants along with Michael Faraday's experimental discovery that polarized light is rotated in the presence of a magnetic field led Maxwell to speculate that light was also an electromagnetic wave. Direct evidence of Maxwell's prediction came in 1888 in experiments performed by Heinrich Hertz (1857–1894) [Buch94].

15 The existence of transverse modes in the propagation of electromagnetic waves was first proposed by Augustin Jean Fresenel (1788–1827). In 1814 Fresenel wrote that he suspected light and heat were connected with the vibrations of a fluid. His concept that light was a form of motion of a medium was basic to his theory of optics. In 1821 he had reformulated his theory of optics in terms of waves propagating in a medium [Harm82]. Fresenel submitted a paper to the Paris Academy prize in 1819 which described the mathematical theory of the interference of light. His theory was confirmed by Thomas Young (1773–1829) through Young's double slit experiment. By 1821 Fresenel had formulated a theory of the polarization of light and realized that the vibrations of the

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In order to develop the description of the refraction of electromagnetic waves in a later section, the influence of a material medium on the propagation of the electromagnetic waves must be addressed. Maxwell proposed that a dielectric constant ε be added to the wave equations such that the propagation velocity is given by 1v = ε . This simple model can be used to describe wave propagation in an isotropic, non–absorbant dielectric, homogeneous medium.

In modern notation, the Traveling Wave equation can be derived from Maxwell's equations. By noting that Maxwell's equations are functions of time, which implies the E E and BB fields are not independent, all four of Maxwell's equations are needed for their solution. The two divergence equations (I) and (II) state that the flux of EE and BB outward through any volume in free space (in the absence of any charges) is zero. The two curl equations (III) and (IV) require that the EE and BB fields are coupled and imply if t∂ ∂B is non zero, then so is ∇×E . The curl of the EE field can only be non zero if EE is a function of position. If BB is a function of time ( t∂ ∂B ) then ∇×E is also a function of time. Eq. (IV) now states that a BB field which varies with time generates an EE field which varies in both time and space.

In a similar manner a non zero t∂ ∂E generates a time and space varying BB field. The coupling of these two fields forms the basis of electromagnetic wave propagation.

The equation relating the spatial variations of the EE field to the time variations of the BB field can be obtained by eliminating BB from the two curl equations (III) and (IV). This is done by taking the curl of Eq. (IV),

( ) ( )t∇× ∇× = ∂ ∂ ∇×E B , using the vector identity 2( ) ( )∇× ∇× = ∇ ∇ ⋅ − ∇E E E and substituting into Eq. (3.30) to produce,

2 0.t

∂−∇ + ∇ × =

∂BE (4.36)

In Eq. (4.36) the ∇ and the t∂ ∂ operations can be interchanged, so that using Eq. (3.24) results in the electric field wave equation,

medium must be purely transverse. If this medium was composed of molecules bound by forces acting at a distance than transverse waves could not be propagated since the aether would have to be rigid. The problem of constructing a model of the aether that could propagate transverse rather than longitudinal waves became a major problem of nineteenth century optical physics [Harm82].

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2

22 0t

∂∇ − =

∂EE . (4.37)

Similarly, taking the curl of Eq. (3.24) and using Eq. (3.8) and Eq. (3.24) gives the magnetic field wave equation,

2

22 0t

∂∇ − =

∂BB . (4.38)

Both vectors for EE and BB satisfy the same differential equation and describe coupled electric and magnetic fields propagating through space at the speed of light. [16]

§4.4.1. Displacement Current in the Field Equations

In §3.4 the displacement current term in Maxwell’s equation,

t

∂∇× = +

∂EB j , (4.39)

describes the displacement of the physical media carrying the electromagnetic force. In the theory of continuous media, such as a material –– either a conducting media or a dielectric.

In free space, the measuring of the displacement current is less clear, since it involves the flow of a current which is caused by the flow of charge somehow connected by the displacement of the media. In Maxwell’s original formulation of the propagation of electromagnetic waves, a mechanical oscillation was visualized which carried the wave. The displacement of this media could account for the displacement current.

16A wave is described as plane homogeneous when it is possible to place a family of

parallel planes so that along each one of these planes the magnetic field strength does not change. Since EE and BB are constant along wave planes, all partial derivatives with respect to z and y vanish. The x–component of the two curl equations and the two divergence equations read ∂ ∂ ∂ ∂EE BBx xt t= = 0 and ∂ ∂ ∂ ∂EE BBx xx x= = 0 while the remaining

components of the curl equations read − = −∂ ∂ ∂ ∂BB EEz yx t , ∂ ∂ ∂ ∂BB EEy zx t= ,

∂ ∂ ∂ ∂EE BBy zt t= − and − = −∂ ∂ ∂ ∂EE BBz yx t . These developments led Maxwell to conclude

that the traveling waves of his electromagnetic theory behaved like the transverse waves of the previously observed light waves.

This discovery was hailed as a dramatic confirmation of Maxwell’s theoretical description of electromagnetism. Within a decade Marconi and others were using Hertzian waves in practical applications. Unfortunately Hertz did not live to see these deices. He died of blood poisoning at age 36 in 1894 [Buch94].

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Once the aether was removed, the displacement current is no longer visible. By using the wave equations the displacement term of Eq. (4.39) can be eliminated.

§4.5. CLASSICAL EXPLANATIONS FOR FORCE FROM FIELDS

Now that the classical theory of fields has been developed and the resulting wave equation derived — what is force that is actually felt by the particle imbedded in the electromagnetic field at a distance from the accelerating charge? By what mechanism is this force carried to the charged particle?

Using the Lorentz force function,

( ) ,q v= + × F E B (4.40)

expressed in terms of the scalar potential φ and the vector potential AA , the electric and magnetic field become,

,t

∂=−∇φ−

∂AE (4.41)

and,

.= ∇ ×B A (4.42)

Since these equations do not uniquely specify φ and AA , Maxwell's equations take their simplest form when the scalar and vector potentials are related through the Lorentz condition,

0,t

∂φ∇ ⋅ + =

∂A (4.43)

which gives the Lorentz force as,

( ) .dq vdt

= −∇ φ− ⋅ −

AF A (4.44)

Maxwell's equations permit, in principle at least, the calculation of the fields EE and BB from arbitrary sources. Since these fields are important because of their actions on charges, the foundations of electromagnetic theory are completed by a description of the Lorentz Force density FF as shown in Eq. (4.44). It should be remembered that Eq. (4.44) is a postulate, but it is illuminating to see its origin [Whit51]. The first terms in Eq. (4.44) extends the definition of EE , as the force exerted on a unit charge, to a force

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exerted on a charge by a time varying field. The second term is the foundation of the postulate – it generalizes the magnetostatic results on the force between stationary currents circulating in loops of wire. Lorentz assumed that the current in a wire was due to the motion of individual, microscopic, charged particles. Formally this assumption is given by the current as I dq dt= , with the current in a loop of wire given as Idl dq dl dt= . Interpreting dl dt as the velocity v of the charge dq , the

force dF on this charge dq in motion is vd dqc×

=BF , from the Biot–Savart

law, which gives the origin of Eq. (4.44). Although the force density formula was inspired by the results of experiments on ensembles of charges that made up stationary currents, has been confirmed for general distributions of charges in arbitrary motion.

§4.6. SUMMARY OF CLASSICAL FIELD THEORY

This classical rationalization of force derived from potential fields provides the explanation for the observed effects of the electromagnetic force. The Lorentz force law describe in Eq. (4.44), plus measurements of the components of acceleration of the test particle, can be viewed as defining the components of the electric and magnetic field. Once the field components are known from the accelerations of a test particle, they can be used to predict the accelerations of other test particles. The Lorentz force law is both the definer of fields and a predictor of particle motions.

Maxwell developed his ideas is a series of papers between 1861 and 1868. Subsequent experimental and theoretical investigations demonstrated a remarkable range of applicability of the theory [Buch85]. Maxwell's equations encompass light waves and the phenomena of optics; they turn out to be consistent with Einstein's special relativity [17] — in 1927 they were put in quantum form by P. A. M. Dirac [Dira27].

17 Although revisionist history has placed Albert Einstein's accomplishments in light of

his failure to unify gravity and electromagnetism his work during the year of 1905 was breathtaking. During 1905 when Einstein was 26 he published his first great work — a paper describing the theory of the photoelectric effect. It was in this paper he formulated the concept that light consists of quanta or photons. In the same year he published the theory of Brownian (Robert Brown (1773–1858)) motion — the movement of fine particles in a liquid — which laid the groundwork for the field of statistical mechanics. A third paper on the special theory of relativity was followed by a fourth paper in which he derived the most popular expression in modern science E mc= 2 .

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What remains — is the explanation for the cause of this force.

The search for this explanation leads to the next level of physical theory, developed at the beginning of the twentieth century — Quantum Mechanics. Before developing the concepts of quantum mechanics a the effects electromagnetic force on a remote charged particle will be examined. This will require a description of the radiated field and antenna theory. This will be the subject of the next section.

Before proceeding with the next section a short summary of the progress made so far is useful. Electrostatics and electromagnetic can be described using Maxwell's equations. From the original four equations electromagnetic waves were deduced which led to the engineering field of radio transmissions.

In the next section, Maxwell's equations will be used to define:

n The energy contained in the electromagnetic field. This field energy will be used to provide the force necessary to move the electrons in the remote antenna.

n The vector and scalar potential fields will be defined. It will be through these new fields that the electromagnetic field will be quantized in later sections.

n A simplistic description of an antenna and its radiation pattern will be developed. The electric and magnetic fields as a function of position and time will serve as the final description of classical electromagnetic — as it applies to the problem of this monograph.

The God said Let there be Light, and there was Light. The Light was made before ether sunne or moone was created therefore we must not attribute that to ye creatures that are Gods instruments.

— The Geneva Bible, 1560

During this year Einstein also held a full–time position at the Bern patent office, attended to his wife and small child and performed his physics research in his spare time. Since he was unable to obtain an academic position Einstein was isolated from the mainstream of the physics community — which may have attributed to his early successes. This position of isolation was repeated in his later life when he objected strongly to the underlying theories of quantum mechanics [Pais79], [Pais82], [Born71]. The argument with Neils Bohr was based on Einstein's contention that quantum mechanics as formulated at the time was not consistent with his principles of objectivity and causality that he found necessary for a sensical explanation of nature [Bohr49]. This argument has been popularized through Einstein's quote that God does not play dice with the Universe.

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§5. THE RADIATED FIELD

In the previous section the Lorentz force equation was developed which describes the force felt by a charged particle in the presence of a vector potential field. If this potential field is varying with time, then the charged particle will feel the oscillating force, which will impart a momentum to the charged particle. If the particle is free to move, as a free electron is in the outer valance shell of a metal, than this motion will cause a current to be generated, which in turn can be amplified and transformed into a signal used by a radio receiver. Some additional work needs to be done though before this phenomenon can be fully explained.

Given the time periodic current ( , )tj r , the radiated field potential given in Eq. (4.23) and the traveling wave equation in Eq. (4.30), what is of interest is the field potential equation at some distance rr from the source of the time periodic current.

From Maxwell's equations it follows that all the quantities that enter into them depend on r r and t. This time dependence can be expressed in a complex exponential notation by writing for any such function ( , ) ( ) i tF t F e − ω=r r . [1]

1 The mathematics of periodic functions can be greatly simplified by the use of the

exponential notation [Lorr70], [Chen83a]. The instantaneous time–dependent expression of a sinusoidal scalar quantity, such as the current flowing in a wire can be written as either a cosine or sine function. Given a function of the form ( ) ( )( )0 cosf x f x t= ω + θ where x0

is the amplitude of x, ω is the angular frequency and t is time. The quantity ( )tω + θ is

the phase, or phase angle with θ being the phase at time t = 0 . The first and second derivatives of the function ( )f x are found in the equations of electrodynamics. The

evaluation of these derivatives are simplified by using the exponential notation. By noting

that e t i ti tω ω ω= +cos sin , then 0 0cos R e : i tx x t x e ω= ω = . This notation can be simplified

by ignoring the Re: operator such that x x ei t= 0ω and the derivatives are then given as,

dx dt i x e i xi t= =ω ωω0 and ( ) ( )2 2 22

0

i td x dt i x e i xω= ω = ω , In this convention the operator

d dt is replaced by the factor iω .

The coefficient before the exponential function can also be complex and be represented

as 0 0Re: sini ti x e x tω = − ω if x0 is real. Since the imaginary number can be defined as

i ei t= ω , ( ) ( )2

0 0 0 0Re: Re: cos 2 sini t i ti x e i x e x t x tω ω + π= = ω + π = − ω .

The most common use of this notation is in the description of plane wave propagation. If a quantity α is propagating with a velocity u is defined at z = 0 by α α ω= 0 cos t then

for any position z in the direction of propagation of the plane wave ( )0 cos t z uα = α ω − .

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§5.1. PLANE WAVES IN FREE SPACE

The wave equations Eq. (4.30) and Eq. (4.31) describe vector waves and can be combined into a simplified scalar wave equation if they are represented in rectangular coordinates, such that.

2

22 0t

∂ ψ∇ ψ − =

∂, (5.1)

for the wave components, , , , , ,andx y z x y zE E E B B Bψ = .

For any two vector field components that lie in a plane, that is any two components of he vector triple x, y, z that are used to form a plane, can be ignored. The remaining component can be used to describe a one – dimensional wave equation,

2 2

2 2 0x t

∂ ψ ∂ ψ− =

∂ ∂, (5.2)

since y and z form the plane and x is considered the propagation vector normal to the plane (formed by y and z). The general solution to Eq. (5.2) can be found using a combination of arbitrary functions,

( ) ( ) ( ),x t f x ct g x ctψ = + + − . (5.3)

The function ( )f x ct+ represents a wave traveling in the negative x

direction and the function ( )g x ct− represents a wave traveling in the positive x direction. By using only the positive propagation direction, the wave equation solution can be expanded as a Fourier series,

( ) ( ), n ni t k x

n

x t e ω −ψ = ∑ . (5.4)

This expression describes a wave with constant amplitude α0 . The wave fronts are surfaces

of constant phase and are perpendicular to the z–axis. The phase angle ( )t z u− is a

constant such that ( )t z u C− = for a point traveling with a phase velocity dz dt u= .

This velocity is the wave's phase velocity because it is the velocity with which the phase ( )[ ]t z uω − is propagating in space. using this notation the wave can be rewritten

as ( )0 cos t kzα = α ω − where k u= ω is the wave number. When this wave travels in a

particular direction specified by the unit vector nn , the wave fronts are then normal to nn

such that the wave is described by, ( )0

i t ke ω − ⋅α = α n r , where z=⋅ rrnn in the case that the unit

vector coincides with the unit vector in the z–direction.

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All the possible frequencies nω are related to the wave numbers (propagation constants) by,

n nk cω = . (5.5)

For a monochromatic wave, the subscript can be dropped, since a single frequency will be considered. This results in the Fourier expansion of the wave front traveling in the x direction of,

( ) ( ), i t kxx t e − ω −ψ ≈ . (5.6)

To generalize this notation, the propagation direction will be denoted by the vector kk , which allows the form of the wave equation to be maintained. The vector kk is a direction normal to the wave front plane, with a magnitude of k=k . For a wave propagating in an arbitrary

direction rr , the expanded Fourier expression is now,

( ) ( ), i tt e ⋅ −ωψ = k rr , (5.7)

where kk is the propagation vector and rr is the propagation direction, relative to the moving plane wave front.

Using this approach the solutions to Maxwell’s wave equations can be developed using the Fourier transform of the vector operators. This allows t he replacement of the vector operators for divergence, curl and gradient. Starting with the following definition in Cartesian coordinates of the expanded plane wave:

i i i ie e e ex y z

⋅ ⋅ ⋅ ⋅∂ ∂ ∂= + +

∂ ∂ ∂k r k r k r k r , (5.8)

the Gradient of the wave is given by,

,

.

i i i ix y z

i

e i k e ik e i k e

e

⋅ ⋅ ⋅ ⋅

⋅

∇ = + +

=

k r k r k r k r

k rk (5.9)

The Divergence of the wave is given by:

( ) ( ) ( ) ( )( )

,

.

i i i iz y y z x z z x y x x z

i

e Aik A ik e Aik A ik e A ik Aik e

e

⋅ ⋅ ⋅ ⋅

⋅

∇× = − + − + −

= ×

k r k r k r k r

k r

A

k A (5.10)

The Curl of the of the wave is given by:

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( ) ( ) ( ) ( ) ,

,

.

i i i ix y z

i i ix x y y z z

i

e A e A e A ex y z

ik A e ik A e ik A e

i e

⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅

⋅

∂ ∂ ∂∇ ⋅ = + +

∂ ∂ ∂

= + +

⋅

k r k r k r k r

k r k r k r

k r

A

k A

(5.11)

These transformation allow Maxwell’s equations to be rewritten in the Fourier space (or momentum space once the quantum mechanical description of the propagating electromagnetic wave has been developed).

The unit vector normal to the wave front is kk and ξ is the distance from the origin of the coordinate system to the plane of the propagation such that,

( )1k

⋅ = ξk r . (5.12)

Using this new notation Maxwell’s equations become,

( Ia ) ∂

∇ ⋅ = ρ → ⋅ = ρ∂ξEE k , (5.13)

( IIa ) 0 0∂∇⋅ = → ⋅ =

∂ξBB k , (5.14)

( IIIa ) 0 0kt t

∂ ∂ ∂∇× − − = → × − =

∂ ∂ξ ∂E B EB j k , (5.15)

( IVa ) 0 0kt

∂ ∂ ∂∇× + = → × + =

∂ ∂ξ ∂ξB E BE k . (5.16)

§5.1.1. LONGITUDINAL PROPAGATION COMPONENTS

The propagating electromagnetic wave can be further separated into longitudinal and transverse components.

If the scalar product of kk is made with Eq. (5.15) the results is,

0kt

∂ ∂ ⋅ × − ⋅ = ∂ξ ∂

B Ek k k . (5.17)

The kk vectors in the first term can be exchanged to allow ×k k to vanish so that Eq. (5.17) can be written as,

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0t

∂⋅ =

∂Ek , (5.18)

Multiplying Eq. (5.18) by dt and the resulting expression from Eq. (5.13) by dξ and then adding the expressions gives,

0d dtt

∂ ∂⋅ ξ + = ∂ξ ∂

E Ek . (5.19)

The sum of the terms inside the parenthesis of Eq. (5.19) is the total differential of the electric field vector EE , which reduces Eq. (5.19) to,

0d⋅ =k E . (5.20)

The same operations can be performed on Eq. (5.16) to produce,

0d⋅ =k B . (5.21)

Eq. (5.20) and Eq. (5.21) require that the components of the EE and BB fields that are normal to the propagating wave front be constant in both time and space. This implies that the longitudinal components of the wave are static. Since such fields do not contribute to the propagation of the wave, they can be eliminated from the wave equations, such that,

0longitudinal ≡E and 0longitudinal ≡B .This leaves the transverse components of the

wave to contribute to the propagation, such that the wave equations are now,

2 2

2 20

t∂ ∂

− =∂ξ ∂E E

, (5.22)

and

2 2

2 20

t∂ ∂

− =∂ξ ∂B B

. (5.23)

The electric and magnetic field are perpendicular to the direction of propagation as well as being perpendicular to each other. The set of vectors EE , BB , and kk constitute an orthogonal set.

§5.2. ENERGY IN THE RADIATED FIELD

As the electromagnetic field propagates through free space is may encounter charged particles in its path. The motion of these particles will be influenced by the electromagnetic field — gaining or losing energy, depending on the direction of motion with respect to the EE field. The law of

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conservation of energy requires that if the charged particles gain energy then the energy must have been present in the electromagnetic field prior to is influence on the particles.

For a collection of particles, each with change q and velocity iv , the Lorentz force felt by the particles is,

( )i iq v= + ×F E B . (5.24)

The rate at which the electromagnetic field does work on a single charge is,

i i i i i iv q v v q v⋅ = ⋅ + ⋅ = ⋅F E B E , (5.25)

where 0i iv v× =B is assumed, where the BB field component does no work since it is perpendicular to the velocity of the charged particles.

The total rate at which the field does work per unit volume is now the sum of all the Lorentz forces felt by all the particles in the volume,

( ) ( ) ( )i x i y i z ix y zi i

dW q v q E v E v E v = ⋅ = + + ∑ ∑E , (5.26)

where ( ) ( )1 i xxi

n v v=∑ is the component of the drift velocity of the charges

and similarly for the y and z components. The rate at which work is done is now,

dW nq v= ⋅ = ⋅E E j , (5.27)

where nqv=j is the current density resulting form the motion of n charged particles with average velocity v. The expression ⋅E j can now be rewritten as,

( )2ct

∂⋅ = ∇× ⋅ − ⋅

∂EE j B E E . (5.28)

Using the familiar vector identity ( ) ( ) ( )∇ ⋅ × = ⋅ ∇× − ⋅ ∇×E B B E E B , gives,

( ) ( ) ( )2 2c ct

∂ ⋅⋅ = ⋅ ∇× − ∇⋅ × −

∂E E

E j B E E B . (5.29)

Using Faraday's Law, Eq. (IV) gives,

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( ) ( )t

∂⋅ = − ∇ ⋅ × − ⋅ + ⋅

∂E j E B E E B B . (5.30)

If there are not charges present in the volume, 0⋅ =E j , there will be no conversion of the field energy into motion and Eq. (5.30) has the same form as Eq. (VII), which expresses the conservation of charge.

§5.3. POYNTING'S THEOREM

Before proceeding with the development of the radiated field potential, the form of the conservation of energy law is important to understand. This law is often called Poynting's theorem [Poyn20]. In a paper published in 1884, John Henry Poynting (1852–1914) examined what the flow of energy must be in the electromagnetic field when the localized energies are altered. [2] Poynting's work, along with Oliver Heavyside (1850–1925) lead directly to a mathematical representation of the conduction current in terms of decaying displacement without requiring a knowledge of the connection between the electromagnetic field and matter. Poynting wrote in [Poyn20]:

... we believe that when it (energy) disappears at one point and reappears at another it must have passed through the intervening space, we are forced to conclude that the surrounding medium contains at least a part of the energy and that it (the medium) is capable of transferring it (the energy) from point to point.

For a single charge q, the rate of doing work by the external electromagnetic fields EE and BB is qν ⋅B , where ν is the velocity of the charge. The magnetic field does no work, since the magnetic force is perpendicular to the charges velocity vector. For a continuous distribution of charge and current, the total rate of doing work by the fields in a finite volume V is

Vdv⋅∫ j E . This power represents a conversion of

electromagnetic energy into mechanical and/or thermal energy.

Using Eq. (5.30) the equivalent equations of Eq. (4.4) and Eq. (4.1), rewritten in terms of the scalar and vector potential, become equations for the field in terms of A A only,

2 John Henry Poynting made several important contributions to electromagnetic field

theory besides his famous Poynting Theorem. His model of the electromagnetic field employed the distribution of lines of force in space and the flux of energy between these lines represented the electromagnetic field [Harm82].

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( )i ikk

= ∇ ∇ ⋅ +E A A , (5.31)

= ∇ ×B A . (5.32)

The Poynting vector representing energy flow for harmonic fields is, [3]

( ) ( ),

4( cos sin ) ( cos sin ).

t tcS e e

t i t t i t

−ω −ω= ×π

= ω + ω × ω + ω

E B

E E B B (5.33)

The physical meaning of the Poynting vector is that the time rate of change of electromagnetic energy within a certain volume, plus the energy flowing out through the boundary surfaces surround the volume per unit time, is equal to the negative of the total work done by the fields on the sources within the volume. This is the statement of the conservation of energy.

Multiplying this express out gives four terms: one each involving 2sin tω and 2cos tω and two involving sin tω and cos tω . These last two

terms oscillate in time and their average value over time is zero, where the first two terms average to ½.

The time average Poynting vector is given by,

( ) ( )8cS i= × + ×π

E B E B , (5.34)

which is equivalent to,

( )8cS ∗= ×π

E B , (5.35)

3 From the system of Maxwell's equations (I) to (IV) it is possible to derive a very

important expression recognized as the energy principle in the electromagnetic field. This principle states that the energy liberated per second in a certain volume shall pass through a close surface surrounding the volume according to the following,

el m n

du u dv dv S dA

dt− + = ψ +∫ ∫ ∫Ñ , where,

1

4elu d= ⋅

π ∫ E D is the electric field density,

1

4mu d= ⋅

π ∫ H B is the magnetic field density and ψ = ⋅jj EE represents the work from the

field expended upon the electric current density. S dAn gives the energy passes per second through the surface element dA , in the direction of its normal nn .. The term S is called the Poynting vector.

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where ∗B is the complex conjugate of B .

§5.4. VECTOR POTENTIAL DESCRIPTION OF THE RADIATED FIELD

Potential problems are, in general, of two types – those in which the charge distribution is known and the goal is to determine the resulting potential. These problems find their solutions in Poisson's equation. The second type are those in which a potential must be determined which satisfies given boundary conditions. The general solution to the second type of problem depends on the geometry of the surfaces on which the boundary conditions are given. In such solutions, it is necessary to find a system of coordinates in which the potential equation separates, in which a solution may be found which is a product of functions of the separate variables. This separation of variables is appropriate for the problems in which the boundary surfaces are parametric surfaces, that is, surfaces in which one of the coordinates is constant. Spherical polar coordinate are suitable for problems involving radiation field potentials in free space.

The field vector potential,

( ) ( ),,

t ct dv

′ ′− −=

′−∫j r r r

A rr r

(5.36)

may be evaluated for sinusoidal waves as,

( ) ( )( )1 ike dv

c

′−

′ ′=′−∫

r r

A r j rr r

(5.37)

where k c= ω is the wave number and the sinusoidal time dependence is

assumed. . The term ( )ike ′− ′−r r r r can be expanded in terms of rr in

spherical coordinates. [4] There are three lengths in the integral and

4 In all potential problems in which there is an axial symmetry, associated Legendre

polynomials and spherical harmonics are used to solve the general purpose boundary value problem. The expansion of the potential rr due to a unit charge at ′r is the most general

form Laplace's equation, where 1

41

2 1 110rr rr

YY YY− ′

=+

′ ′+

∗

=−=

∞

∑∑π θ φ θ φl

r

r

l

l lm lmm

l

l

( , ) ( , ) . The

spherical harmonics YYlm are explicitly defined by YYlm lm

lm imN P e( , ) ( cos )θ φ θ φ= , where the

associated Legendre polynomial is given by

( )( ) ( )( )2 2 2( ) 1 2 ! (1 ) ( 1)m l m l m l m l

lP l d d+ +µ = −µ µ µ − and the normalization factor is given

by ( ) ( )[ ]1 / 2( ) 2( 1) 2 1 4 ( ) ! ( )!m m m

l l l m l m+= − + π − +N . The peculiar factor ( )( )/− +

12m m

in the

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different expansions can be used depending on the relative values of the lengths. These lengths are: the wavelength 2 /cλ = π ω ; the distance r to the field point; and the size a of the source. It is always assumed that

′>r r . The relationships between the three lengths result in three unique time varying potential equations.

There are also three regions in the radiation field that produce different expressions for the time varying potential:

n Near (quasi stationary) r aλÓ Ó ,

n Intermediate (multipole) r aλ ≈ Ó ,

n Far (radiation) r aλÓ Ó .

The fields have very different properties in the different zones. The near zone the fields have the character of static fields with radial components and the variation with distance depends on the properties of the source. In the far zone the fields are transverse to the radius vector and fall off as 1 r . In the intermediate zone, both dipole and quadrupole approximations are in effect.

§5.4.1. QUASI–STATIONARY EXPANSION

In the near zone, which is call the Quasi-Stationary expansion, the relationships between the wavelength, distance from the source and source size are rλÓ and aλÓ , resulting in a potential equation as a function of the distance, rr , of:

10

( )1 ( ) 1 4( ) lim ( ) ( ) ( , )

2 1llm

lmlkr

dvr dv

c c l r∗

+→

′ ′ θπ′ ′ ′ ′= ≡ θ φ

′− +∑∫ ∫Yj r

A r A r j r Yr r

. (5.38)

The Quasi–Stationary expansion is like the expression for AA in magnetostatics where the current at a point is stationary, i.e. independent of time. The exponential in Eq. (5.38) can be replaced by unity. Then the vector potential is of the form where the near field oscillates harmonically as i te − ω , but is otherwise static. [5]

normalization constant NNlm is the conventional phase factor of the spherical harmonic.

These phase factors become important when adding linear combinations of spherical harmonics in determining interference effects. [Hobs31], [Arfk85].

5 Although the Quasi–Stationary approximation derives from assuming that ik cEE jjω ωπ<< 4 and ikAA ω ω<< ∇φ , i.e. assuming k is small, the approximation cannot be

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Copyright 2000, 2001 5–11

§5.4.2. MULTIPOLE EXPANSION

In the Multipole expansion, the relationships between the wavelength, distance from the source and source size are aλÓ and r a> , resulting in a potential equation as a function of the distance, r, of:

ed md eq= + +A A A A , (5.39)

in which the electric dipole potential, edA is given by,

ed ikne dv′= ∫rA j , (5.40)

the magnetic dipole potential, mdA is given by,

1 1

2

ikmd e ik n dv

r r c ′ ′ ′= − × × ∫

r

A j r , (5.41)

and the electric quadrupole potential eqA is given by,

( ) ( )1

2

ikeq e

ikcr r

′ ⋅ ⋅ ′= − + ∫

r r jr rA j r

r r. (5.42)

The multipole approximation is used in atomic and nuclear distances, where the conditions for its validity are often satisfied. In the Quasi-Stationary expansion the parameter a r , the ratio of the source size to the distance to the field point, and the fields of higher multipoles fall off with successfully higher powers of a r . In the multipole expansion, all multipole fields drop off as 1 r in the radiation zone and the parameter that defines the relative importance of successive multipoles is ka not a r .

§5.4.3. RADIATION EXPANSION

In the far zone or Radiation expansion in which r aÓ and r >> λ , the exponential in Eq. (5.38) oscillates rapidly and determines the behavior of the vector potential. In this region r′ ′− ≅ − ⋅r r n r where nn is a unit vector in the direction of rr . This results in a potential equation of,

completely characterized for k ≈ 0 . In the equation ∇ × = −EE BBω ωik , the right hand side

cannot be set to zero without reverting to the completely time stationary case for the potential fields.

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( )( ) ( ) ( )( )

!

ik ikr nik n

n

e e ikr e dv dvcr cr n

− ⋅ −′ ′ ′ ′= ≡ ⋅∑∫ ∫r

n rA r j j r n r . (5.43)

If only the first expansion terms in kr are considered, than the inverse distance 1 ′−r r can be replaced by 1 r to give,

( ) ( )limikr

ik

kr

e e dvc

′− ⋅

→∞′= ∫ n rA r j r

r. (5.44)

The radiation zone expansion is useful in calculating the fields of an arbitrary current distribution at large distances. These far fields fall off as

1 r , since the original expansion 3

1 1 1r rr r r

′⋅= + + ≈

′−r rL , [6] is all that is

needed for calculating the rate of radiation energy by means of the Poynting vector. The far zone potential describes an outgoing spherical wave with an angular dependent coefficient. The calculated values from

= ∇ ×B A and = ∇ ×E B are transverse to the radius vector and fall off as 1 r . If the source of the radiation has dimensions small compared to the wavelength, then Eq. (5.44) can be expanded in powers of k to give,

( ) ( ) ( )( )lim!

nikrn

krn

ike dvc n→∞

−′= ⋅∑ ∫A r j r n r

r. (5.45)

In the radiation expansion, Eq. (5.40) and Eq. (5.41) and Eq. (5.42) are used to evaluate the time averaged Poynting Vector, such that,

( ) ( ) ( ) ( )4 2

3

,

2 .3

i iS ik i ikk kp

c

= ∇ ∇ ⋅ + × ∇× + ∇ ∇⋅ + × ∇ × ω

=

A A A A A A (5.46)

6 This expression is derived through a Taylor series expansion of 1

r r− ′, in increasing

powers of the components of ′r , such that 0 0

1 1 1

r rx x x′ ′= =

∂ ∂ ∂= − = −

′ ′ ′ ′∂ − ∂ − ∂

r r r r r

,

giving, the expansion as,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2

2 2 2

2 2 2

1 1 1 1 1 1 1 1 11 1 1

2x y z x y z x y y z z y

x y z x y z x y y z z y

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′= + + + + + + + + +

′− ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

r r r r r r r r r

r r rL

The first expression in brackets decreases like 1 2r with increasing r, the next two terms like 1 3r while further terms decrease with higher powers of r.

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where p is the dipole moment of the radiation field. [7]

§5.5. POLARIZATION OF THE RADIATED FIELD

The wave equation developed in the previous section describes the propagation plane waves of electromagnetic energy through free space or a conductive media. The concept of polarization may be familiar to anyone has worn glasses coated with a Polaroid film. This film consists of an array of polymer molecules which has a preference for absorption of light along one specific axis. If two such Polaroid filters are place, one on top of the other, and illuminated from behind, the amount of light passing through the two filters will be a function of how the filters are oriented relative to each other. If the filters are arranged so that the maximum amount of light is transmitted — then one filter of rotated so that the minimum amount of light is transmitted, the intensity of the light passing through the two filters is given by 2

0 cosI I= θ , where I is the transmitted intensity,

0I is the intensity of the light transmitted through the first filter and θ is the angle between the two polarizes. This behavior is Malus’ Law, when it was discovered in 1809 by Ètienne Louis Malus (1775–1812) that light becomes partially or completely polarized by reflection [Hall60].

Malus wrote in his memoir in 1809 that...

...light reflected by a surface of water at an angle of 5245′o with the vertical, has all the characteristics of one of the beams produced by double refraction ... above and below this angle a part of the ray is more or less modified in a way analogous to that which occurs when light passes through two crystals whose principal sections are neither parallel nor perpendicular [Harm82].

Until the discovery by Faraday in 1846 that a magnetic field can alter the polarization of light, there was little evidence that light and electromagnetism were connected. In 1865 Maxwell published an important paper describing the connection between electromagnetism and light, laying the theoretical groundwork for his prediction of electromagnetic radiation [Maxw65].

Since light is an electromagnetic wave the radiation of electromagnetic

7 The dipole moment of the radiation field represents the simplest charge distribution,

which at large distances r a>> leads to a field consisting of two charges (poles) of equal

strength but of opposite signs. The dipole moment is defined as ( )p p dv′ ′ ′= ∫ r r .

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waves resulting from Maxwell's equations satisfies the general wave equation,

2

22

1 0,uuv t

∂∇ − =

∂ (5.47)

where,

c

v =µε

. (5.48)

represents the constant velocity of the wave determined by the characteristics of the medium. The Wave Equation in Eq. (5.47) has the general solution,

ik i tu e ⋅ − ω= r , (5.49)

where the frequency ω and the magnitude of the propagation constant k are related by,

kv cω ω

= = µε . (5.50)

If waves propagating only in the x–direction are considered, the solution to Eq. (5.50) is,

( ), i k x i t i k x i tu x t Ae Be⋅ − ω − ⋅ − ω= + , (5.51)

using Eq. (5.50) gives,

( ) ( ) ( ), ik x vt ik x vtu x t Ae Be⋅ − − ⋅ += +k . (5.52)

If the velocity v is not a function of the propagation vector k, that is the propagation media is nondispersive, with the permittivity and permeability, µε independent of frequency the linear superposition of the wave equation results in a general solution of the form,

( , ) ( ) ( )u t f vt g vt= − + +r r r , (5.53)

where ( )f z and ( )g z are arbitrary functions. Eq. (5.53) represents waves traveling in the rr direction with phase velocity v.

With the convention that the physical electric and magnetic fields are obtained by taking the real parts of the complex quantities, the plane waves take the form of,

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( )( ),,

ik i t

ik i t

t et e

⋅ − ω

⋅ − ω

=

=

n r

n r

E rB r

EB

, (5.54)

where E, B and nn are vectors that are constant in time and space. Each component of the EE and BB field satisfies the wave equation,

2

22 0,u u

cω

∇ + = (5.55)

provided,

2

22kcω

⋅ =n n . (5.56)

It is useful at this point to introduce a set of mutually orthogonal unit vectors 1 2( , , )ε ε n where 1 0E= εE and 2 0E= ε µεB . The plane waves in Eq. (5.55) is a wave with its electric field vector in the direction 1ε . Such a wave is linearly polarized with polarization vector 1ε . A second wave which linearly independent of the first can be linearly polarized with polarization vector 2ε . The plane waves are now given as,

1 1

2 2

ik i t

ik i t

ee

⋅ − ω

⋅ − ω

= ε

= ε

n r

n r

EB

EB

, (5.57)

with,

, 1, 2jj j

×= =k E

Bk

. (5.58)

These equations can be combined to describe a homogeneous plane wave propagating in the direction k=k n , such that,

( ) ( )1 1 2 2, i i tt E E e ⋅ − ω= ε + ε k rE r . (5.59)

The amplitudes 1E and 2E are complex numbers which allow the description of a phase difference between waves of different polarization. If

1E and 2E have the same amplitude Eq. (5.59) represents a circularly

polarized wave with its polarization vector having an angle 12 1tan ( )E E−θ =

with 1ε and a magnitude 2 21 2E E E= + , as shown Figure 1.0,

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E2

E1

EE

ε2

ε1

θ

Figure 1.0 — Circular Polarization of the Electric Field occurs when a plane

wave composed of the components 1E and 2E has a constant phase θ and equal amplitudes.

If 1E and 2E have different amplitude Eq. (5.59) represents a elliptically polarized wave as shown in Figure 2.0,

E2

E1

EE

ε2

ε1

θ

Figure 2.0 — Elliptical polarization of the Electric Field occurs when a plane

wave composed of the components 1E and 2E has a constant phase θ but different amplitudes.

It is the circularly polarized wave equation that is of interest in this

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Copyright 2000, 2001 5–17

monograph. To illustrate this, if 1E and 2E have the same magnitude, but differ in phase by 90°, the electric field equation is given by,

( ) ( )0 1 2, i i tt E e ⋅ − ω= ε ± ε k rE r , (5.60)

with 0E the common real amplitude of the propagating waves.

The two circularly polarized waves given in Eq. (5.60) form a set of basic fields for the description of the general state of polarization. Further on the this monograph an expression for the spin of photons carrying the electromagnetic force will be developed. This concept is directly related to the polarization of propagating electromagnetic waves [Bagg92].

A notation will now be introduced which will later be used in the quantum mechanical description of the photon force particle. By convention, if the electric vector, or the polarization vector rotates clockwise when viewed in the direction of propagation, the wave is said to be right circularly polarized. The polarization vector can be given as,

( )1 2

12

iε = − ε + εR , (5.61)

and the left circular polarization vector can be given as,

( )1 2

12

iε = − ε − εL . (5.62)

The two complex orthogonal unit vectors in Eq. (5.61) and Eq. (5.62) can be written as,

( )( ) (1) ( 2 )12

i±ε = ε ± ε , (5.63)

with,

( ) ( )

( ) ( )

0

1

± ∗

± ∗ ±

ε ⋅ε =

ε ⋅ε =

∓

. (5.64)

The general representation equivalent to Eq. (139) is now given by,

( ) ( ) ( ) ( )( , ) i i tt E E e+ + − − ⋅ − ω = ε + ε k rE r , (5.65)

where ( )E + and ( )E − are complex amplitudes.

The direction of rotation is defined as right–handed when viewing the wave along its direction of propagation, the electric vector is rotating

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counter–clockwise. The waves shown in Figure 1.0 and Figure 2.0 are right–handed polarizations. These definitions are used in modern optics and particle physics, the right–handed photon having a positive helicity and spin vector in the direction of motion [Dobb85].

It has been seen that, on the electromagnetic theory of light, the propagation of waves of light “in vacuo” ought to take place with a velocity equal, within limits of experimental error, to the actual observed velocity of light.

— Sir James Hopwood Jeans (1877–1946) [Jean25] §589, pp. 532

Copyright 2000, 2001 6–1

§6. ANTENNAS AND RADIATED FIELDS

Now that Maxwell's equations for the radiated field have been developed a practical result is appropriate. When Maxwell's equations are written in terms of EE and BB , they result in eight simultaneous scalar first–order partial differential equations. In order to solve these equations in practice, the vector potential A is used.

Given a simple dipole antenna element, which is far removed from the ground and other objects consists of two straight wires with a generator connected between them.

I e i t0

ω

λ/2

Figure 3.0 — Simple Dipole Antenna driven by the current source 0i tI e ω . The

current distribution along the antenna’s wires is known from the current sources time behavior. The radiated field produced by the flowing current follows the time behavior of the driving source.

If the current distribution along the antenna element is known, the far field radiation pattern can be found by integrating over the length of the antenna. Although the evaluation of the simple dipole may appear to be a straight forward problem, it is a very difficult boundary value problem even of the antenna wire is assumed to be a perfect conductor. For the antenna to radiate properly, the current must be zero at the ends of the wire where charges are deposited and the tangential electric field due to all currents and charges must vanish. In order to simplify further the evaluation of the dipole antenna it is assumed the current flowing in the antenna element varies sinusoidally with time as expressed by 0

i tI e ω ,

where is the phasor current value. [1]

1 The Phasor representation is derived from the representation of time-varying fields

through a Fourier series. Rather than use sinusoidal functions directly, it is convenient to introduce the complex exponential ei tω . The advantage of this representation is that derivatives and integrals of ei tω are proportional to ei tω , so that the function can be eliminated from all equations. Given a complex expression, ρ ρ ρ= +real imagi , the

instantaneous value, as a function of time, of ρ is then given by the expression ρ ρ ρ ρ ω ρ ω( ) Re[( ) ] cos sint i e t treal imag

iwtreal imag= + = − .

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A simplified model of a dipole antenna will first be used. In this model the antenna of length l, will lie along the z–axis with the x–axis projected horizontally into free space. To further simplify the model the charge produced by the driving current is zero everywhere except at the ends of the antenna 2z l= ± where there is a time–dependent charge, ( )Q t , produced by the changing antenna current, ( )I t . This charge can be derived from the current as,

( ) ( ) 0 0cosd Q t I t I t

dt= = ω

, (6.1)

which can be integrated to give,

( ) 0 00

1sinQ t I t= ω

ω. (6.2)

The dipole moment pp is this charge expression times the physical separation of the charges. [2] The dipole moment is now given as,

( ) 00

0

2 sin2

I llQ t t= = ω

ωp z zp z z . (6.3)

In order to further simplify the expression for the radiated field, the Lorentz gauge will be used in which the radiated field potentials satisfy,

0t

∂φ∇ ⋅ + =

∂AA . (6.4)

The current density jj in the antenna is given by,

( ) ( ) ( )0 0cos , 2 20, otherwise

I t x y l lx

ω δ δ − < <=

zzjj , (6.5)

and the vector potential is,

( ) ( ) 31 i k x xex x d xc x x

′−

′ ′=′−∫A j . (6.6)

If x l@ — far field radiation zone — then x x′− can be approximated

2 In a more realistic description of the antenna, the current distribution along the

antenna wire would not be constant and the charge density would be nonzero along the wire as well as the ends.

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by x′= ⋅r n where nn is a unit vector along the x–axis and x = r . Substituting this approximation into the field potential and rearranging the integral terms gives,

( ) ( ) 3ik

ik xex x e d x

c′⋅′ ′≈ ∫

rrnnA jA j . (6.7)

Integrating over both x′ and y′ and letting cos⋅ = θn z where θ is the angle between nn and the z–axis gives the vector potential as,

( ) 00

0

2sin cos cos

cos 2

ikI e klx t ≈ θ ω ω θ

rr

A zA zrr

. (6.8)

For small values of kl this expression reduces to the far field approximation given in Eq. (6.8) as,

( ) 0 0cosikI le tx

cω

=rr

AArr

. (6.9)

By using spherical coordinates, this simplified example can be expanded further to illustrate some of the complexities of the simple dipole. The current element is in the z–direction with its location at the origin of a set of spherical coordinates. By evaluating the electric field potential given by Eq. (6.9) for any point Q at radius rr , in the z–direction,

( / )0

4i r

zaI er

− ω ν=π

A , (6.10)

given the potential in spherical coordinates,

0cos cos4

ikrr z

aI er

−= θ = θπ

A A , (6.11)

0sin sin4

ikrz

aI er

−θ = − θ = − θ

πA A , (6.12)

where 2k = π λ .

There is no φ component of AA and there are no variations with φ in any expressions because of the symmetry about the axis. The electric and magnetic field components may be found from Eq. (6.11) and Eq. (6.12) resulting in,

02

1 sin4

ikraI iker r r

−φ

= + θ π B , (6.13)

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02 3

2 2cos

4ikr

r

aIe

r r i r− η

= + θ π ω E , (6.14)

03 2

1sin

4ikraI i

er r i r r

−θ

ω η= + + θ π ω

E , (6.15)

where 20η = µ ε = π Ω is the impedance of free space. [Sche52]

Evaluating the electric and magnetic field equations in the radiation zone, results in terms which vary as 1 r .

0 sin4

ikrikaI er

−φ = θ

πB , (6.16)

0 sin4

ikri aI er

−θ φ

ω= θ = η

πE B . (6.17)

Using the developments of §5, Eq. (6.17) exhibits the characteristics typical of uniform plane waves, with the Poynting vector in the radial direction,

2 2 2

2 202 2

1( ) sin , watts m2 32r

k I aSr

−θ φ

η= × = θ

πE B . (6.18)

The total power is given as the integral is the Poynting vector over any surrounding surface,

2

0

2 2 230

0

2 222 20

0

2 sin ,

sin ,16

40 watts.3

rSW S dS S r d

k I ad

I a aI

π

π

= ⋅ = π θ θ

η= θ θ

πηπ = ≅ π λ λ

∫ ∫

∫

Ñ (6.19)

§6.1. TIME-DEPENDENT FIELDS IN CONDUCTORS

The distinction between a conductor and a nonconductor of electricity was first made through experiments in the early eighteenth century. When an electric charge was applied to a nonconducting material the charge remained for some time. However when a charge was applied to a conducting material it rapidly spread over the body of the material. This spreading action is due to the mobility of charge–carrying particles. In conductors the charge–carrying are electrons or ions. In nonconducting

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materials or insulators the electrons are bound to atoms and only a very strong force can pull them away. [3]

Using this simple understanding of conduction an expression for the current in a conducting wire induced from the external electromagnetic field will be developed. The receipt of the electromagnetic wave by the conducting wire antenna will also be developed, to complete the communication path. This formulation will take a simplified approach to the description of the motion of the electrons in the conductor. A full description of the interaction of electromagnetic waves with a conductor is beyond the scope of this monograph, but can be found in [Pari69], [Jack62], [Eyge72].

The basic concept is that all matter contains charged particles — free electrons, bound electrons, ions, etc. When an electromagnetic wave impinges on a material the charged particles at set in motion, resulting in a spatial distribution of charge. These distributions of current will vary depending on the different macroscopic properties of the material. These behaviors can be summarized as:

given certain approximations, a time harmonic wave of frequency ω will propagate in the material with an complex propagation constant k′ that differs from the free space propagation constant k c= ω [Eyge72], [Ramo84].

Materials react to applied electromagnetic fields in a variety of ways. For a metal or a semiconductor, where there are mobile electrons present, the electric polarization or displacement current given in Maxwell's Eq. ( III ) as well as the Lorentz force results in the motion of these electrons in the presence of a time varying electromagnetic field. [4]

3 It was Stephen Gray who first observed that there were two kinds of matter, one that

could be given an electrical charge and one that could not. He also observed that some electrified bodies would repel each other while others attracted each other. The attraction was in direct conflict with the effluvial theories popular in Newton's time [Will66].

4 In a dielectric the electrons are bound to the material's atoms. The a conductor the electrons are free to move in response to electromagnetic force. The number of conduction electrons varies with the conductor. In noble metals such as copper, silver and gold, the number of electrons is ≈ −1028 3m . In semiconductors such as germanium the electron density is in the range of 1020 3m − to 1024 3m− . In a weak plasma such as the ionosphere and outer space the electron density is 106 3m − to 1011 3m− .

A parameter in the propagation of electromagnetic waves is the relaxation time τ between collisions of the charge carriers (electrons) which determines the conductivity and the frequency of the propagating waves in the conductor.

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In most cases this motion can be treated as a linear response — proportional to the applied field. Often the response is also independent of the direction of the applied field, which allows the material to be considered isotropic. The response of these linear, isotropic materials to time–varying fields may also depend on the frequency of the field, where the permittivity ( )ε ω and permeability ( )µ ω are functions of frequency ω . Common dielectrics and conductors behave in this manner in normal applications.

In solids and liquids the permittivity differs significantly from the free space permittivity due to the behavior of the bound electrons in the materials atoms. In a conductor the movement of the free electrons in response to the applied electric field produces a current that overwhelms the electromagnetic field of the bound electrons [Ramo84].

§6.1.1. WAVE PROPAGATION IN A CONDUCTION MEDIA

A conductor placed in an oscillating electric field, ( )0( , ) i tt e ⋅ −ω= k rE r E , with

a sufficiently low frequency of oscillation will produce an oscillating current density given by,

( , ) ( ) ( , )t t= σ ωj r E r , (6.20)

where the frequency dependent conductivity is given by,

0( )1 i

σσ ω =

− ωτ, (6.21)

The expression is Eq. (6.20) is Ohm's Law. As a result a wave propagating in the conductor, in the z–direction, resulting from the external radiation is given by,

/ ( / )0

z i z te e− δ δ−ω=E E , (6.22)

and,

(1 )i z−

= ×ωδ

B E , (6.23)

where,

2

0

0 0 0

2 2ccn

εδ = = = ω σ ω µ σ ω

, (6.24)

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which is the skin depth. [5]

For classical materials (non–solid state devices) the motion of the charged particles in the material is determined by classical forces. Four different material states will be considered: a plasma, a conductor, a dielectric and a lossy dielectric.

In each material an electron of mass m and charge e is driven by an electric field, 0

i tE e− ω=E . It will be assumed that the charge density is dilute enough so that no other forces are acting on the electrons. The electrons and surrounding positive charge form a plasma with the electron's equation of motion given as,

5 In most physics and engineering texts the skin depth term is usually given as a

tautology with no further background. It can however be derived directly from Maxwell’s equations. Starting with the homogeneous equations 1 µ∇ ε ∂ ∂× = +BB jj EE t and ∇ × = −EE BB∂ ∂t and Ohm’s Law j E= σ a one dimensional example can be constructed. By using only one dimension Maxwell’s equations for a particle moving in the z–direction can be written starting with ∂ ∂ ∂ ∂x y= = 0 so that the current is moving only in the z–

direction with ∇ × =EE ∂ ∂E zx which gives ( ) ( )1 2 21

21

2= → =ωµσ δ δ ωµσ and

∇ × = −BB B zy ∂ which gives −∇ = +B z E E ty x x∂ µσ µε ∂ ∂ . These two Maxwell equations

have a solution of the form ( )E E exi t z= − −

0ω kk and ( )B B ex

i t z= − −0

ω kk . By defining the terms ∂ ∂z i= − kk and ∂ ∂ ωt i= − the differential equations can be changed to the algebraic equations i E i Bx ykk = ω and ( )− = −i B i Ey xkk µσ ωµε . This homogeneous system of equations

can be solved if the determinant of the coefficients vanishes such that −−

=i i

i i

kkkkω

µσ ωµε0.

Expanding the determinant gives ( )k i i2 0− − =ω µσ ωµε . This expression is usually called the dispersion equation because it described the relationship between the frequency ω and the wave number kk , which is related to the phase velocity of the wave vphase = ω kk . Unless ω and kk are linearly related different frequencies will propagate at different velocities. A medium in which the conductivity is zero σ = 0 and µ and ε are independent of frequency

is nondispersive. The relationships between kk and ω is given by k = ω µε . Solving the

dispersion equation gives ( )kk = +ω µε ωµσ21

2i . Whenever the conductivity is nonzero, the

wavenumber is complex. The spatially varying part of the wave equation is then ( )e e e ei z i k ik z ik z ik zreal imag real imagkk = =

+ −. If the imaginary part of kk is positive the amplitude of

the wave declines exponentially. If the conductivity of the medium is large, the second term

of the dispersion equation dominates giving ( ) ( ) ( ) k i i≈ = ± +ωµσ ωµσ1

21

21 2 . The rate at

which the electromagnetic field decays in a good conductor is given by this expression.

Since ( )kimag = ωµσ 21

2 , the electric field amplitude varies as ( )E E exz= −

02

12ωµσ . The

distance at which the amplitude decays to 1 e is defined to be the skin depth and is given

by ( ) ( )1 2 21

21

2= → =ωµσ δ δ ωµσ . [Soly93], [Zima72], [Jack75], [Pano55].

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0i k i tm eE e ⋅ − ω= rr&& . (6.25)

The ionosphere is an example of such a free electron material.

In a conductor there is a force applied to the free electron from the collisions it makes with the impurities or with the vibrating lattice of ions in the material. These collision forces can be described as a damping force, γ , proportional to the velocity of the electron ν and is written as mγν . The equation of motion of an electron in a conductor becomes,

0i k i tm m eE e ⋅ − ω+ γ = rr r&& & , (6.26)

which is the equation of motion for a damped oscillator.

For a pure dielectric, there is no damping but rather the free electrons are now bound to an origin with a natural frequency. This frequency is a crude counterpart of the natural frequencies of electrons bound in atoms. In this model the electrons experience a restoring force which involves this frequency given by, 2

0m− ω r , which gives the electrons equations of motion as,

20 0

i k i tm m eE e ⋅ − ω+ ω = rr r&& . (6.27)

The final model of a material is a lossy dielectric in which the equations of motion involve both a damping term and a harmonic restoring force, which gives the equations of motion as,

20 0

i k i tm m m eE e ⋅ − ω+ γ + ω = rr r r&& & . (6.28)

In the lossy dielectric and dielectric model any displacement of the dilute charge density from the central ion force produces a restoring force [Hipp54]. This restoring force interacts with the inertia of the moving charge cloud to produce a resonance similar to a mass–spring mechanical system. The displacement of one ion from another also produces a resonance in the ionic polarizability. This resonance however appears at lower frequencies than the purely electronic contribution because of the larger masses of the individual ions [Ramo84].

There are also losses or damping in each of the resonances rising from radiation from the free electrons. The motion of the electrons described in the last four equations produces a point current. In order to consider the effects of this current the steady–state solution to the equations of motion must be found.

If the equation of motion for a lossy dielectric is considered the most general of the four models, the steady–state solution can be found by

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solving for the position and velocity variables, using the complex notion of i t

real e− ω=r r , which gives,

( )0

2 20

realeE

m i=

ω − ω − ωγr . (6.29)

From the relation real realiν = − ωr , the velocity becomes,

( )0

2 20

reali eE

m i− ω

ν =ω − ω − ωγ

. (6.30)

The equations of motion for a plasma, conductor or dielectric can now be found by setting γ or 0ω or both to zero.

In order to properly explain the motion of charges in materials an extensive development of the underlying theory of solids is necessary. As an alternative a heuristic approach will be taken using the description of electrons and their motion which results in a current, that has been presented above. In this heuristic description many electrons are put in motion by an external electric field. The resulting motion produces a current jj based on the number electrons per unit volume passing through a surface N, such that the average current is given by,

Ne= νj . (6.31)

There are several assumptions made here that simplify the description, but would require more complex descriptions in order to properly explain actual phenomenon.

Using the expression for the velocity of an individual electron, the conductivity constant σ can be given as,

( )2

2 20

iN em i

ωσ = −

ω − ω − ωγ. (6.32)

In conductors the natural frequency 0ω is zero so that,

( )2

2

iN em i

ωσ = −

ω − ωγ. (6.33)

It has been experimentally that the damping constant γ is of the order 17 110 sec − , which has the dimensions of frequency. This means that for

electrons driven at frequencies less than this, 2ω in the denominator is negligible and the conductivity constant is given approximately as,

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2e N

mσ =

ωγ. (6.34)

Which allows a crude derivation of Ohm's Law to be restated as,

E= σj , (6.35)

which embodies Drude's model of conductivity [Egye72] [6]. The complex propagation constant k′ is now given as,

2

22

41k i

c ω πσ

′ =µε + ωε . (6.36)

The first term corresponds to the displacement current and the second term to the conduction current. [7] Assuming σ , µ and ε are real the propagation constant can be simplified to,

2

k i α= β + ,

where the real part is given by,

( )( )21 4 1

2c

+ πσ ωε +ωβ = µε , (6.37)

and the imaginary part is given by,

( )( )21 4 1

2 2c

+ πσ ωε −α ω= µε . (6.38)

6 This simple model of free electron conductivity was first constructed by Paul Karl

Ludwig Drude (1863–1906), in 1900.

7 One of the notational complexities of electromagnetic theory is the definitions of the various constants and propagation vectors. It seems that each author of the various bibliographical sources has chosen a different notation. In this monograph the following notation is used in the context of classical electrodynamics. The italic letter k represents the propagation constant and the bold faced letter rr represents the propagation vector. In the quantum mechanical description of electrodynamics the bold face letter kk represents the propagation vector and the propagation constant, which contains information regarding the medium's characteristics is absent from the expression.

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The exponential form of the propagation vector k now becomes,

( ) 2i k t t i te e e⋅ −ω −α −ω − β ⋅ − ω≡r r (6.39)

In metals σ is again 17 110 sec − which means that 24 cπσω is much

larger than 2 2cω which allows the skin depth to be restated as,

2c

δ =πµσω

, (6.40)

using the relation ( )1 2i i= + and ( )1k i′ = + δ .

The electric and magnetic fields propagating in the metallic conductor are now given as,

2 2

2 2

0,

0.

k

k

′ ∇ + =

′∇ + =

E E

B B (6.41)

§6.2. ELECTROMAGNETIC WAVES INCIDENT ON A CONDUCTOR

The study of uniform plane waves propagating in a region which has an abrupt change in conductivity, permittivity and permeability is referred to as the study of reflection and refraction. The analysis of the effects on the plane wave as it crosses various media boundaries is complex. Several generalizations will be used, as in the previous section, to illustrate the principles without undue detail. [8]

Suppose that an electromagnetic wave is propagating in one medium (1) and encounters a discontinuity of a second medium (2). If the dimensions of the second medium are large compared to the wavelength of the propagating wave a fraction of the incident energy will be reflected from the surface of medium (2) and the remaining energy will be transmitted into the second medium. The direction the electromagnetic energy takes in the second medium will be different from the original direction in medium (1). This change in direction is referred to as

8 The purpose of this portion of the monograph is to provide the reader with a

background in the issues of reflection and refraction of electromagnetic waves. One major simplification is done by ignoring the effects of the orientation of the electric vector on the reflection and refraction of the incident wave. Two orientations are possible, the EE vector normal to the plane of incidence and the EE vector in the plane of incidence. For a detailed discussion of the complex subject the reader is referred to [Jack75], [Lorr70] whose source material was taken from the classic text on the subject of electromagnetism [Stra41].

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refraction. The laws governing reflection and refraction of electromagnetic waves at the surfaces of an infinite dimension discontinuity are relatively simple [Eyge72], [Jack75], [Beck64]. If the dimensions of the discontinuity are on the order of the wave length of the radiation, the mathematical description of the wave's behavior becomes difficult [Jack75], [Stra41]. The disturbance to the propagating electromagnetic field in this case is called diffraction. In both cases — refraction and diffraction — the electromagnetic field in medium (2) induces the conducting charges into motion. Both situations are classified as inhomogeneous boundary–value problems. [9]

A situation is that of a conductor embedded in a dielectric medium. Charges in medium (2) are displaced from their equilibrium distribution of the surface of the conductor. The resulting oscillations produce oscillations in the surrounding field. This induced field can be represented as a superposition of the characteristic waves functions whose form is determined by the configuration of the conducting body and whose relative amplitudes fixed by the initial conditions [Stra41]. This is called a homogeneous boundary–value problem.

Associated with each characteristic of the conducting medium is a characteristic number that determines the frequency of that particular oscillation. The oscillation are damped, partly due to the finite conductivity of the material and partly because the energy dissipated in radiation. In this situation the positions of the conductor and the dielectric relative to the surface of separation can be interchanged. The electromagnetic oscillations then take place within a dielectric cavity formed by the boundaries of the conducting material [Stra41].

9 The origins of the theory of reflection and refraction can be traced to Newton's

Opticks [Cohe52]. In Opticks Newton attempted to describe light rays as very small Bodies emitted from shining Substances. Newton postulated that these Bodies…

… will pass through uniform Mediums in right Lines without bending into the Shadow, which is the Nature of the Rays of Light.

In Newton's theory light was a particulate of nature just as ordinary matter.

In order to explain reflection and refraction Newton theorized that the corpuscles of light were repelled by…

…some power of the Body, which is evenly diffused over the surface and by which it acts upon the Ray without immediate Contact. The power could repel the corpuscles of lights during the reflection and pull them through …

…the glass during refraction.

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The general law of reflection and refraction describing the behavior of plane waves between two linear, homogeneous, isotropic media is called Snell's Law [10] and is given as,

sinsin

i i

r r

vv

θ=

θ, (6.42)

where the angle if incidence of the incoming plane wave is iθ and the angle of refraction of the incoming plane wave is rθ and iv and rv is the phase velocities of the plane wave in the medium of incidence and refraction.

In geometric optics these laws were derived through intuitive observations, but they can also be derived by using the Maxwell equations to evaluate Transverse Electromagnetic Waves (TEM) [Pari69].

For an electromagnetic wave with a propagation vector 1r incident on a conductor with conductivity 2σ , Figure 4.0 describes the reflection and refraction effects.

kk 0

kk1

kk 2

θ0

θ θ1 0=

ε1 ε2

µ 2µ1

σ1 σ 2

z

y

θ2

10 This law was apparently discovered by Willebrord Snell around 1621. However Snell never published his result which lead Descartes to attempt to take credit for Snell’s work in his Dioptrique in 1637. Descartes derivation was deductive and incorrect. It was Fermat that first derived the correct formulation in 1661, using the Principal of Least Time. Since Fermat’s principal was based on metaphysical rather than physical reasoning it had no lasting influence. Hamilton provided the physical foundation for Snell’s Law using his variational principals in 1834. See §8 for the development of Hamilton’s approach to mechanics.

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Figure 4.0 — Spatial relationships for the description of reflection and refraction at a dielectric–conductor boundary. An incoming plane wave with a propagation vector 0k is incident on a boundary at 0z = . A reflected wave 1k and a transmitted

wave 2k result from the incoming wave.

The incident wave can be written using the exponential notation as,

( )0 00ˆ i t k

i iyE e ω − ⋅= rE , (6.43)

and,

( )0 00

ˆ i t ki izB e ω − ⋅= rB . (6.44)

where z and y are unit vectors in the z and y direction. The reflected wave cane be written in a similar manner as,

( )1 10ˆ i t k

r ryE e ω − ⋅= rE , (6.45)

and,

( )1 10

ˆ i t kr rzB e ω − ⋅= rB . (6.46)

and the transmitted wave as,

( )2 20ˆ i t k

t tyE e ω − ⋅= rE , (6.47)

and,

( )2 20

ˆ i t kt tzB e ω − ⋅= rB . (6.48)

The propagation constants for the wave traveling in the refracted medium is given by,

1 1 1 1k i i= β = ω µ ε , (6.49)

and wave traveling in the transmitted medium can now be given as,

2 2 1k i= α + β , (6.50)

where,

( )( )2

2 2

2 2 2

1 4 1

2c

+ πσ ωε +ω β = µ ε

, (6.51)

and,

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( )( )2

2 22

2 2

1 4 1

2 2c

+ πσ ωε −α ω = µ ε

. (6.52)

Within the conducting medium the transmitted wave is now given by,

( ) ( )2 2 2 2 2sin cos2 2

k i t k y z i tt e e− ⋅ + ω − θ − θ + ω= =k rE E E . (6.53)

The transmitted waves propagation constant is complex and requires further expansion. Using Snell's Law

1 12 0 1

2 2 2

sin sin sink ik i

βθ = θ = θ

α + β, (6.54)

resulting in,

2

2 212 2 1

2

cos 1 sin 1 sin2

ii

βθ = − θ = − θ α + β

. (6.55)

It is convenient to set,

2cos ie δθ = ⋅r (6.56)

in terms of the spatial component of Eq. (195) becomes,

( ) ( )

( ) ( )( )

( )

12 2 2 2 2 1

2 2

1 1 2 2

2 2

1 1

sin cos sin ,

sin cos sin

sin cos ,

sin .

iik y z i y z ei

i y z

i

pz i y qz

δ βθ − θ = α + β θ − α + β

= β θ − α δ − β δ −

− α δ − β δ

= − + β θ −

r

r

r (6.57)

where,

( )2 2cos sinp = α δ − β δr , (6.58)

and,

( )2 2sin cosq = α δ − β δr , (6.59)

are real quantities, dependent on the parameters of the medium and the angle of incidence of the incoming wave, 1θ .

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Substituting these expressions into Eq. (195) gives an expression for the transmitted wave as,

( )1 1sin2

pz i y qz tt e− + β θ − +ω =E E . (6.60)

This refracted wave has planes of constant amplitude parallel to the boundary of the plane, constant,z = and has planes of constant phase inclined to its normal kk , ( )1 1sin constant,y qzβ θ − = at an angle which is no longer equal to 2θ , as shown in Figure 5.0.

kk0

kk1

z

y

kkψ

kk2

θψ

θ0

θ θ1 0=

Figure 5.0 — Spatial relationships for the description of reflection and refraction at a dielectric–conductor boundary, with the refracted wave ψk which is different

from the transmitted wave 2k .

The true angle of refraction ψθ is given by the imaginary component of

Eq. (72) as,

( )

( )( )

( )( )

1 1

2 2 1 11 1 2 22 2

1 1 1 1

sin

sinsinsin sin

y qz t

qzq y tq q

β θ − + ω =

β θ − β θ + × − + + ω β θ + β θ +

(6.61)

By defining two terms,

( )

1 12 2

1 1

sinsinsin q

ψ

β θθ =

β θ +, (6.62)

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and,

( )2 2

1 1

cossin

q

qψθ =

β θ +. (6.63)

Eq. (203) can be rewritten as,

( ) ( ) ( )2 22 21 1 1 1sin sin cos sinq y z t q tψ ψ ψ− β θ + − θ + θ + ω = − β θ + ⋅ + ωk r ,

(6.64)

where,

sin cosψ ψ ψ= − θ + θk . (6.65)

Evaluating for the refraction angle ψθ gives,

1 1 1sintanq

−ψ

β θθ = . (6.66)

In a good conductor where 2 2 1σ ωε @ ,

2 22 2 2

ωµ σα ≈ β ≈ . (6.67)

With the condition that 2σ is large in a conductor and 1 2µ ≈ µ Snell's law results in,

( )

1 1 412

1 22 2

sin 0sin 2 1

iie

iπω µ ε ωεθ

= = →θ σωµ σ +

. (6.68)

With this result,

2 22 2

pωµ σ

≈ α ≈ , (6.69)

and,

2 22 2

qωµ σ

≈ β ≈ , (6.70)

and,

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1 1 11

2 2

sintan 0

2−

ψ

ω µ ε θθ = →

ωµ σ. (6.71)

In the incident medium, when the conductivity increases, the true angle of refraction tends to zero 0ψθ → , and the planes of constant phase

are oriented parallel to the reflecting plane and to the planes of constant amplitude. The result is that in the receiving antenna, the incident electromagnetic wave generates a propagated wave of the form,

( )2 1 1 2sin2

z i y z tt e−α + β θ +β +ω =E E . (6.72)

§6.3. SUMMARY OF MAXWELL'S CLASSICAL FIELD THEORY

By the end of the 19th century Maxwell's four equations described all known electric and magnetic phenomena, which in turn could be used to describe all macro behaviors in chemistry and biology. The equations Maxwell formulated in 1861 are still valid today, it is their interpretation that has changed. The next sections will lay the foundation for the modern theory of electromagnetic radiation — Quantum Electrodynamics (QED). In this theory the electromagnetic force is carried through space by the quanta of the electromagnetic field — the photon. It is at this point the classical description of the electromagnetic field must be extended to include quantum mechanics and special relativity. In this monograph the formulation of this theory starts with the classical description of a charged particle moving in a potential field. In this way, the Lagrangian and Hamiltonian dynamics can be developed, then quantized and expanded to the 4–dimensions of special relativity.

Curiouser and curiouser, cried Alice...

— Through the Looking Glass [Dodg60]

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§7. PRINCIPLE OF RELATIVITY

At the end of the 15th century, the geometry of Euclid dominated the concept of time, space, and motion [Jamm54]. The physics of Aristotle and the Astronomy of Ptolemy (100–170 BC) stated that every body in the universe had its natural place and natural motion [Guth39], [Sedg39], [Hope61].

According to Ptolemy...

... all sublunar bodies were composed of one of four elements of earth, fire, air and water. The celestial bodies were composed of a fifth element — the ether, which was described as a transparent rigid, all pervasive and changeless substance [Guth39].

In the 19th century, physicists believed in the existence of a medium for the propagation of the electromagnetic field. This medium pervaded all of space and behaved as an elastic aether through which electromagnetic waves traveled. With the theoretical predication by Maxwell and experimental verification by Hertz of electromagnetic waves, this belief was strengthened. It was difficult to envision the propagation of waves without the existence of medium. The subsequent description of the properties of this medium became the primary effort of physics at the end of the 19th century [Whit60], [Will60], [Scha72], [Jung86], [Hert93], [Elli93].

One of the questions raised regarding the properties of the aether was — when material bodies move, does the aether remain stationary or is it carried along with the bodies? In 1726 James Bradley (1693–1762) observed that the aberration of star light could not be explained as due to parallax and seemed to demand an aether which was not carried along with the earth’s motion through space [Brad28]. He found that the altitude of a star above the horizon varied with the position of the earth in its orbit around the sun. The greatest apparent stellar displacement occurred when the earth was moving directly toward or away from the star [Tayl75]. Another experiment in 1851 by Armand Hippolyte Louis Fizeau (1819–1896), [Monk37] to measure the velocity of light in flowing water indicated that the aether is dragged along with the water though it does not have the same velocity.

During the late 19th century many experiments were performed in an attempt to detect the aether [Shan55], [Shan64], [Shan64a], [Mill33]. The most famous of these was the Michelson–Morley experiment performed in 1881 and more again carefully in 1887 [Mich81], [Livi73], [Jaff60]. Once

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Michelson and Morley performed their experiment in 1887 the results indicated that the relative velocity of the earth and the aether did not exceed one fourth of the earth’s velocity. Although this was not a strictly a null result, it has been interpreted so. The theoretical implications of this result is that the equations for the electromagnetic field must by their very nature reflect the indifference to the aether’s motion [Angl80]. This implies that Maxwell’s equations must remain invariant under the transformation from one reference system to another.

The objective world simply “is”, it does not “happen”. Only to the gaze of my consciousness, crawling along the life of my body, does a section of this world come to life as a fleeting image of space which continuously changes in time.

— Hermann Weyl (1885–1955) [Weyl63].

§7.1. ORIGINS OF RELATIVITY THEORY

Although Albert Einstein is credited with the formulation of special as well as general relativity, there were others who could have laid claim to the formulation of special relativity [Clar71], [Swen79], [Hoff83], [Holt83], [Pyen85]. The one coming closest to this claim is Henri Poincaré (1854–1912). [1] Poincaré could not abandon the concept of the aether as the carrier of the electromagnetic force and the light waves which were manifested by electromagnetism. This was asserted in 1899 and expanded in another paper in 1904 [Poin04]. In this paper later, Poincaré considers relativistic topics such as the Lorentz contraction, the increase in the mass of a moving body, the problem of comparing clocks in different moving reference frames and the limits on the speed of light.

When Lorentz (1853–1928) developed his transformation mathematics in 1908, he also did not question the existence of the aether which carried the electromagnetic waves. [2] It was not until 15 years later, in 1923, that

1 Jules Henri Poincaré was a French mathematician who with Lorentz laid down the equations duplicated by Einstein. These equations described the transformation between two coordinate systems moving with uniform motion relative to each other.

2 Hendrik Antoon Lorentz (1853–1928) was a Dutch physicist who developed a theory of relativity, which used Maxwell’s equations as a foundation. Einstein’s mathematics for Special Relativity made use of these transformations, but Einstein had not heard of Lorentz’s work when he published his original paper [Eisn09].

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Lorentz attached some value to the concept of absolute space, in the absence of the aether.

There were other contributors to the relativity foundations including Voight, who in 1887 published a paper describing the Doppler effect, (Christian Johan Doppler (1803–1853)) which can be regarded as the problem of observing a propagating wave from different inertial reference frames. The foundations of time dilation were formulated by Sir Joseph Larmor (1857–1942) in 1900 as a explanation of the FitzGerald–Lorentz contraction, if the invariance of the equations of motion for the properties of the ether were taken into account.

It was Einstein who combined all these disparate theories and speculations into a single concept — the Special Theory of Relativity. Einstein had given thought to the problem of measuring time starting at the age of 16 [Schi70], [Eins79]. From this early work the use of the Lorentz transformation and the dynamics of a moving mass were formed. In his 1905 paper, [Eins05] Einstein described his theory without a single reference to the scientific literature. [3]

3 Since Einstein derived the Lorentz transformation from scratch in the 1905 paper

there is an important question that remains unanswered: On what basis did Einstein build his theory? When asked this question many years later, he did he was influenced by the aberration of starlight and the propagation of light in flowing water, but he did not recall having had any prior knowledge of the Michelson–Morely experiment [Shan63]. This experiment was performed in 1887, 18 years prior in direct response to Maxwell’s speculations that the motion of the earth through the aether could be detected [Maxw52], [Maxw65b], [Shan64]. For the ordinary man it seems unlikely that Einstein could not have known of the experimental and theoretical work that came before. Einstein was however no ordinary man. With a few insights, he could derive the underlying principals of nature. He proceeded in the same manner in the formulation of the theory of the photoelectric effect. It was this discovery that earned him the Nobel Prize in 1921, not the Theory of Relativity, since Alfred Nobel stated that the prize should be awarded for ... discoveries conferring benefit on mankind. The award read:

For his contributions to mathematical physics, and especially for his discovery of the law of the photoelectric effect.

There is one author, Edmund Whittaker, [Whit51] that questions the extent to which Einstein formulated his work solely on original ideas. Whittaker dismissed Einstein’s 1905 paper as one ...

... which set forth the relativity theory of Poincaré and Lorentz with some amplifications, and which attracted much attention.

Much has been written about Einstein’s development of the Theory of Relativity [Holt53], [Holt60], [Holt73], [Fran47], [Infe50], [Whit55], [Reic49]. Although this controversy may seem arcane by today’s standards, the origins of scientific theories and their development and presentation to the general public is a complex and convoluted

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§7.1.1. Invariance of Newton’s Equations of Motion

Newtonian mechanics defines a set of reference frames — inertial frames — in which the laws of nature take the form described in the Principia [Cajo62], [Wein72]. The invariance property for Newton’s equations of motion can be derived using a set of point masses whose interaction in a gravitational field is,

( )2

32

i j j iii

jj i

mm x xd xm Gdt x x

−=

−∑

(7.1)

where im is the mass of the thi particle and ix is its coordinate vector at time t. If the motion of these particles are now observed from a different coordinate system, one which is moving at a velocity v with respect to the first system, then between the coordinate of the mass points in the first coordinate system and the second coordinate system, there is the relationship,

,

.x Rx vt dt t′ = + +′ = + τ

(7.2)

where R is some real orthogonal matrix, v, d and τ are real constants. If the two coordinates systems are labeled S and S ′ , then the observer in S ′ sees the coordinate system in S rotated by the Euler angles contained in R, moving along the direction of the x axis with velocity v, displaced on the x axis at time 0t = by d. This observer also sees the clock in the coordinate system S running behind a clock in S ′ by τ . The transformations given in Eq. (7.2) form a 10–component group, the 3 Euler angles of rotation in R, the 3 components each of v and d and the 1 component of τ . This group is named the Galileo Group and the laws of motion described by Newton [Cajo66] are invariant under this Galilean Transformation.

These equations can maintain their invariance only under a limited number of transformations. For example accelerating or rotating coordinate systems do not leave Eq. (7.2) invariant. The equations of motion do maintain their form in inertial frames of reference. Newton determined than an inertial reference frame was one that was at rest in absolute space or was moving with uniform motion with respect to absolute space. Newton stated...

process. The classical reference for the this process is [Holt73].

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Absolute space in its own nature and with regard to anything external always remains similar and unmovable. Relative space is some movable dimension or measure of absolute spaces; which our senses determine by its position to other bodies; and which is commonly taken for immovable space ... [Cajo66, pp. 6]

The theory of electrodynamics as formulated by Maxwell in 1864 did not satisfy the principal of Galilean Relativity. The primary difficulty was that Maxwell’s equations predict that the velocity of electromagnetic waves in a vacuum are a constant whose value is c. For this prediction to be valid in one reference frame S, it could not be valid in a second moving reference frame S ′ as defined by the Galilean transformation in Eq. (7.2).

When the equations for electromagnetic processes are considered in such a transformation, the two coordinate system are no longer equivalent, since the universal constant of the velocity of light, c, in the primed system would have the velocities c v± , depending on which direction the expanding light wave was traveling — making it no longer a universal constant. [4]

Maxwell proposed that the luminiferous aether conveyed these electromagnetic waves. Maxwell’s equations would then obey the Galilean transformation for those coordinate systems that were at rest with respect to the aether. All attempts to measure the velocity of the earth with respect to the aether failed [Möll52], [Shan55], [Shan64], [Whit60]. These early experiments conducted in 1887, produced a null result which showed that the velocity of light is the same within 5km sec for light traveling along the direction of the earth’s orbit to the velocity of light transverse to the earth’s orbit. This small difference is compared to the velocity of the earth in reference to the sun, 30km sec and 200km sec relative to the center of the galaxy. Experiments conducted recently have improved this accuracy of this measurement to 1km sec [Jase64].

4 The measurement of the velocity of light and the attempt to determine the relative

velocity of the earth through space was performed by several well known experiments by Fizeau in 1849 and 1851 and Foucault in 1865. The results from these experiment showed that the motion of the earth had no effect on the speed of light, in agreement with the principal of relativity. The understanding the results of these and other experiments resulted in many explanations.

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§7.2. VELOCITY OF LIGHT AND ABSOLUTE MOTION

In the current theory of electromagnetism, light is considered as a stream of particle like concentration of energy, called photons. These photons travel at a velocity, c. which denotes the velocity of light [5]. Since these particles of light can be propagated in a vacuum, they can not be considered physical particles like marbles.

The finite velocity of light was considered heresy by Aristotle (384–322 BC) who censured Empedocles (490–430 BC) for considering that light could take time to go from one place to another [Robi90]. The Aristotelian view that light was a quality of the medium through which it traveled pervaded for many centuries. Ibn al–Haythan (965–1039) asserted that light required a finite, though imperceptible, time to travel a distance. Although Roger Bacon (1220–1292) argued in favor of the finite velocity of light, most 17th century authors assumed an instantaneous velocity for light. Galileo attempted to measure the velocity of light in 1607. His use of lanterns as signals failed to detect any delay in the arrival of light over terrestrial distances.

The first real measurements came from Olaf Rømer (1644–1710), who used the eclipse of the moon of Jupiter as his measuring rod [Cohe44]. In a 1½ page paper titled “Proof of the Movement of Light,” published in Journal des Scavaus, December 7, 1676, Rømer showed that light propagates at approximately 214,000km s . By measuring the time between the eclipses of the moons of Jupiter, when the Earth was closet and furthest from Jupiter, a difference of 22 minutes was observed. The extra time needed for light to travel from Io to Earth is equal to the Earth’s orbital diameter. Since Rømer only seemed interested in proving it was finite, he did not actually calculate the velocity of light.

Charles Huygens (1622–1695) using the known orbital diameter of Earth, concluded that 283,000,000km 22sec 214,000km sc = = . This result allowed James Bradley (1693–1762) to measure the aberration of star light. A. H. L. Fizeau (1819–1896) made the first terrestrial measurements, using a technique first proposed by Galileo. Two lamps are separated by a large distance (by 17th century terms). When the first lamp was shut off, the second lamp (remote) observer shut off her lamp. The first observer then notes the time it took for the second observer to see the absence of light. If this time interval were to be proportional to the

5 The symbol c comes from the Latin celeritias, which means velocity.

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separation of the lamps, then the velocity of light would be finite. Because of the limited terrestrial distances and the time it takes the observers to react, Galileo could not produce a valid result. Galileo stated…

In fact I have tried the experiment only at short distances less than a mile, from which I have not been able to ascertain with certainty whether the appearance of the opposite light was instantaneous or not; but if not instantaneous it is extraordinarily rapid… [Gail65].

Fizeau replaced the second lamp of Galileo’s experiment with a mirror. Leon Foucault (1819–1868) improved Fizeau’s mechanical device with a rotating mirror. By the time Maxwell presented his theory of electromagnetic fields, a measured velocity of approximately

300,000km secc = was the agreed upon value. At the end of the 19th century A. A. Michelson had improved upon Foucault’s apparatus [Mich79], [Jafe60]. [6] Michelson’s careful measurements in 1882 resulted in a speed of 299,820km secc = in a vacuum. This figure held for 45 years. When it was replaced in 1923 Michelson was again involved. Michelson placed two stations 22 miles apart, one on Mt. Wilson and the other on Mt. San Antonio (both in Southern California). After many observations, Michelson determined the velocity of light to be 299,798km secc = in the air of California [Jafe60]. The final measurement performed by Michelson was published in 1935 which determined the velocity to be

299,774km secc = in vacuo [Mich35]. The current value adopted by the International Union of Geodesy and Geophysics and by the International Radio Union is 299,792.458km secc = [IAU91], [Cohe86].

In Maxwell’s Treatise, first published in the early 1860’s, the electromagnetic waves traveled at a finite velocity given by 0 01c = ε µ .

6 Albert A. Michelson (1852–1931) was a Polish born scientist who attended the

United States Naval Academy on a direct presidential appointment. Upon graduating Michelson became an instructor at the Academy started on the first determinations of the velocity of light. Michelson was a preeminent builder of optical instruments, After traveling and working in Europe he returned to the United States to join Professor Morley to improve on the construction of the interferometer. With this instrument they both repeated his original experiments that was the experimental foundation of the Special Theory of Relativity. Together they also repeated Fizeau’s moving water experiment as well as lay the foundations for the establishment of the wave length of sodium light as the standard of length.

Michelson’s skills in the manufacture of instruments was recognized by his receipt, as the first American, of the Nobel prize in 1907.

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Using the values of the permittivity and permeability of free space available at the time, Maxwell concluded that his electromagnetic waves traveled at the speed of light 300,000km s≈ . This is a tremendous velocity compared to human or celestial velocities. A typical space craft travels at 12km s . The Earth revolves around the Sun at 30km s .

The problem of defining absolute motion was addressed by Newton through the application of a force acting on a body.

A Body A may be in motion relative to B, even though no force is acting on A. It suffices that somehow or other B moves. B’s motion means that A has relative motion. But if some force acts on A, then A must really move, not just relative to B.

The causes by which true and relative motion are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, by some force impressed upon the body moved; but relative motion may be generated or altered without any force upon the body [Rich46], [Adle60].

Whenever a body moves in a non–inertial manner, some external force is being applied to the body.

The effects which distinguish absolute from relative motion are, the forces of receding from the axis of circular motion. For there are no such forces in circular motion purely relative, but in a true and absolute circular motion, they are greater or less, according to the quantity of the motion [Rich46], [Adle60].

It is this search for the means to detect the difference between absolute or relative motion that leads to the theory of relativity. The fundamental fact that observations can be made across the distances of free space only with electromagnetic radiation leads to a serious problem in the formulation of the meaning of the words simultaneity. The time of arrival of electromagnetic waves, the angle of the arrival of these waves and the frequency and polarization of the waves can be observed, but no other observations of distant events are possible. In many texts the constancy of the speed of light is discussed. Experiments can be conducted which show that the speed of light is constant. There is however a deeper process at work here.

The scale of time in which all the measurements are made is established by the frequency standards (time) is use in the reference frames laboratory. A measurement of the frequency standard in another reference frame is not possible, since in order to observe the frequency in

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the other frame of reference requires transmitting information about wavelength across space using electromagnetic waves.

This results in the second problem, that the speed of light can not be determined if there are no standards of length independent of the transit times of light waves. Since the standard of length is defined in terms of the time passage of light signals, an independent standard of length can not established for any distance greater than a few meters. The result is that all times and distance measure over large distances must be made in the local observes time scale — the principal of relativity [Kenn32]. Using this approach the speed of light must be assumed to be constant, since it cannot be established that it is not [7].

One way to illustrate the paradox of the absolute speed of light to two moving reference frames is through the following example. [8] Suppose there is a source of electromagnetic radiation located at position 0S which emits a pulse of energy which travels to a mirror at location 0M . At the mirror the electromagnetic radiation is reflected to the source. The time it takes for the radiation to travel to the mirror and back again divided by two will be called the distance 0x . This distance will then be referred to as

the measuring rod. [9]

7 Einstein argued that the speed of light is not necessarily constant everywhere and in

general the metric coefficients used in the Lorentz transform are dependent on the distribution of matter. From this approach he derived the gravitational red shift, the deflection of light by the Sun, and the advance of the perihelion of Mercury [Ande75]. Special relativity remains appropriate for the observation of local events and for regions far from great masses. At large distance and in locations near large masses, such as the Sun, the speed of light is independent of its position.

8 The presentation of the special relativity arguments is not meant to be an adequate description of the theory. Such descriptions can be found in numerous texts, many of which have been referenced in this work. This explanation is presented here to lay the groundwork for the Lorentz transformation of Maxwell’s equations, which in turn are needed for the further development of the relativistic description of the electromagnetic interaction between matter and the electromagnetic field.

9 This concept of defining a measuring rod using the radar rule, that is the out and back time for an electromagnetic signal can be ascribed first to Einstein [Brid83]. There is an interchangeable argument between the radar rule and the Lorentz transformation. If any distant event is to be dated, that is the time of the event is to be determined using the finite velocity of light (or radar) as the communication signal, then the following logic can be used.

Suppose a signal is sent from a location at time t. The sending of this signal can be labeled as event e1 . At the remote location the arriving signal is recorded as event e2 . The

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x0

S0

x0

vc

S M0 0

M0

M1

S1

S2

vc

M S1 2

S0

(a)

(b)

(d)

(c)

M

arriving signal is retransmitted with now delay back to the originator, who records the arrival of the retransmitted event as e3 which arrives at time t3 according to the clocks at

the originating location. One question raised by Special Relativity is what is the value of t 2

assigned to event e2 measured on the clocks at the originating location?

Einstein argued that t2 could not have a value earlier than t1 or later than t3 . Since the event e2 is the effect of the event e1 and the cause of the event e3 , t2 must occur

between t1 and t3 , such that ( )2 1 31t t tε ε= + − , where 0 1< <ε . Einstein used ε = 1 2 by

convention giving ( )2 1 31 2t t t= + , which results in the radar rule definition of the distance

as ( )0 1 32x c t t= + .

There are several assumptions necessary to conclude that ε = 1 2 . The first is that the velocity of the electromagnetic source is independent of the velocity of the source of the electromagnetic radiation, which implies that the velocity of light is the same in all uniformly moving reference frame — which has been shown experimentally to be true. This empirical claim is the foundation of Special Relativity and Einstein’s assumption that ε could be any value between 0 and 1 is based on the symmetry of space leaving ε = 1 2 .

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Figure §7.1 – The radar rule method of explaining the time dilation of a signal sent from a source iS , moving with a uniform velocity to right. Both the source

and the reflective surface iM and moving with velocity v.

Consider a measuring rod of length 0x defined in a rest frame (a). The length of the measuring rod is defined by the time it takes a pulse of light to travel from the source 0S to a mirror 0M and back again. This distance is given by,

0 0x xtc+

∆ = . (7.3)

In a moving frame, (b) the length of the measuring rod must now be measured in two pieces. If the reference frame of moving to the right with velocity v, the length measured by the outbound pulse of light is,

1 0 0 1vt x S Mc

∆ = + , (7.4)

since the mirror, originally located at 0M has now moved to a new position

1M . The reflected light now returns to the source by traveling,

2 0 1 2vt x M Sc

∆ = − . (7.5)

The total distance traveled and therefore the total length of the measuring rod, is now given by,

( )

0 0 1 0 1 20 1 1 2

02

,

2 .1

v vx S M x M SS M M S c ctc c

xc

+ + − + ∆ = =

=− β

(7.6)

where 2 21 v cβ = − .

If the length of the measuring rod in the rest frame is compared to the length of the rod in the moving frame, then,

2resting moving 1x x= − β (7.7)

and is the basis of the Lorentz contraction of measuring rods in moving reference frames. A similar experiment can be performed using clocks to

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show that time measured in the resting frame differs from time measured in the moving frame by,

movingresting 21

tt

∆∆ =

− β. (7.8)

The concept of time dilation seems foreign to our everyday experience. There is direct experimental evidence that this event exists. One example is the π –meson. The life times of π –mesons have been studied in the laboratory (stationary) reference frames as well as reference frames moving at near relativistic speeds of 0.755 c. π –mesons are unstable subatomic particles which decay into µ–mesons and a neutrino with a probability distribution of 0

tN N e − τ= is a rest frame, where 0N is the number of π –mesons at time t = 0 and x ict4 = , τ is the characteristic lifetime of the decay process. This decay lifetime has been established experimentally to be τ = × −256 10 8. sec for a π+ at rest.

A beam of π+ –mesons traveling at 0.775c relative to a rest frame exhibits a decay rate of ≈ × −25 10 8 sec [Fris63], [Penm61], [Easw91]. Since the π+ –meson is traveling at nearly 70% the speed of light, its local clock, that is the mechanism which causes the π+ –meson to decay, is also traveling at nearly 70% the speed of light. The result is a dilation in the time based decay mechanism given by,

( )21rest movingt β∆ − = ∆ (7.9)

This situation can be better understood through the view of the π+ –meson. If the frame of reference is from the laboratory this is outside looking in, to the resting frame of the π+ –meson. In this reference frame the π+ –meson ticks an average for τ = × −256 10 8. sec before it decays. If the frame of reference is viewed from inside looking out, that is viewed from the traveling π+ –meson, the time measured for the decay rate is still the same. But from the outside looking in the traveling π+ –meson has

experienced a time dilation of ( )21 1 β− . This result is called the Lamor

dilation [Lamo04].

The alternative point of view can also be taken. When the π+ –meson is traveling at 0.7c, it covers a distance of 07 256 10 8. .× × − light seconds when viewed from the outside., but only 07 25 10 8. .× × − light seconds, using its internal clock. This contraction of distance traveled is given by Eq. (7.7) and called the FitzGerald contraction, named after George Francis FitzGerald (1851–1901).

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All of this seems like nonsense when first presented to the classical world. The answer to this nonsense is a third principal (always unexpressed) to go along with the first two principals of relativity given in §2.

(1) Local laws of physics have the same form in frames of reference that are elated by a uniform relative velocity.

(2) The invariance or frame independence of the speed of light, which states that the propagation speed of energy has a finite upper bound.

(3) Measurements are always confines to certain recognized inertial reference frames which introduce an asymmetry to the measurement. In the case of the π+ –meson, the instruments measuring the decay rate if the traveling particle are favored since they are stationary.

The π+ –meson travels through the space attached to the rest frame of the instruments. From the view of the π+ –meson, the instruments and the attached space are moving toward it at 0.7c. However the π+ –meson is at rest relative to its inertial reference frame and therefore decays in τ = × −256 10 8. sec . When viewed from the point view of the measuring instruments, the π+ –meson is traveling toward them at 0.7c and is subject to the Lamor and FitzGerald equations in Eq. (7.7) and Eq. (7.8).

§7.3. THE LORENTZ TRANSFORMATION

The Lorentz transformation has been derived in many ways. Albert Einstein was led to the Lorentz transformation by observing that Maxwell’s equations were left unaltered in form as they were transformed from once reference frame to another. Several significant events occurred prior to Einstein’s Special Theory of Relativity, based on the Lorentz transformation. It was these events that laid the ground work for the introduction of the covariant form of Maxwell’s equations. There is a rich body of literature describing the foundations of Special Relativity and its connection with the Lorentz transformation [Frie86], [Paul58], [Born24], [D’abr39], [Fren68], [Gold84], [Molr52], [Breh62], [Gold87], [Gelf63]. The most compelling approach to the Lorentz transform is through the physical evidence produced by the null experiment performed by Michelson and Morley. This experiment forced the abandonment of the theory of the aether and the acceptance of the finite speed of light. It was

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the experimental result, not a theoretical conjecture that paved the way for the Special Theory of Relativity. [10]

§7.3.1. The Components of Relativity

An event is described by the place and time it occurred [Land71], defined by the three coordinates and time. The interval between events is given as,

( ) ( ) ( ) ( )2 2 2

2 1 2 1 2 1 2 1

spatialdistance propagation distance

x x y y y y c t t− + − + − = −144444424444443 14243 (7.10)

in the first reference system. In the second reference system the same set of events define the interval using that reference systems coordinate system, such that,

( ) ( ) ( ) ( )2 2 2

2 1 2 1 2 1 2 1

spatialdistance propagation distance

x x y y y y c t t′ ′ ′ ′ ′ ′ ′ ′− + − + − = −144444424444443 14243 (7.11)

The distance or interval between the two events is given by,

( ) ( ) ( ) ( )2 2 2 22 1 2 1 2 1 2 1s x x y y z z c t t= − + − + − − − (7.12)

and

( ) ( ) ( ) ( )2 2 2 22 1 2 1 2 1 2 1s x x y y z z c t t′ ′ ′ ′ ′ ′ ′ ′ ′= − + − + − − − (7.13)

§7.3.2. Relativity Principals Formally Stated

The expression of Maxwell’s equation in a form that is invariant between two moving coordinate system requires the use of a Lorentz transformation between these two systems. The details of the Lorentz

10 It is often written that Einstein developed the Special Theory of Relativity to

account for the null result from the Michelson–Morely experiment. The historical facts do not support this. Einstein thought in a general manner about physical problems and sought the beautiful simplicity that must lie beneath the phenomena of experience. Only after working out the underlying mathematics of a position would he search for the experimental evidence which might test his theory [Mack82], [Holt73], [Pais82], [Eins79], [Angl80].

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transformation are given in several texts [Beck64], [Køll52], [Land72], [Land71], [Lévy76]. [11] This section will summarize the results of this material and restate Maxwell’s equation in their 4–vector invariant form.

The theory of relativity states that:

The characteristic equations of motion of a particle traveling in the x direction when viewed from a primed coordinate system has coordinate values given by:

11Starting with the understanding that each Galilean frame has its own private time.

A generalization of the Galilean transformation between

( )( ) ( )2 2 2 21t t v c x v c′ = − − x y z t, , , and ′ ′ ′ ′x y z t, , , can be constructed. Starting with the

experimental evidence that the speed of light is the same in all reference frames, four assumptions are required to proceed [Møll55]. (1) the transformation between any two reference frames is linear. This is necessary in order to avoid giving the origin of the coordinate system preference over any other location. (2) The motion between the two reference frames is restricted to the x axis so that y y= ′ and z z= ′ . This assumption allows x and t to be stated as linear functions of ′x and ′t , such that ′ = +x ax bt and ′ = +t dx ft. (3) At some point the origin of the two reference frames coincide, such that

x x= ′ = 0 at time t t= ′ = 0 , with the unprimed system moving relative to the primed systems with velocity v. This allows the origin in the primed system ′ =x 0 to be given in the unprimed system as x vt= , which results in b va= − and ( )x a x vt′ = − . This

assumption characterizes the Special Theory of Relativity. If the moving reference frame was accelerating, then the General Theory of Relativity would be needed to described the transformations between coordinate systems. (3) It is not possible by any physical measurements to establish any fundamental differences between the two moving reference systems. (4) The velocity of light is the same in both systems, a spherical pulse of light originating at location x x= ′ = 0 at time t t= ′ = 0 can be described by a spherical wave equation, x y z c t2 2 2 2 2+ + = and ′ + ′ + ′ = ′x y z c t2 2 2 2 2 . Both of these equations must be valid in their respective reference frames, since they both describe the same wave front. By substituting ′ = +t dx ft and ( )x a x vt′ = − into the unprimed wave equation. Giving,

( ) ( )2 22 2 2 2a x vt y z c dx ft− + + = + or, ( ) ( ) ( )2 2 2 2 2 2 2 2 2 2 2 2 22a c d x va c fd xt y z c f a v t− − + + + = − .

By equating the coefficients of this expression with those of the spherical wave equation, results in, c f a v c2 2 2 2 2− = , a c d2 2 2 1− = and va c fd2 2 0+ = . By eliminating the first d from the second and third coefficient expressions and eliminating a 2 from the same coefficient expressions gives, ( )2 2 21 1f v c= − and the first expression gives a f2 2= .

By taking the positive square root of f and a and allowing the Galilean reference frame

velocity go to zero, v → 0, produces in the Galilean limit f v c= −1 1 2 2 , a f= and

( )2 2 21= − −d v c v c . When these expressions are substituted into the original equations

for ′x and ′t , the result is the Lorentz transformation ( ) ( )2 21x x vt v c′ = − − , ′ =y y,

′ =z z and ( )( ) ( )2 2 2 21t t v c x v c′ = − − .

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( )2 21

x vtx

v c

−′ =

−, y y′ = , z z′ = ,

( )( )

2

2 21

t v c xt

v c

−′ =

− (7.14)

Solving these equations of the unprimed quantities gives:

( )2 21

x vtx

v c

′ +=

−, y y′= , z z′= ,

( )( )

2

2 21

t v c xt

v c

′ +=

− (7.15)

The relationship between these two equations is the Lorentz Transformation [Beck64].

This transformation can be restated in the form of a matrix multiplication by defining the following,

γβ

=−

=−

1

1

1

12 2 2v c, (7.16)

where β = v c, which allows the following relationships to be defined, [12]

( )x x vtγ′ = − , ′ =y y , ′ =z z, 2

vt t x

cγ ′ = −

(7.17)

In matrix form, these relationships can be expressed as,

0 0

0 1 0 0

0 0 1 0

0 0

x x

y y

z z

t t

γ βγ

βγ γ

′ − ′ = ′ ′ −

, (7.18)

which can be expressed as,

4

1

x a xµ µν νν =

′ = ∑ , (7.19)

where the coefficients of Λ ij are represented by the elements of the 4 4× matrix in Eq. (7.18).

The Lorentz Transformation replaces the Galilean transformation developed in Eq. (7.2). In the limiting case where c = ∞, the Galilean and Lorentz transformations are equivalent. The Lorentz transformation is

12 Einstein used the notation β to abbreviate 1 2 2− v c , but now γ has become the

common notation.

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the key to understanding the Theory of Special Relativity. This transformation can be stated in words as...

... the change from one frame of reference to another moving at a uniform translational [13] velocity with respect to it is like a change from one set of coordinates to another set of coordinates with some angle between them... [Luca90].

The Lorentz transformation is the consequence of combining space and time into a single description of spacetime. It connects the spatial and temporal coordinates of a point in one frame of reference with those in another frame of reference moving relative to the first frame with a uniform velocity. In the previous example, the motion of the second reference frame was restricted to the x direction.

If the Lorentz transformation concept is now applied to an event which is represented by a light source which emits a pulse of light, in an expanding spherical wave whose equation of motion is given by,

x y z c t2 2 2 2 2+ + = , (7.20)

and a similar expanding wave of light described in a second reference frame,

′ + ′ + ′ = ′ ′x y z c t2 2 2 2 2 . (7.21)

By using two references to the time variable t and ′t it can be emphasized that the possibility exists that the time scale will also be transformed between the two moving reference frames. By treating time explicitly as a coordinate the equations of motion for the expanding sphere of light can be equated as,

x y z c t x y z c t2 2 2 2 2 2 2 2 2 2+ + − = ′ + ′ + ′ − ′ ′ (7.22)

If the spatial coordinates x y z, , are equated to the generalized coordinates x x x1 2 3, , a fourth imaginary coordinate can be introduced, x ict4 = . [14] This approach allows the transformation to be generalized as,

13 In the original Einstein paper [Eins05], the word translational was used to

distinguish motion in a straight line from angular velocity, or rotation, under which the laws of physics are not covariant. The current practice in the description of Special Relativity is to omit the word translational and assume the velocity is in fact translational unless it is stated to be angular [Luca90].

14 There are at least three methods of formulating the relativistic equation for a spherically expanding light wave, marking the location of an event in space–time. The

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x x x x x x x x12

22

32

42

12

22

32

42+ + + = ′ + ′ + ′ + ′ , (7.23)

or in then terms of the 4–vector notation,

x xµµ

µµ

2

1

42

1

4

= =∑ ∑= ′ . (7.24)

The form of this transformation corresponds to a rotation in a 4–dimensional space. This space is known as Minkowski space, with the Lorentz transformations represented as orthogonal transformation in Minkowski space. [15] All the mathematical apparatus for spatial; orthogonal transformations is available for use in special relativity.

The Lorentz transformation connects points in one reference frame with similar points in another reference which is moving relative to it with uniform motion. If the points in both reference frames are represented in a four dimensional Euclidean space by the coordinates ( )1 2 3 4, , ,x x x x where

x x x y x z x ict1 2 3 4= = = =, , , and , then a rotation in the ( )1 4,x x plane

through an angle θ transforms the point ( )1 2 3 4, , ,x x x x to the point

( )1 2 3 4, , ,x x x x′ ′ ′ ′ , where,

method used here is taken from [Menz53], and represents the simplest approach for the purpose of this monograph.

The straightforward method of describing the geometric interpretation of the invariance of the interval s2 is to introduce the variable x y z, , ,τ τ = ict, then the original

interval ( )2 2 2 2 2s c t x y z= − + + becomes s i x y z= + + +2 2 2 2τ . This interval is related

to the radius vector in the vector space, x y z, , ,τ by the constant i.

15After the publication of Einstein’s paper on Special Relativity in 1905 [Eins05], Hermann Minkowski (1864–1909) published a paper in which the fourth dimension x ict4 = was used to construct a geometry of space–time. Using Minkowski’s notation, the fundamental derivative with respect to x4 assumes the same form as those with respect to the 3–dimensional spatial derivatives. The laws of dynamics and electrodynamics take on a simple and elegant form [Fano60]. In the preface of Arnold Sommerfield’s (1868–1951) Electrodynamics, [Somm52] he states...

... after I had heard Hermann Minkowski’s lecture on Space and Time in 1909 in Cologne, I carefully developed the four–dimensional firm of electrodynamics as a apotheosis of Maxwell's theory...in return, this has always met with an enthusiastic reception on the part of my audience.

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′ = +

′ =

′ =′ = − +

x x x

x x

x x

x x x

1 1 4

2 2

3 3

4 1 4

cos sin

sin cos

θ θ

θ θ

(7.25)

Since the second and third coordinates are unchanged during the rotation, they will be left out for the following development. The equations that described the rotation in the ( )1 4,x x plane can be represented in a

matrix form as,

1 1

4 4

cos sin

sin cos

x x

x x

θ θθ θ

′ = ′ −

(7.26)

These relationships can be shown graphically in the following figure.

x4

′x4

′x1

x1

X1

X 4

′X 4

′X1

θ

A

B

C

D

P

O

θ

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Figure §7.2 – A trigonometric illustration of the spatial contraction. This method can also be used to construct a hyperbolic translation of the spatial and time coordinates of the moving reference frame.

Substituting the generic coordinates ( )1 2 3 4, , ,x x x x , for the space and

time coordinates ( ), , ,x y z ict which results in,

′ = +′ = +

x x ict

ict x ict

cos sinsin cos

θ θθ θ

(7.27)

with the latter being written as,

′ = − +tx

ict

sincos

θθ . [16] (7.28)

§7.3.3. Structure of Space–Time

In Galilean relativity an event occurring at the origin of a coordinate system t t= 0 and x x= 0 can cause any simultaneous or later event with t t≥ 0 and may be caused by a previous event for t t≤ 0.

[17] In Special Relativity all events ,t x that causally relate to an event occurring at the

origin 0 0,t x are given by ∆ ∆x c t≤ , such that,

( ) ( ) ( ) ( )2 2 2 2

0 0 0 0x x y y z z c t t− + − + − ≤ − (7.29)

16 In advanced treatments of the Special Theory of Relativity, the approach taken here

is extended by considering the trigonometric functions to be similar to hyperbolic function. The angle ( ) ( )( )′ ′= − = − +x x vt t x c tγ γ β θ is replaced by iφ , such that cos coshiφ φ= and

sin sinh sinhi i iφ φ φ= = −1 . The Lorentz transform then becomes ′ = −x x ctcosh sinhφ φ and ′ = − +t x c tsinh coshφ φ . In matrix form this transformation is given as,

cosh sinh

1 sinh cosh

x c x

t c t

′ φ − φ = ′ − φ φ

. The pseudo–angle φ can be used to represent the

velocity of the moving reference by the relationship tanh φ = v c , which in most texts is

abbreviated β = v c . Since 2cosh 1 tanhφ = − φ it can be abbreviated as

γ φ φ= = −cosh tanh1 2 and sinh tanh tanhφ φ φ= −1 2 which results in the Lorentz

transformation of ( ) ( )( )x x vt t x c t′ ′= γ − = γ −β +

17 The equality in each of the terms holds only for Newtonian action at a distance. The inequality holds for events which are spatially local [Doug90] connected by a finite, but possibly unbounded propagation velocity.

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By suppressing one of the coordinates in Eq. (7.29), the space–time relationships can be illustrated in a 3–dimensional surface called a light cone centered on the origin of the event as shown in the following diagram,

ct

x

z

Figure §7.3 – A light cone centered on the origin of the event in the moving reference frame.

The separate between two events can be given as,

ds x x x x x x x x

dx dx dx dx

= − ′ + − ′ + − ′ + − ′= + + +

( ) ( ) ( ) ( ),

.1 1 2 2 3 3 4 4

1 2 3 4

(7.30)

The term ds represents the separation between two events in the four dimensional space–time. Using this approach space and time can no longer be considered separately, but are fused into one geometry by means of the Lorentz transformation. The term for the coordinates in this four dimensional frame is the proper system of coordinates.

The relationship between the two coordinate systems takes on an interesting property. For two events separated by a spatial distance x x x1

222

32+ + and a time interval x4

2, can be placed into one of three Lorentz invariant classifications, [18]

18 The order of the spatial and time terms given here may be different than those

found in other texts. This causes great confusion, until it is realized that the invariant term for the separation of two events can be written either as c t x y z2 2 2 2 2− − − or

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Light–like x x x x s12

22

32

42 2 0+ + + → = (7.31)

Space–like x x x x s12

22

32

42 2 0+ + + → > (7.32)

Time–like x x x x s12

22

32

42 2 0+ + + → < (7.33)

When a cause propagates more slowly than the speed of light, that is the space–time interval is s2 0< , then the relation between the cause and effect is time–like, when viewed from an arbitrary reference frame, that is causality of Lorentz invariant. This is required if the cause is to occur before the effect, regardless of the reference frame. Past and future are invariant with respect to the observer if the past can be causally related to the future. Therefore the cause produces the effect and the space–time interval is time–like.

If the proper distance cause and effect is s2 0> , then no light signal can pass from the cause to the effect faster than the speed of light. Therefore the cause can not produce the effect and the space–time interval is space–like. Events separated by space–like intervals can be seen indifferent time orders by different observers. How ever this type of time reversal of past and future does not violate causality.

The four dimensional coordinate system described above represents the space–time frame. A line in this space represents the progress of a point through ordinary space and time and is called a world line. A point that is stationary in space will move parallel and define the x ict4 = axis in the four dimensional space.

§7.4. COVARIANT NOTATION

In the previous section the Special Relativity theory was formulated in terms of t and the three dimensional vector coordinates x, y and z. An alternative formulation can be used which eliminates the imaginary coordinate x ict4 = . There are several motivations for this change, one of which pertains to the use of the complex conjugate in quantum mechanics. In the formulation of Quantum Field Theory the complex conjugate is the mechanism used to manipulate the fields which represent charged particles. A second motivation allows the use of hyperbolic coordinate transformations in place of trigonometric forms. This will provide for the covariant transformation of the vector potential. See Endnote 16 for details.

x y z c t2 2 2 2 2+ + − .

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The coordinates of an event ( ), , ,ct x y z can be considered as

components of a 4–vector is a four dimensional space, such that,

x ct x x x y x z0 1 2 3= = = =, , , . (7.34)

The length of the 4–vector is defined as,

( ) ( ) ( ) ( )2 2 2 20 1 2 3x x x x− − − . (7.35)

In general a set of 4–quantities, A A A A0 1 2 3, , , , which transform like components of the 4–vector coordinate system is called a 4–dimension vector, A i . There are tow types of 4–vectors, denoted by A i andAi , which are related by,

A A A A A A A A00

11

22

33= = − = − = −, , , . (7.36)

The quantities A i are called contravariant and the quantities Ai are called covariant components of a 4–vector. All sums of 4–vectors are written as A Ai

i with the summation symbol omitted for clarity. This convention of summation uses one subscript and one superscript. Also by convention Latin letters i j k, , ,… are used for 4–vector indices and take on values 0 1 2 3, , , .

In a manner similar to the square of a 4–vector, the scalar product of two different 4–vectors is defined as,

A B A B A B A B A Bii = + + +0

01

12

23

3. (7.37)

The notation A Bi i or A Bii produces the same results. The result of this

product is a 4–scalar and is invariant under rotation of the 4–dimension coordinate system.

In enumerating the components of the 4–vector the A0 is the time component and the A A A1 2 3, , components are the space components. In the context of relativity the square of a 4–vector can be positive, negative or zero corresponding to timelike, spacelike, and null 4–vectors.

Often the 4–vector will be written as,

( )0 ,=iA A A . (7.38)

The covariant components of this contravariant vector are,

( )0 ,iA A= −AA . (7.39)

and the square of this 4–vector is,

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( )20 2iiA A A= − AA . (7.40)

For the position vector, rr , the resulting notation is,

( ),ix ct= rr , (7.41)

( ),ix ct= −r , (7.42)

and

x x c tii = −2 2 2rr . (7.43)

§7.4.1. Covariant Transformations

A 4–dimensional tensor of second rank is a set of sixteen quantities, A ik which transform like the products of the components of two 4–vectors. These components can be written in three forms, covariant, Aik , contravariant, A ikand mixed, Ak

i . The term covariance is taken to mean unchanged in form and can be used in place of invariant.

A tensor of rank k is associated with point in space–time and is defined by the transformation properties under the transformationx x→ ′.

A scalar is a tensor of rank 0 and is a single quantity whose value is unchanged by the above transformation. For tensors of rank one, vectors, contravariant transformations are defined as,

′ =′

Axx

Aii

kk∂

∂. (7.44)

The repeated index k implies the summation over k = 0 1 2 3, , , . This can be shown explicitly here as,

′ =′

→′

+′

+′

+′∑A

xx

Axx

Axx

Axx

Axx

Aii

kk

k

i i i i∂∂

∂∂

∂∂

∂∂

∂∂0

01

12

23

3 . (7.45)

A covariant vector Ai transforms as,

′ =′

Axx

Ai

k

ik

∂∂

, (7.46)

or,

′ =′

→′

+′

+′

+′∑A

xx

Axx

Axx

Axx

Axx

Ai

k

ik

k i i i i

∂∂

∂∂

∂∂

∂∂

∂∂

0

0

1

1

2

2

3

3. (7.47)

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The contravariant tensor of rank two transforms according to,

′ =′ ′

Fxx

xx

Fiki

j

k

ljl∂

∂∂∂

. (7.48)

A covariant tensor of rank two transforms according to,

′ =′ ′

Fxx

xx

Fik

j

i

l

k jl

∂∂

∂∂

, (7.49)

and a mixed tensor or rank–2 transforms as,

′ =′

′F

xx

xx

Fki

i

l

j

k jl∂

∂∂∂

. (7.50)

The scalar product of two vectors is invariant under a transformation and is given as,

A B A Bxx

xx

A B A Bii

k

i

i

jk

j jk k

j≡ ⋅ =′

′=

∂∂

∂∂

δ . (7.51)

The above expressions are general in nature. The specific geometry of the space–time of special relativity is defined by the invariant interval s2. In differential form, the infinitesimal interval ds defines the norm and is defined as,

( ) ( ) ( ) ( ) ( )2 2 2 22 0 1 2 3ds dx dx dx dx= − − − . (7.52)

The norm or metric is a special case of the general differential length element,

( )2 i kikds g dxdx= , (7.53)

where g gik ki= is called the metric tensor. For flat space–time of special relativity, the metric tensor is diagonal with elements,

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

ikikg g

− = = − −

. (7.54)

Using the metric tensor, the covariant 4–vector A i can be obtained from the contravariant 4–vector A i by a contraction with gik ,

A g Ai ikk= , (7.55)

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and its inverse,

A g Ai ikk= . (7.56)

§7.4.2. Divergence and Curl in 4 Dimensions

The scalar product of two 4–vectors can now be written in the form,

A A g A A g A Aii ik

i k iki k= = . (7.57)

The partial derivative operators with respect to x i and x i can be defined using the rules of implicit differentiation,

∂

∂∂∂

∂∂′

=′x

xx xi

k

i k. (7.58)

This form shows that differentiation with respect to a contravariant component of the coordinate vectors, transforms as the component of a covariant operator. The notation for these partial derivatives is then given as,

∂ ∂ ∂ ∂ = −∇ = −∇ ∂ ∂ ∂

0 , ,i

ix x t, (7.59)

and

∂ ∂ ∂ ∂ = ∇ = ∇ ∂ ∂ ∂ 0

, ,i ix x t. (7.60)

The 4–Gradient can now be written as,

∂φ ∂φ ∂φ = ∇φ = ∇φ ∂ ∂ ∂ 0

, ,ix x t

. (7.61)

In general, operators of differentiation with respect to the coordinates x i , ∂ ∂xi , should be regarded as the covariant components of the operator 4–vector. In the literature, as well as this monograph, partial derivatives with respect to the coordinates are abbreviated using the symbols,

∂∂

∂i

ix= and ∂

∂∂i ix

= . (7.62)

In this form of the differentiation operators, the covariant or contravariant character of the quantities resulting from the operation is made explicit.

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The 4–Divergence of a 4–vector is the invariant

∂

∂ = ∂ = + ∇ ⋅ ∂

0

0i i

i i

AA A

xAA , (7.63)

which is familiar form of the continuity of charge and current density.

The 4–dimensional Laplacian operator is defined as the invariant contraction,

o= ∂ ∂∂

∂∂∂i

i

x t= − ∇ = − ∇

2

0 22

2

22

( ), (7.64)

which is the operator of the vacuum wave equation.

§7.5. LORENTZ TRANSFORMATION IN COVARIANT FORM

Now that the covariant notation has been established the Lorentz transformation can be restated in a more compact manner. A Lorentz transformation can be described as a transformation from one system of space–time coordinates x i to another ′x i such that,

′ = +x x aij

i j iΛ , (7.65)

where a i and Λ ji are constants of the same form as Eq. (7.19), which are

restricted by the condition,

Λ Λki

lj

ij klη η= , (7.66)

with,

1; 1,2,3

1; 00;

ij

i j

i j

i j

+ = =

η = − = = ≠

. (7.67)

In this notation i, j, k and l assume the values 0, 1, 2 and 3 with x x x1 2 3, and representing the Cartesian coordinates and x0 representing the time t. This allows Eq. (7.65) to be an abbreviation for,

′ = + + + +x x x x x ai i i i i iΛ Λ Λ Λ00

11

22

33 . (7.68)

The fundamental property of the Lorentz transformation is that the proper time defined by,

( ) ( ) ( )2 2 22 1 2 3 i jijd dt dx dx dx dxdxτ ≡ − − − =−η , (7.69)

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is invariant under the transformation.

In the new coordinate system an interval is given by,

dx dxik

i k′ = Λ (7.70)

so that the new time coordinate will be,

d dx dx

dx dx

dx dx

d

iji j

ij ki

lj i l

ijk l

′ = − ′ ′

= −

= −

=

τ η

η

η

τ

2

2

,

,

,

.

Λ Λ (7.71)

It is this property that produces the observations generated from the Michelson–Moerly experiments — The speed of light is the same in all inertial reference frames.

§7.6. MAXWELL'S EQUATIONS IN 4–DIMENSIONS

When a permanent magnet is moved while through a coil of wire, the area integral over the magnetic field BB is changed. The term ∂ ∂BB EEt = −∇ × describes the induced electric field in the loop of wire caused by the magnetic field in the permanent magnet. If the bar magnet is held stationary and the coil of wire moved towards it with velocity v, there is no electric field EE induced in the wire. Instead the Lorentz force qv × BB moves the electrons in the wire generating a current. When this process is described from the relativistic view point, the processes are equivalent. The fields EE and BB are only different manifestations of the same electromagnetic field.

Using explicitly 4–vector representation of Maxwell’s equations it can be shown that they remain invariant in form under a spatial transformation that is equivalent to a rotation of their axes. In 1904 Lorentz developed the mathematics which allowed this transformation to be described.

In many texts Maxwell's equations are given in 4–dimensions as is, with little or no development of the underlying mathematics. This section provides a detailed description of Maxwell's field equations in 4–dimensions, taken from [Eise69].

Given the inhomogeneous pair of Maxwell's equations that involve current density jj and charge density ρ ,

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∇ × = +BB EE jj∂∂t

, (7.72)

and,

∇ ⋅ =EE ρ, (7.73)

these equations can be rewritten in 4–dimensional notation. Given the 4–vector components,

x x x y x z x ict1 2 3 4= = = =; ; ; , (7.74)

and the 4–vector potential AA as,

A A A A A A A ix y z1 2 3 4= = = =; ; ; φ , (7.75)

the familiar expression,

BB AA= ∇ × , (7.76)

can be written in its 4–vector component form as,

BAy

A

zAx

Ax

f fxz y= − = − = = −

∂∂

∂∂

∂∂

∂∂

3

2

2

323 32 ,

BAz

Ax

Ax

Ax

f fyx z= − = − = = −

∂∂

∂∂

∂∂

∂∂

1

3

3

131 13 , (7.77)

BA

xAy

Ax

Ax

f fzy x= − = − = = −

∂∂

∂∂

∂∂

∂∂

2

1

1

212 21.

The second vector potential expression,

EE AA= −∇φ −

∂∂t

, (7.78)

can be written in 4–vector component form as,

114 41

1 4

xx

A i AE i if if

x t x x

∂∂φ ∂ φ ∂= − = − = = − ∂ ∂ ∂ ∂

,

224 42

2 4

yy

A i AE i if if

y t x x

∂ ∂φ ∂ φ ∂= − = − = = − ∂ ∂ ∂ ∂

, (7.79)

334 43

3 4

zz

A AiE i if if

z t x x

∂ ∂∂φ ∂ φ= − = − = = − ∂ ∂ ∂ ∂

.

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where i Aφ = 4. Arranging these expression into a square array of fi k, gives,

12 13 14

21 23 24

31 32 34

41 42 43

0000

0000

z y x

z x y k iik i k k ii k

y x z

x y z

B B iEf f fB B iE Af f f A

F A AB B iEf f f x xiE iE iEf f f

− − − − ∂ ∂ = = = − = ∂ − ∂ − − ∂ ∂

, (7.80)

which implies F F00 00= − and the same for the F11 , F22 , and F33 components. The remaining off diagonal components form into six independent sets of two and their transpose. Only one element in each set is independent, the other being its negative. This symmetric for has 10 independent components, four diagonal and six off diagonal which are independent.

The elements of F form an antisymmetric 4–dimensional tensor of rank 2. The purely spatial part is a 3 3× matrix in which ( )23 31 12, ,F F F=BB and

( )41 42 43, ,i F F F=EE

Before proceeding with the development of Maxwell's four field equations, the equation of continuity will be considered in 4–dimensions. Starting with the familiar equations for the conservation of charge and current,

∂ρ∂

ρφt

+ ∇ ⋅ = 0, (7.81)

results in the 4–dimensional notation of,

∂ ρ∂

∂ρ∂

∂ρ∂

∂ρ∂

∂∂

µ

µ

icict

vict

v

ictv

ict

j

xx y z+ + + = =

=∑

41

4

0, (7.82)

where ( ) ( )1 2 3 4, , , , , ,x y zj j j j j icµ = = ρφ ρφ ρφ ρ .

The Lorentz gauge condition,

∇ ⋅ + =AA 10

c t∂φ∂

, (7.83)

can also be written in 4–dimensions as,

∂∂

∂∂

∂∂

∂ φ∂

∂∂

µ

µ

Ax

A

yAz

iict

A

xx y z+ + + = = 0. (7.84)

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Maxwell's equations can now be rewritten starting with the 3–dimensional form,

∇ ⋅ =BB 0, (7.85)

resulting in the 4–dimensional form of,

∂∂

∂

∂∂∂

∂∂

∂∂

∂∂

Bx

B

xBx

fx

fx

fx

x y z+ + =

+ + =

0

023

1

31

2

12

3

,

. (7.86)

which can be simplified by using the notation ∂ ∂ ∂i ix≡ to give,

∂ ∂ ∂1 23 2 31 3 12 0f f f+ + = . (7.87)

Rewriting,

∇ × + =EE BB10

c t∂∂

, (7.88)

gives the x, y, and z components as,

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂

Ey

E

z cBt

Ex

E

x ciBict

ifx

ifx

ifx

fx

fx

ifx

Ez

Ex c

B

tEx

Ex c

iB

ictifx

ifx

ifx

fx

fx

ifx

E

x

z y x z y x

x z y x z y

y

− + = − + = − + = + + =

− + = − + = − + = + + =

−

1 10

1 10

2 3

34

2

24

3

23

4

34

2

24

3

23

4

3 1

14

3

34

1

31

4

14

3

34

1

31

4

,

,

Ey c

Bt

E

xEx c

iBict

ifx

ifx

ifx

fx

fx

ifx

x z y x z

∂∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+ = − + = − + = + + =1 1

01 2

24

1

14

2

12

4

24

1

14

2

12

4

.

(7.89)

The two expressions for ∇ ⋅ =BB 0 and ∇ × + =EE BB10

c t∂∂

can be collected

into a single 4–dimensional expression,

∂∂

∂∂

∂∂

f

x

f

xfx

jk

i

ij

k

ki

j

+ + = 0 , (7.90)

where i, j, and k are never equal and always vary in cyclic order from 1 to 4, while omitting some value l. As i, k, j and l assume all values from 1 to 4, four l equations are generated of the form given above, which are the homogeneous Maxwell equation in 4–dimensional form.

The inhomogeneous Maxwell equations can now be developed in a similar manner. Starting with,

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∇ ⋅ =EE ρ, (7.91)

the 4–dimensional form is given by,

∂∂

∂∂

∂∂

ρ∂∂

∂∂

∂∂

ρ∂∂

∂∂

∂∂

Ex

E

yEz

ifx

ifx

ifx

i i cfx

fx

fx

jx y z+ + = = + + = − = + + = −14

1

24

2

34

3

14

1

24

2

34

34b g (7.92)

The second inhomogeneous Maxwell equation,

∇ × − =BB EE jj∂∂t

, (7.93)

can be written in 4–dimensional form as,

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

By

B

ziEict

fx

fx

fx

j

By

B

ziEict

fx

fx

fx

j

B

xBy

iEict

fx

fx

fx

j

z y x

z y x

y x z

− − = − − − =

− − = − − − =

− − = − − − =

21

2

23

3

41

41

21

2

23

3

41

41

13

1

23

2

43

41

,

, (7.94)

The two inhomogeneous Maxwell equations can be collected into,

∂∂

∂∂

∂∂

fx

f

xfx

jil

i

jl

j

kl

kl+ + = − . (7.95)

§7.7. LORENTZ TRANSFORMATION OF MAXWELL’S EQUATIONS

To ensure that Maxwell’s equations remain invariant under the Lorentz transformation, the potential form in which the vector potential, AA , is treated as a 4-vector, such that,

( ) ( )0 1 2 3 0,1,2,3, , , i i

A A A A A=

= =AA (7.96)

where i = 1 2 3, , represents the 3 spatial coordinate indices and i = 0 represents the time index.

If v is the relative velocity of the observer and i = 1 direction is parallel to v, then the transformation matrix is given by,

cosh sinh . .sinh cosh . .

. . 1 .

. . . 1

ξ ξ ξ ξ

(7.97)

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where ξ is the ( )arctan v c . This matrix can now be rewritten as,

. .

. .. . 1 .. . . 1

v

v

γ γ γ γ

(7.98)

where ( )2 21 1 v cγ = − . This form results in cosh ξ γ= and sinhξ γ= v . In

both forms v2 is the magnitude of the velocity vector vv .

The electromagnetic vector potential now transform as,

′ = +

′ = +

′ =

′ =

A A A

A A A

A A

A A

i0 0

1 1 0

2 2

3 3

cosh sinh

cosh sinh

ξ ξ

ξ ξ (7.99)

or

( )( )

0 0 1

1 1 0

2 2

3 3

A A vA

A A vA

A A

A A

′ = γ +

′ = γ +′ =′ =

(7.100)

These equations provide the rules for the transformation of the vector potential between one reference frame and another. What is needed next is the rules for transforming the differential operators used in Maxwell’s equations. These are given by,

′ = +

′ = +

′ =

′ =

∂ ∂ ξ ∂ ξ

∂ ∂ ξ ∂ ξ∂ ∂

∂ ∂

0 0 1

1 1 0

2 2

3 3

cosh sinh

cosh sinh (7.101)

or

( )( )

0 0 1

1 1 0

2 2

3 3

v

v

′∂ = γ ∂ + ∂

′∂ = γ ∂ + ∂′∂ = ∂′∂ = ∂

(7.102)

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The magnetic field can now be transformed using the vector potential and the differential operator transformations. Using the vector potential definition for the magnetic field, BB AA= ∇ × whose individual components are given by,

( )i i k k jB A A= ∂ − ∂ , (7.103)

where the indices i, j, and k are permuted indices. The electric field components are,

( )0 0i i iE A A= ∂ − ∂ . (7.104)

The transformation of the electric and magnetic filed components can now be developed using the differential operators and the individual vector components.

Starting with the magnetic filed components,

( )( ) ( )

1 1

2 1 3 3 1

1 0 3 3 1 0

2 3

3 3 2

.

,

cosh sinh cosh sinh ,

cosh sinh .cosh sinh .

B B

B A A

A A A

B E

B B E

′ =

′′ = ∂ − ∂

= ∂ ξ + ∂ ξ − ∂ ξ + ξ

= ξ + ξ′ = ξ − ξ

(7.105)

These transformations can be rewritten as,

( ) 1 1

1,2 1,2

.

.

B B

B v

′ =

= γ + ×B EB E (7.106)

The electric field components transform in a simpler manner,

( ) ( )( )( )

1 1 0 0 1

0 0 1

0 1 1 0

,

cosh sinh cosh sinh

sinh sinh cosh sinh ,i

i

E A A

A A

A A

E

′ ′ ′ ′ ′= ∂ − ∂

= ∂ ξ + ∂ ξ ξ + ξ

− ∂ ξ + ∂ ξ ξ + ξ

=

(7.107)

The transformation of Maxwell’s (III) equation,

∇ × − =BB EE∂0 0 , (7.108)

can be given as,

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( )2 3 3 2 0 1 0B B E′ ′ ′ ′ ′ ′∂ − ∂ − ∂ = (7.109)

which transforms as,

( ) ( ) ( )1 3 2 3 2 3 0 1 1 0B vE B vE v Eγ∂ − −γ∂ + − γ ∂ + ∂ = (7.110)

which can be rewritten as,

( )0 1,2,30vγ ∇ × − ∂ − γ ∇ ⋅ =B E EB E E (7.111)

Since ∇ × − =BB EE∂0 0 and ∇ ⋅ =EE 0 .

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Out yonder there is this huge world, which exists independently of us human beings and which stands before us like a great, eternal riddle, at least partially accessible to our inspection and thinking.

— Albert Einstein [Eins79]

Copyright 2000, 2001 8–1

§8. HAMILTONIAN MECHANICS

In order to proceed from the classical formulation of Maxwell's electrodynamics to the quantum mechanical description a new mathematical language will be needed. In the previous sections the electromagnetic field was described using partial differential equations — Maxwell's equations — for the field components and their vector and scalar potentials. These equations provided the basis for the development of the equations of motion of charged particles embedded in the electromagnetic field. However these equations of motion were simplified descriptions of the actual motions of large numbers of charges in a conducting material. In this previous formulation, the electromagnetic field was an abstract mathematical entity. This approach was a consequence of the classical nature electromagnetism since the field is treated as an ethereal entity that serves as the medium to carry electromagnetic waves, their energy and momentum.

Special methods have been developed by Lagrange in order to deal with the large — possibly infinite — number of particles. By formulating Newton’s 2nd Law in terms of the kinetic and potential energy as functions of the coordinate system in which the particles are moving. Lagrange succeeded in generalizing the use of the coordinates. [1] This approach allows the equations of motion to be isolated from a specific coordinate system, which in terms allows the variational principal to be applied to a variety of problems including the description of the electromagnetic field and it quantum mechanical formulation.

The formulation of the electromagnetic field can be restated in terms of Hamilton's theory of mechanics using the electromagnetic field's vector potential as a starting point [Heil81]. This method provides a classically consistent transition to the quantum mechanical description of the effect of the electromagnetic field on charged particles. In order to proceed with this formulation several new concepts must be presented. The quantum nature of matter will be briefly described followed by the description of Hamiltonian mechanics. The formulation of the equations of motion was first used in classical mechanics [Gold55], but now serves as the introductory method to quantizing the electromagnetic field.

1 Joseph–Louis Lagrange (1736–1813) was a professor of mathematics in Turin

Italy at age 19 he succeeded Leonhard Euler (1707–1783) in the chair of mathematics at the Berlin Academy in 1766 and moved to the Paris Academy in 1787. his monumental work Mécanique Analytique published in 1788 was a comprehensive treatise of Newtonian mechanics. Lagrange showed that the general solution of an nth order linear homogeneous differential equation is a linear combination of n–independent solutions.

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§8.1. NEWTON’S EQUATIONS IN LAGRANGIAN FORM

Special methods have been developed by Lagrange in order to deal with the large – and possibly infinite – number of particles to be described by the equations of motion. By formulating Newton'’ 2nd Law in terms of the kinetic and potential energy as functions of the coordinate system in which the particles are moving. Lagrange succeed in generalizing the use of coordinates.

This approach allows the equations of motion to be isolated from a specific coordinate system, which in turn allows the variational principal to be applied to a variety of problems including the description of the electromagnetic field and its quantum mechanical formulation.

Isaac Newton formulated the laws of motion using a calculus of his own invention. Using the Cartesian coordinate system, Newton’s equations for the thi particle with mass im are:

⎫= ⎪

⎪⎪

= =⎬⎪⎪

= ⎪⎭

K

2

2

2

2

2

2

1,2, ,

ii i

ii i

ii i

d xm X

dtd y

m Y i ndtd z

m Zdt

(8.1)

where, iX , iY and iZ are the three components of the force acting on the thi particle.

The transformation of the equations from Newtonian form to the Lagrangian form will be make use of both the kinetic and potential energy definition in the Cartesian coordinate system. The kinetic energy, T, is defined as:

( ) ( )

( )=

= + + + + + +

= + +∑

& & & & & &L

& & &

2 2 2 2 2 21 1 1

2 2 2

1

1,

2 21

.2

nn n n

n

n i i ii

mT x y z x y z

m

m x y z (8.2)

If only conservative systems of particles are considered, then the potential energy, V, can be defined as a function of the coordinates

K1 1 1, , , , , ,n n nx y z x y z of all the particles. In this approach the force experienced by each particle is equal to the partial derivative of the potential energy, such that,

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⎫∂= − ⎪∂ ⎪

⎪∂ ⎪= − =⎬

∂ ⎪⎪∂

= − ⎪∂ ⎪⎭

K1, 2, ,

ii

ii

ii

VX

x

VY i n

y

VZ

z

. (8.3)

These equations can now be used to restate Newton’s equation of motion. By removing the references to the individual coordinates, the notation for the equations of motion can be simplified. Assuming that the force applied to the particle can be found from a potential, which is a function of both position and time, ( ),V tr , according to the relationship

( ) ( )= −∂ ∂, ,F t V tr r r. Substituting this expression into Newton’s

equation of motion gives,

( )

−

∂+ =

∂&&

14 2 43

,0

F

V tm

rr

r. (8.4)

Since the momentum of the particle is = &&p mr, Newton’s equation can be rewritten as,

( )

−

∂+ =

∂&&

&14 2 43

,0

mxF

V tp

r

r. (8.5)

Since the mass of the particle, m, is a constant, the momentum can be rewritten in terms of the particle’s kinetic energy,

( )( )∂⎛ ⎞∂

= = =⎜ ⎟∂ ∂⎝ ⎠&

&&& && &1 4 2 431 4 2 43

22 2

2pm

md d m dp m

dt dt dtr

rrr

r r (8.6)

By defining the kinetic energy as, = &2 2T mr , and using this expression to eliminate the momentum p from Newton’s Law gives,

−

∂ ∂+ =

∂ ∂&

&1 2 30

Fp

d T Vdt r r

. (8.7)

The kinetic energy is now a function of &r but not r. The potential energy V as a function of r but not &r . This allows the Lagrangian to be

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defined again as using the relations ∂ ∂ = ∂ ∂& &L Tr rand ∂ ∂ = −∂ ∂L Vr r as,

( ) ( ) ( )= −& &,L T Vr r r r . (8.8)

This now allows Newton’s Law to be stated as the Euler–Lagrange equation (in Cartesian coordinates),

( ) ( )∂ ∂

− =∂ ∂&

& &&1 4 2 4 3 14 2 43

, ,0

m F

L L rddt

r

r r r

r r. (8.9)

So far the changes from the description of Newton’s Law as the simple equations of motion, to the Lagrangian description has not simplified anything. In the next sections, the Lagrangian description of the motion of a particle will be used to remove the dependence on the Cartesian coordinate system. In addition the Hamiltonian description of the particles motion will be developed. This description will used as the basis for the quantum mechanical description of the electromagnetic field interacting with charged matter.

§8.2. VARIATIONAL DESCRIPTION OF THE EQUATIONS OF MOTION

The equations of motion of an object moving in a Cartesian coordinate system was first described by Isaac Newton. In Newton’s mechanics the motion of a particle is uniquely determined by the vectorial force acting on the particle at every instance of time [Lanc70], [Byro70], [Byro69], [Chan95]. In Newton’s mechanics the action of a force is described by the momentum produced by that force. There are other descriptions of the action of a force. One such description was provide by Gottfried Wilhelm Liebniz (1646–1716), who was a contemporary of Newton’s. Leibniz’ formulation included a quantity knows as vis viva (Latin for living force) which in modern terms is call the kinetic energy [Asim66]. Leibniz replaced Newton’s momentum by the kinetic energy and replaced Newton’s force by the work of the force. This work of the force was later replaced by the work function. Leibniz is now credited with founding a second branch of mechanics — analytical mechanics, which is based on the maintenance of the equilibrium between the kinetic energy and the work function. In modern terms the force function is replaced by the potential energy. This approach laid the foundation for the Principal of Least Action.

It is convenient to divide the development of classical mechanics into three periods, the first based on Newton's Philosphiae Naturalis Principia Mathematica published in 1687 [Cajo62], the second based on

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Lagrange's Mécanique Analytique (Analytical Mechanics) published in 1788 [Lagr88] and the third based on Hamilton's General Method of Dynamics published in 1834 and 1835 [Hami34] and well as Carl Gustav Jacob Jacobi's (1804–1851) Vorlesunger über Dynamics published by Clebsh in 1866. These works established mechanics as a mathematical science complete with theoretical explanation of the behavior of objects and like the previous descriptions of Fourier's mathematical physics works, formed a paradigm for the methods used by Maxwell and the description of electromagnetic phenomenon.

On New Year’s Day 1697, Johann Bernoulli (1667–1748) of the University of Basal posed the question to the sharpest mathematicians in the whole world — given two points A and B in a vertical plane, find the path A–M–B which the movable particle M will traverse in the shortest time, assuming the acceleration on M is due solely to gravity. Using Bernoulli’s description [Stru86], the curve ACEDB shown in Figure 8.1, has a path of least time from A to B. Letting C and D be two points on the curve, Bernoulli said CED must have the same path of least time.

This is the essential point of Bernoulli’s argument and the power of this development in modern physics. Any curve which has a minimum property globally (in the large) must also have this property locally (in the small). If it were not the least time path than there would be some other path CFD which would be faster. If that were the case, the new path ACFDB would be faster than the path ACEDB, which would be contrary to the original hypothesis.

The result is Bernoulli’s contribution to modern physics..

…the path quickest overall must be the quickest in between any intermediate points, and the property which holds globally most also hold locally.

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A

C

D

E

F

B

Figure §8.1 — Bernoulli’s curve which describes the least time path between to points A and B. The mathematical problem was posed to the world’s mathematicians as a challenge. Although the solution was already known to Bernoulli, a simple solution was also known to Newton, who did not respond to Bernoulli’s challenge.

This problem is known as the brackistochrome problem — brachistos = shortest and chronos = time — and it marks the beginning of the general interest in the calculus of variations [Byro69], [Chan95], [Stru86], [Reid69]. [2] Using Bernoulli’s approach the local principal

2 When Bernoulli first issued the challenge there were no responses. He forwarded

the problem to Charles Montagu ( – ), who was the president of the Royal Society. Isaac Newton responded to the question with an anonymous solution in a letter dated Jan 30, 1697. The results were published in Philosophical Transactions, for January 1696/7. Although Bernoulli is given credit for the solution to his problem, Newton’s solution was recognized by Bernoulli in a letter to Basange de Beauval…

…although it’s author, in excessive modesty, does not reveal his name, we can be certain beyond any doubt that the author is the celebrated Mr. Newton… [Chan95]

When Bernoulli solved the brachistochrone problem he boasted of having discovered a wonderful solution, but did not publish it immediately. Instead he proceeded to challenge other mathematicians, especially his elder brother, Jacob (1654–1705). Bernoulli carried on a bitter feud in which he publicly characterized his brother as incompetent. He finally published his solution in 1697 which described the motion of a bob traveling on a cycloid path. Before Bernoulli published his work Huygens had discovered that a mass point oscillating without friction under the influence of gravity on a vertical cycloid has a period independent of amplitude. This cycloid was called a tautochrone with Bernoulli's discovery, this curve was renamed the brachistochrone [Cour56].

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allows the integral equations of motion to be transformed into differential equations of motion. The result is a greatly simplified method of analyzing the motion of the particle M along the path of least time.

In any change which occurs in nature, the sum of the product of each body multiplied by the space it traverses and by its speed (referred to as “the action”) is always the least possible.

— Pierre Louis de Maupertuis [Maup46], [Doug90].

The development of analytical mechanics is associated with Leonhard Euler (1707–1783), Joseph Louis Lagrange (1736–1813), Simeon Poisson (1781–1840) and William Rowan Hamilton (1805–65). It is essentially a reformulation of Newton's mechanics which allows many problems to be solved more simply.

Newtonian mechanics was founded on the concept of point masses, that is objects with no dimensional form. Newton's equations of motion are stated in terms of the Cartesian coordinates of the particle in motion. While the problems of dynamics can theoretically be solved by such means, in systems containing large numbers of particles, the integration of the equations of motion is generally too complex. Special methods were developed to deal with this complexity.

Lagrange’s approach makes use of an integral equation containing the potential and kinetic energies. The kinetic energy (T) depends on the object’s velocity =v dx dt , while the potential energy (V), depends only on the object’s position x. The form of Lagrange’s solution is the difference between the kinetic and potential energies.

Lagrange formulated the solution to the equations of motion by means of generalized coordinates, i.e. any set of variables sufficient in number to define unambiguously the configuration of the system. The generalized coordinates in the Lagrange and Hamilton descriptions of motion utilizing the expressions for kinetic and potential energy as functions of these coordinates.

In the classical description of motion, two measurable quantities of a particle in motion are its spatial position and momentum. If these values are known for any point in space and time, the particle’s motion or path can be calculated from Netwon’s second law of motion and knowledge of the external force law acting on the particle. If the particle’s motion is observed over a small portion of its path its momentum is nearly constant. The product of the particle’s momentum and small distance is called the increase in the particle’s action. This action is a scalar quantity

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that the particle carries with it and increases as the particle moves along its path. [3]

§8.3. CALCULUS OF VARIATIONS

The principal of stationary action appeared in Hero of Alexandria’s (62 A. D.) Cataptrica (Optics) which described the reflection of light from a plane mirror as the shortest path taken. Pierre de Fermat (1601–1665) reformulated this concept as the principal of Least Time in 1657. Fermat stated that a light ray required the least time even if deviated from the shortest physical path,

... nature operates by the simplest and most expeditious way and means.

Fermat’s principal was capable of producing the correct law of reflection and lead to the law relating the angle of incidence and reflection at an interface to the ratio of the refractive indices of the media. The relationship was confirmed experimentally by Willebrord Snell van Royen (1591–1626) in 1621 and is known as Snell’s law.

The calculus of variations and the principle of least action combine to form a powerful method of investigating problems in dynamics. Pierre–Louis Moreaude Maupertuis (1698–1759), the author of the Principle of Least Action in 1774, declared it to be a metaphysical principle on which all canons of motion are based. [4] The Newtonian equations of motion can

3 This description of action differs from the original concept developed by P.

Maupertuis who proposed that brachistochrome problem could be better solved by not considering the transit time of the movable particle, but rather by a quantity called action. Maupertuis incorrectly defined this action as the product of the distance the particle travels and its speed [Motz89]

4 Eighteenth century philosopher scientists learned to compute the paths taken by planets and objects using Newton's equations of motion. A French geometer and philosophea, Pierre–Louis Moreau de Maupertuis [Maup46] along with Joseph Louis Lagrange showed that the paths taken by these objects are always the most economical when the kinetic and potential energy are computed as a single quantity. In the way the moving object minimizes it action — a quantity based on the objects velocity, mass and the space through which it travels. No matter what forces were applied to the object, it somehow choose the cheapest of all possible paths. Unlike the total energy of an object — its kinetic and potential energy — which are always conserved, the quantity of action is constantly changing. No matter what value the action may assume during the objects flight, at the destination the action will a minimum of all the possible actions that could have occurred. In this view of mechanics, the object seems to choose its path, with the knowledge of all possible paths — at the beginning of the motion. Maupertuis wrote...

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be written in a form which makes the transition to quantum mechanics appear natural [Byro69].

The concept and principle of least action were generalized by Hamilton to include the propagation of light as well as the motion of particles. By placing a restriction on the definition of action Newtonian mechanics can be transformed into quantum mechanics. Newtonian mechanics assumes that a particles motion can be followed in infinite detail and infinite precision. If this were possible than the motion of a particle could be described by the particles position and momentum at a single point in space and time.

This process would be observable if all physical entities were infinitely divisible. However if nature is somehow limited in its divisibility than the action during a process can change only by a finite amount h , than the precise determination of a particle’s motion can never be determined. In order to determine the particles motion precisely, the momentum and position must be known at the same point in space and time.

Since the action is the product of momentum and a measured spatial interval that must be taken as infinitesimal, the action becomes infinitesimal and thus smaller than some limit h . The result is that the momentum becomes infinite, losing all knowledge of the particles action. The result is that the particle’s action becomes quantized so that its position and momentum can not be simultaneously known. The full impact of this result will be developed in later sections.

§8.4. ORDINARY MAXIMUM AND MINIMUM THEORY

The calculus of variations has been an important branch of mathematical physics for nearly three centuries. The task of finding points at which a functions possesses a maximum or minimum is common in the analysis physical problems. In the calculus of variations, functional forms are found in which integrals assume maximum or minimum values. These forms may contain several variables and describe multidimensional processes. Before considering maxima and minima of an integral function, the theory of the calculus of functions of a single variable will be examined.

It is not in the little details ... that we must look for the supreme

Being, but in phenomena whose universality suffers no exception and whose simplicity lay them quite open to our sight. [Glei92], [Feyn64], [Your68].

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Let ( )f x be a continuous function of a single variable, x, having a maximum or minimum value at =x a . The for a sufficiently small ε , there is a maximum at,

+ ε − <( ) ( ) 0f a f a , (8.10)

and a minimum at,

+ ε − >( ) ( ) 0f a f a . (8.11)

Taking the maximum case and assuming + ε( )f a can be expanded in positive integral powers of ε , by Taylor's theorem, gives,

+ ε − = ε + ε + ε& &&2 3( ) ( ) ( ) ½ ( ) ( )f a f a f a f a O . (8.12)

The Landau symbol, O, has the meaning: ε3( )O possesses the property that as ε → 0 , the quantity ε ε3 31 ( )O is bounded. From Eq. (8.10) and Eq. (8.12) at a maximum or a minimum the sign of

+ ε −( ) ( )f a f a is independent of the sign of ε , and so from Eq. (8.12)

=&( ) 0f a .

From Eq. (8.10) and Eq. (8.12) it follows that at a maximum &&( )f a is

negative and from Eq. (8.11) and Eq. (8.12) that at a minimum &&( )f a is

positive. Alternatively at a maximum &( )f a is a decreasing function of a and at a minimum it is an increasing function of a.

It is possible that =&( ) 0f a and that ( )f a is neither a maximum or

minimum of ( )f x . Such a condition occurs when =&( ) 0f a and =&&( ) 0f a ,

and ≠&&&( ) 0f a . It is then customary to say that ( )f a is a stationary value

of ( )f x . In general all roots of =&( ) 0f x are said to give rise to stationary values of ( )f x . With this brief background the Lagrangian formalism will developed in the next section.

§8.4.1. Lagrangian Formalism and the Calculus of Variations

The Lagrange formalism will be developed through a simple example — the motion of a particle with mass m in a harmonic oscillator potential given by = 2( ) / 2V x kx . According to Newton's second law of motion, the acceleration of the particle is determined by,

= −&&mx kx , (8.13)

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which has the well known solution,

= ω + φ0 cos( ),x x t (8.14)

where ω = k m , is the angular frequency, the constants 0x and φ are determined by the initial conditions.

Consider two times 1t when the particle is at position 1x and 2t when the particle is at position 2x . The path the particle follows between times

1t and 2t can be described by the quantity,

⎛ ⎞= ⎜ ⎟

⎝ ⎠∫2

1

,t

t

dxS x dt

dtL , (8.15)

where the difference between the kinetic energy T and the potential energy V, = −T VL is called Lagrangian. The quantity S, which in the past was called Hamilton’s Principal Function, but is now called the action function. Dimensionally, the action is an energy times a time and has similar dimensions as Planck's constant.

The action, S, is a functional of x, that is it is a function of the function ( )x t , which describes the path satisfying the two constraints that ( )x t at time = 1t t assumes the value 1x , while ( )x t at time = 2t t assumes the value 2x . Apart from these constraints the path may be arbitrary.

The action S is then a function of the different paths satisfying the two constraints. In order to formulate the extremum on S, a family of functions is considered, given by,

α = +αη( , ) ( ,0) ( ),x t x t t (8.16)

where the function ( ,0)x t is the one corresponding to the extremum. The function η( )t is arbitrary, except that is satisfies the constraints η = η =1 2( ) ( ) 0t t .

The action S is then a function α( )S of the parameter α ,

( )α = α α∫ &1

( ) ( , ), ( , ), .t

S dt x t x t t2t

L (8.17)

This expression allows for the possibility that the extremum may depend explicitly on time with the force constant of the harmonic oscillator being a function of time, = ( )k k t .

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The extremum condition is given by,

α=

∂=

∂α 0

0,S (8.18)

By differentiating Eq. (8.18) with respect to the parameter α gives,

( )⎛ ⎞∂ ∂ ∂⎛ ⎞= − η⎜ ⎟⎜ ⎟∂α ∂ ∂⎝ ⎠⎝ ⎠∫ &

2

1

t

t

S L d Lt dt

x dt x (8.19)

Integrating by parts [5] in order to replace η byη&, results in,

⎛ ⎞∂ ∂ ∂⎛ ⎞

= − η⎜ ⎟⎜ ⎟∂α ∂ ∂⎝ ⎠⎝ ⎠∫ &

2

1

( ) ,t

t

S dt dt

x dt xL L (8.20)

since,

∂ ∂ ∂⎛ ⎞ ⎡ ⎤ ⎛ ⎞η = η − η⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂⎝ ⎠ ⎣ ⎦ ⎝ ⎠

∂⎛ ⎞= − η ⎜ ⎟∂⎝ ⎠

∫ ∫

∫

&&& &

&

22 2

1 11

2

1

( ) ,

0 ( ) .

tt t

t tt

t

t

ddt t dt

x dt x

dt dtdt x

L L Lx

L (8.21)

The extremum condition Eq. (8.21) then becomes,

∂ ∂⎛ ⎞

− =⎜ ⎟∂ ∂⎝ ⎠&0,

ddt x x

L L (8.22)

since η( )t is arbitrary except for the condition η = η =1 2( ) ( ) 0t t .

Eq. (8.22) is named the Lagrange equation. The derivative of Lagrange's equation started from a consideration of the instantaneous state of the system and small virtual displacements about the instantaneous state, i.e. from a differential principle such as D'Alembert's principle. [6] It is also possible to obtain Lagrange's equations from a

5 The method of integration by parts employs the identity

( ) ( ) ( )f dg dt d fg dt g df dt= − , were f and g both are functions of t. When both sides of

this equation are integrated with respect to t over the interval from t1 to t2 the result is

( ) [ ] ( ) 1

1 1

tt t

f dg dt dt fg df dt g dt= −∫ ∫2 2

2

t tt

.

6 It is surprising that there can be several formulations of the principals of mechanics. Once it is understood that mechanics is a description of motion, then

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principle which considers the entire motion of the system between times 1t and 2t and small virtual variations of the entire motion from the

actual motion.

different methods of describing this motion can serve different purposes. Although Newton’s method of describing the motion of particles has long been the most useful simple approach other formulations have been created which attempt to simplify the solutions of various types of problems. These alternative formulations differ considerably in their concepts of mass and force.

Some are restatements of Newton’s laws, while others introduce new concepts. D’Alemberts’s principal is a restatement of Newton’s Laws which seeks to reduce dynamics to statics using Newton’s concept of mass and force. D’Alembert formulated his principal in 1743 in the work Traitê de Dynamique, which was revised in 1758 as A general principal for finding the motions of several bodies which react on each other in any fashion. In D’Alembert’s formulation the concept of virtual displacements is used to describe the motions of particles in the presence of external forces. This principal can be stated in a general form as:

... if there are n particles 1, 2, 3, ..., n acted on by forces 1 2 3, , ,KF F F ,

respectively, and if these are given arbitrary (virtual) displacements

1 2 3, , ,Kd d dr r r , where r is the position vector of the particle, the condition of equilibrium under the action of the forces is

1 1 2 2 3 3 0n n⋅ + ⋅ + ⋅ + + ⋅ =LF d F d F d F dr r r r [Lind56]

The second type of formulation employees the concept of energy, Hamilton’s principal being the one utilized here. In 1894 Heinrich Hertz published Principals of Mechanics in which he re–established the principals of mechanics with logical a consistency not found in the usual Newtonian presentations of the day. Such late 19th century works usually contained metaphysical uncertainties and vagueness. Hertz attempted to reduce dynamics to kinematics, avoiding concepts like force, mass and energy. The fundamental principal of Hertzian mechanics is:

Every free system remains either is a state of rest or in uniform motions along a straightest path [Lind56].

Since most system encountered in practical situations are non–free Hertz assumes that every part of a non–free system is part of a free system. Every motion of a free system or its non–free parts obeys the fundamental principal called natural motion, and Hertzian mechanics is only concerned with natural motion. There were serious problems with the description of motion, since in order to make the principal work it was necessary to invoke the existence of other particles, which may not be immediately discernible. The words concealed became associated with Hertz’s principals.

Even though Hertz’s concept did not lead to a particle method of computing the motions of particles it did lay the foundation for Hamilton’s principal, in which the concealed aspects of the motion becomes the energy of the system which is minimized during the particles motion.

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§8.5. GENERALIZED COORDINATES

It is not always convenient to use Cartesian coordinates when solving problems in Newtonian mechanics. Alternative coordinate systems result in simpler solutions. The analysis of the motion of a pendulum is an example. The Lagrangian formulation of the equations of motion is well suited for these non–Euclidean or constrained dynamical variables. The generalized coordinates presented in this section are not alternatives to Euclidean coordinate systems, but are descriptions of the configuration of the mechanical system with iz degrees of freedom. An example of such a system is n particles each with mass im and coordinates iz , where

= K1, 2, ,i n . By choosing any independent functions of the original 3n

dynamical coordinates iz , ( ) ( ) ( )( )= = =i j iq q t q t q z t , where

= K1, 2, , 3j n , the new dynamical variables can be defined. These new variables are the generalized coordinates, ( )= iq q t and the generalized

velocities, ( )=& iq dq t dt [Doug90].

It is straightforward to generalize the Lagrange formalism to systems with more degrees of freedom than are found in classical mechanics. This may be done by considering a system described by the set of generalized coordinates rq , where r assumes the values K1, 2, , s . As before, the Lagrangian is the difference between the kinetic and potential energies is given by = −T VL .

The Lagrange equations are derived by requiring the action,

( )= ∫ & & &K K K K1

21 2 1 2, , , , ; , , , , ;

t

r rtS q q q q q q t dtL , (8.23)

to have an extremum. The equations of motion or Lagrange's equations, then become,

⎛ ⎞∂ ∂

− =⎜ ⎟∂ ∂⎝ ⎠&

0r r

ddt q q

L L , (8.24)

where the time derivative of the generalized coordinate is now given by,

=& dqq

dt [Whit37]. [7] This formulation of Lagrange's equations is most

7 For a particle moving in a time independent potential, the Lagrangian does not

depend explicitly on time. If the system is isolated in such a manner so that time transformation invariance is preserved, the Lagrange equations can be shown without an explicit time dependence and written as, L( , ! )q qi i .

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powerful for theoretical purposes and will be used in the development of the electromagnetic field Hamiltonian. The most significant property of Eq. (8.24) is its invariance with respect to arbitrary coordinate transformations. Proof of this important attribute is given in [Your68], pp. 37–39.

The Lagrangian of a system is not unique. The total time derivative of an arbitrary function can be added to the Lagrangian L to give a new Lagrangian ʹ′L , such that,

( ) ( ) ( )ʹ′ = + && &, , , , ,r r r r rq q t q q t f q tL L , (8.25)

which has the same properties as the initial Lagrangian L with respect to the principle of least action. The action function ʹ′S relative to the Lagrangian ʹ′L is given as,

( ) ( )ʹ′ ʹ′= = + −∫2

1

2 2 1 1( ), ( ),t

r rt

S dt S f q t t f q t tL . (8.26)

Since the initial and final positions are fixed S and ʹ′S have the same extremum and differ only by a constant.

§8.6. HAMILTONIAN FORMALISM

If the only use of the Lagrangian action principle is to regenerate the equations of motion it would be considered interesting but redundant. However, the Lagrangian action principle provides a description of the dynamics of a system which contains more information than supplied by Newton's equations of motion. First the action function, S, is a global statement about the system, from which a local differential equation can be derived by imposing the extremum condition. The action is global in the sense that it receives contributions from the entire trajectory of the particle in motion. As such the action records the history of the particle's motion. Second, although the action is extremized by the set of classical trajectories that are the solution to Eq. (8.24), the action can be evaluated for any trajectory. Third, the action approach allows the definition of the canonical coordinates of position and momentum to be generalized by describing the energy of the system through the Hamiltonian.

The solution of a dynamical problem by Lagrange's method requires the integration of n second–order differential equations in the n unknowns K1, , nq q . An alternative system proposed by Hamilton consists of 2n first–order differential equations in 2n unknowns, and has the

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advantage that it is simple and concise in its formulation [Hami35] [8]. In Hamilton's equations, the canonical equations constitute the basis for the quantum mechanical formulation of electrodynamics. An original concept introduced by Hamilton is the generalized momentum, which is defined as,

δ ∂

≡ =δ ∂& &

( )( ) ( )rr r

Sp t

q t q tL

. (8.27)

Defining the generalized momenta as = &p mx allows the introduction of the Hamiltonian by the transformation,

( ) ( ) ( )= −& &, ( , )r rp q p q p q q q p qH , L , . (8.28)

The differential ddtH

is,

∂

= + − = − + = −∂

& & & &dpdq qdp d pdq qdp

dt tH LL , (8.29)

since,

∂

=∂

&pqL

. (8.30)

It follows from Eq. (8.29) that the equations of motion for the system may be written as,

8 Hamilton was an astronomer and mathematician in Dublin, Ireland. As a child

prodigy he was able to translate from Latin and Greek at age 5 and had mastered 13 languages by age 13. He studied at Trinity College, Cambridge and was appointed professor of astronomy at age 22. Hamilton’s works include mechanics and optics as well as the discovery of quanternions, which generalize complex numbers to a non–communtative algebra. Although Hamilton died 35 years before Planck published his theory of quantum mechanics, Hamilton has been immortalized through his association with the energy operator in Schrödinger’s wave equation.

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⎛ ⎞∂ ∂≡ ⎜ ⎟

∂ ∂⎝ ⎠

⎛ ⎞ ⎛ ⎞∂ ∂= −⎜ ⎟ ⎜ ⎟

∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂= − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂= − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

= −

∑

∑ ∑

∑ ∑

&

&

& &&

& &&

&

,

,

,

,

.

r r p

ss

s r rp p

s ss

s ss r s rq p

s ss

s ss r s rq p

r

q q

qp

q q

q qp

q q q q

q qdp

q dt q q q

p

H H

L

L L

L L

[9] (8.31)

⎛ ⎞∂ ∂≡ ⎜ ⎟

∂ ∂⎝ ⎠

⎛ ⎞ ⎛ ⎞∂ ∂= + −⎜ ⎟ ⎜ ⎟

∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂= + −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

=

∑

∑ ∑

&&

& &&&

&

,

,

,

.

r r q

sr s

s r rq q

s sr s

s ss r s r qq

r

p p

qq p

p p

q qq p

q p q p

q

H H

L

L L

(8.32)

which constitute Hamilton's equations. The change of variables from &( , )q q to ( , )q p results from the transformation in Eq. (8.31) which is

known as a Legendre transformation. [10]

9 In this derivation of the Lagrange equations the index subscript r or s is used to

indicate the rth or sth coordinate, in order to distinguish between each generalized coordinate in a multidimensional coordinate space. The suffix to the bracketed derivative indicates that the q's are kept constant while the index is summed over the s's.

10 The change in basis from ( , ! , )x x t to ( , , )x p t is accomplished through the Legendre transformation. Consider a function of the variables f x y( , ) so that a differential of f has the formdf udx vdy= + , where u f x= ∂ ∂ and v f y= ∂ ∂ . To change the basis of the description from x y, to the independent variables u y, , so that differential quantities are expressed in terms of the differentials du and dy . Let g be a function of u and y defined by the equation g f ux= − . A differential of g is then given as dg df udx xdu= − − which has the desired form. The quantities x and v are now functions

of the variables u and y by the relations ( )x g u= − ∂ ∂ , ( )v g y= − ∂ ∂ , which are the

converse of the above relations for u and v.

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Restating the equations of motion in terms of the coordinates and the momenta results in the Hamiltonian of the system,

= −∑ & &( , , ) ( , , )r r r rp q t p q q q tH L , (8.33)

which describes the dynamics of the system in terms of the sum of the kinetic energy and the potential energy. Hamilton's principle requires that the path taken by any physical system between two states at specified times and with fixed values of the variables must be such that

the value of the function δ −∫1( )

o

t

tT V dt must be an extremum. In this form

Hamilton's principle is sufficient to generate both the equations of motion of the system and the boundary conditions for any continuous field with localized forms of energy. [11]

For the actual solution of problems, the equations of Lagrange are more convenient than those of Hamilton, since the first step in integrating Hamilton's equations is to reduce their number by half, an operation which leads back to the original Lagrange equations.

The dominate position of the equations of Lagrange in the history of dynamics can best be cited in Hamilton's own words.

The theoretical development of the laws of motion of bodies is of such interest and importance, that it has engaged the attention of all the most eminent mathematicians, since the invention of dynamics as a mathematical science by Galileo... Among the successors of those illustrious men, Lagrange has perhaps done more than any other analyst, to give extent and harmony to such deductive researches, by showing the motions of systems of bodies may be derived from one radical formula; the beauty of the method so suiting the dignity of the results, as to make his great work a kind of scientific poem. [Hami34].

§8.6.1. Canonical Coordinates and Poisson Brackets

The Hamiltonian and the Lagrangian are related as,

⎛ ⎞∂ ∂⎛ ⎞

= −⎜ ⎟ ⎜ ⎟∂λ ∂λ⎝ ⎠⎝ ⎠ & &.

,, p qp q

H L (8.34)

11 For physical applications, L( , ! , )x x ti i must be chosen so that the Euler–Lagrange

equations represent the correct equations of motion.

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The Hamiltonian equations of motion are also known as the canonical equations, resulting in,

∂ ∂ ∂ ∂= = − = −∂ ∂ ∂ ∂

& &; ; .r rr r

q pp q t tH H H L (8.35)

The Hamiltonian formulation provides an elegant description of mechanics in which the position and momentum of each particle is treated as though they were independent quantities. The coordinates rq and the momentum rp are actually allowed to be more general than just Cartesian coordinates. In particular Hamilton's equations form the basis of the quantum formulation of Maxwell's equations, since the field potential becomes an integral part of the canonical momentum and is treated as if it were a generalized coordinate in the Lagrange equations. [12]

Hamilton’s equations can be restated using the Poisson bracket notation. For a system of s generalized coordinates and s generalized momenta, the Poisson bracket can be defined for any two functions ( ),u p q and ( ),v p q is an antisymmetric operation given as,

=

∂ ∂ ∂ ∂= −

∂ ∂ ∂ ∂∑1

,s

i i i i i

u v v uu v

q p q p (8.36)

This form can be rewritten as,

=

⎛ ⎞∂ ∂= +⎜ ⎟

∂ ∂⎝ ⎠∑& & &1

S

i ii i i

u uu q p

q p (8.37)

for the variation of any physical quantity u. Using Hamilton’s equations,

12 Equations Eq. (8.31) and Eq. (8.32) are also valid for a system of N particles with

coordinates ( , , , )x x x N1 2 3… . The forces between the particles can be represented by the potential energy V x x x N( , , , )1 2 3… . The Newtonian equations of motion in Eq. (8.33) may be considered as Euler equations corresponding to the requirement that the function S should be an extremum. This alternative concept is important because it enables the equations of motion to be expressed in a form that is invariant with respect to the coordinates. The extremum requirement (Hamilton's Principle) contains only physical quantities such as kinetic and potential energy, which are independent of the coordinate system. For any arbitrary coordinate system, the momenta p xj j= ∂ ∂L ! do not in general

have dimensions of true momentum, with the coordinate x j being a dimensionless

angular quantity. The product of any momentum pj with its associated coordinate x j

always has the dimensions of action (energy time× ). The momentum pj and the

coordinate x j are said to be canonically conjugate.

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⎡ ⎤= ⎣ ⎦& ,u uH . (8.38)

Substituting the generalized coordinates of position and momentum gives,

=& ,i iq q H , (8.39)

and

=& ,i ip p H . (8.40)

Several identities result form the Poisson bracket notation that will be useful in the formulation of quantum mechanics,

First,

= δ ,,i i i jq p . (8.41)

The Poisson brackets involving only p or q vanish as,

= =, , 0i i i ip p q q . (8.42)

Quantities whose Poisson brackets are zero, commute, and those whose Poisson brackets are equal to 1 are canonically conjugate. From Eq. (8.39), it can be seen that any quantity that commutes with the Hamiltonian does not vary with time.

Using Eq. (8.36), if the Poisson bracket of a function u with a constant c gives,

=, 0u c , (8.43)

and,

= −, ,u v v u . (8.44)

Using the rules of differentiation,

+ = +, , ,u v w u w v w , (8.45)

and,

= +, , ,u vw u v w v u w . (8.46)

Using the Poisson brackets and Hamilton's equations of motion,

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∂= +∂ ,

,p q

du uu

dt tH (8.47)

and

= δ ,,,i j i jp q

q q (8.48)

It is the Poisson bracket formalism that will serve as the basis of the quantum mechanical commutator algebra developed in the subsequent sections.

§8.7. STANDARD LAGRANGIAN OF CLASSICAL ELECTRODYNAMICS

To this point in the monograph the radiation field density has been treated in form described by Eq. (4.33). As such the details behind this equation have not been developed. Before proceeding with the operator approach to the quantum field equations, the Hamiltonian form of the field equations will be addressed.

The field energy described in Eq. (4.33) can be derived through the expansion of Maxwell's equations using Hamilton's equations of motion with equations ( VI ) and ( VII ) representing the electric and magnetic energy densities of the field proper. These energies are considered to reside in the field and to be localized by ( VI ) and ( VII ) in every volume element, such that a volume dv contains field energy in the amount

+∫ 2 212 ( )V

dvE B . The development of the Hamiltonian form of the field

energy, starting from Maxwell's equations is one approach. [13] Another approach is to construct the field equations and their associated Hamiltonian, by searching for the Lagrangian that results in the proper field equations. That is the approach taken here and by [Your68]. [14]

The Lagrange and Hamilton equations of motion developed in the previous section will be used to derive the Hamiltonian for the radiation field, which in turn will be used to derive the operator formulation of the electromagnetic radiation field. This approach will be worked out in detail starting with the equations of motion for a charged partial in an

13 For a more rigorous development of the radiation field energy and the associated

conservation laws derived from Maxwell's equations rather then the Lagrangian, see §11 of [Eyge72].

14 This approach depends on developing the detailed Lagrangian for the radiation field equations in both vacuum and source forms. This section and the reference [Your68] provides the lowest details of the Lagrangian approach to quantum field problems. As such it gives insight to the current research activities in particle physics and quantum field theory.

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electromagnetic field and concluding with the Hamiltonian for the same field.

In this case the forces on the particle are not derived from the field potential, but rather arise from the velocity of the particle as it travels through the field. The acceleration (change in momentum) of the charged particle is given by the Lorentz equation,

( ) ≡ = + ×&F p e vE B (8.49)

where v of the velocity of the particle, e is the particle's charge and E and B are the electric and magnetic fields. The E and B fields are derived from the vector potentials in the usual manner,

∂ ⎫

= −∇φ − ⎪∂ ⎬

⎪= ∇× ⎭

,

.tA

E

B A (8.50)

The acceleration equation, Eq. (8.49), becomes,

∂⎧ ⎫= −∇φ − + ×∇×⎨ ⎬

∂⎩ ⎭

∂⎡ ⎤= −∇φ − + ∇ ⋅ − ⋅∇⎢ ⎥∂⎣ ⎦

& &

& &

( ) ,

( ) ( ) .

p e qt

e q qt

AA

AA A

(8.51)

In order to express the equations in the Lagrangian form, Eq. (8.28), the following Lagrangian function is used,

= − φ + ∑ &r rr

T e e A qL (8.52)

The generalized momenta, using Eq. (8.32), as,

∂

=∂

ʹ′= +

&,

.

rr

r r

pq

p eA

L (8.53)

The form of Eq. (8.53) is similar to the Gauge transformations given in Eq. (4.7) and Eq. (4.8) and developed further .

Finally the Hamiltonian is still equal to the total energy, using Eq. (8.52),

= −∑ & .r rr

p qH L (8.54)

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Moving to the Lagrangian for a charged body rather than a singular point, the Lagrangian is given by,

⎛ ⎞= + ρ − φ⎜ ⎟

⎝ ⎠∑∫ .r rr

T v dVAL (8.55)

where v is the velocity of the charge at any point, ρ is the charge density and dV is an element of volume. To obtain Eq. (8.55) the charged body is considered as a system of mutually attracting particles. The v's and ρ are treated as functions of the generalized coordinates used to define the system. According to Eq. (8.55), the generalized momenta will be determined by,

∂

ʹ′= + ρ∂∑∫ &

,sr r s

s r

vp p dV

qA (8.56)

and the Hamiltonian, using the notation of Eq. (8.33) is given as,

⎛ ⎞∂

= − ρ + ρφ⎜ ⎟∂⎝ ⎠

∑∫ ∫&, .s

r r ss r

vq p dV dV

qAH H (8.57)

§8.7.1. Time Independent Lagrangian

The development of the actual electromagnetic field equations depends on a fundamental difference between the pervious equations. Up to this point the field equations contained one independent variable, t, and several dependent variables, rq . In the electromagnetic field equations both the rq 's and t are independent variables, and the quantities specifying the field are the dependent variables.

This situation can be described by considering a field defined by the quantities ( , )r rf q t . A Lagrangian, L, can be found which is a function of

the rf 's, the ∂

∂r

r

fq

's and the &rf 's, so chosen that Lagrange's differential

equations,

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⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎜ ⎟ ⎜ ⎟+ + − =

∂ ∂∂ ∂ ∂⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟∂ ∂⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎜ ⎟ ⎜ ⎟+ + − =

∂ ∂∂ ∂ ∂⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟∂ ∂⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎜ ⎟ ⎜ ⎟+ + −

∂ ∂∂ ∂⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟∂ ∂⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

L

L

L

1 1 1

2 2 2

0

0

r r

f f fq qs t qs t

f f fq qs t qs t

f fq qs ts t

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪∂

= ⎪∂ ⎪

⎪⎭

0r

fq

(8.58)

for the integral ∫ rdq dtL to be stationary resulting in the equations of the

field. Rewriting Eq. (8.58) by considering rf 's to be functions of the generalized coordinates evaluated at a specific point in space, results in the ordinary Lagrange equations for L and Eq. (8.58) for L,

∂⎛ ⎞

⎛ ⎞∂ ∂ ∂ ∂⎜ ⎟⎛ ⎞+ − =∂⎜ ⎟ ⎜ ⎟∂∂ ∂ ∂∂ ⎜ ⎟⎝ ⎠ ⎜ ⎟∂⎝ ⎠⎝ ⎠

∑& 0rs s rr

r

ft q ff q

LL L

, (8.59)

are now equivalent, that is Lagrange's equations are valid for both point charges and distributed charges moving in the electromagnetic field. The Lagrangian L, which is a function of the dynamical variables can be rewritten as,

= ∫ 3rd qL L , (8.60)

where L is now called the Lagrangian density. [15]

15 In the development of electrodynamics, the Lagrangian density is a function of

the dynamical variables ( )iA r and ( )idA dtr where i describes all the individual points

in the discrete space and r describes all the possible coordinate values. The Lagrangian density function of the coordinates ( )iA r and the velocities ( )idA dtr and the spatial

derivatives, denoted by ∂ j iA , whose presence shows that the motion of the coordinate

( )iA r is coupled to the motion of a neighboring point in the same manner the discrete

variable qi depends on qi−1 and qi+1 [Cohe89].

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§8.7.2. Lagrangian Density

In order to develop the underlying mathematics necessary for the quantum field description of electromagnetism, the standard Lagrangian will be extended. What is needed is a formalism that describes the observed phenomena of the radiation interaction with charged matter. This description must define the total Lagrangian L and maintain internal consistency, when the number of degrees of freedom becomes infinite.

The Lagrangian formalism for systems of point particles and the derivation of the Hamiltonian provides an easy transition to quantum mechanics. The systems presented so far consist of a finite number of variables. Although there are many physical systems with a finite number of degrees of freedom, the electromagnetic field is not one of them. There are other cases such as gases or liquids all of which have one or more variables which are functions of continuous variables. There are various methods of transforming a discrete system to a continuous one. One method is to consider a continuous linear elastic structure as the limit of a system of point particles and then to generalize the results. A second method is to construct a generalized variational principle and the third method is to employ the Fourier transform to construct a generalized set of variables in Fourier space. There are several advantages to the third approach. First the continuous system which was a function of the continuous variable x is transformed to a discrete system of variables with an index of k, as long as the system is enclosed in a finite volume.

By combining the first and third approach — using the Fourier spatial description of a linear elastic medium — a transition to the quantum field description can be made. The starting point starting point for this new Lagrangian will be the same as the transition from classical to quantum mechanics — Classical Hamiltonian dynamics. A generalized approach can be formulated by letting ( )φ r represent the displacement

These spatial derivatives are not new independent variables but are linear

combinations of generalized coordinates. In the study of electromagnetic theory the Lagrangian density takes on the form ( ), ,i i j iA dA dt A∂L .

The Lagrangian density that is used in electrodynamics contains generalized coordinate derivatives. Such a structure allows Maxwell's equations to be describe the motions of fields coupled from point to point in space. The absence of these spatial derivatives in the Lagrangian density would lead to a theory where the electromagnetic field evolves independently at each point in space. Since Maxwell's equations involve the spatial derivative of the field, the Lagrangian density also depends on spatial derivatives.

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amplitude of the field at a point r. This results in the field having an infinite number of degrees of freedom which must be specified at each point where ir where →∞i .

For a continuous space the summation in Eq. (8.59) becomes infinite and must be replaced by the integral in Eq. (8.60). In this way ( )rL

depends on the amplitude of the field at or near the point r. This amplitude might be a function of ( )φ r itself and must contain the time

derivatives of φ , just as the Lagrangian of a particle contains the kinetic energy, which is a function of velocity. The Lagrangian density L must also depend on the spatial derivatives of ( )φ r , otherwise there would be

no connection between the field amplitudes at neighboring points in space.

The system is more easily quantized as a discrete formulation, since the Fourier coefficients can be directly introduced as creation and annihilation operators. There are some difficulties with this approach, but they will be dealt with in the section on gauge theory. Because of the difficulty a simplified mechanical example will be used in which a longitudinal wave in one dimension is used to illustrate the idea.

Starting with the one dimension longitudinal wave described by the wave equation,

∂ φ ∂ φ

ρ −µ =∂ ∂

2 2

2 2 0t x

, (8.61)

where ( )φ ,x t is the displacement of point x at time t. The density of the

elastic medium is ρ and the restoring force of the media is µ . The length of the one dimensional media is L which requires ∂φ ∂x vanishes at the boundaries. Given these conditions, the displacement function can be expanded in a Fourier series as,

( ) ( )ʹ′φ = φ∑, sinkx t t kx , (8.62)

where k has the periodic values π = K, 0, 1, 2,n L n The periodic boundary condition ∂φ ∂ = 0x can now be replaced by,

( ) ( )φ + = φ, ,x L t x t . (8.63)

Another simplification is to expand the Fourier series is an exponential such that,

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( ) ( )φ = φ∑1

, ikxk

k

x t t eL

, (8.64)

where k is now the wave number which can be positive as well as negative. Since the Fourier expansion given in Eq. (8.64) involves complex numbers and ( )φ ,x t is a real quantity in terms of φk and −φ k

are related to each other through this complex conjugate ∗−φ = φk k

resulting in k independent variables.

The thk instance of φ can be obtained from the continuous function through Fourier transform,

( )φ = φ∫1 ikx

k x e dxL

. (8.65)

By letting the extent of the medium tend to infinity →∞L and using the following limits,

→π∑ ∫2k

Ldk , (8.66)

and,

( )πφ → φ

2k k

L. (8.67)

The Fourier transform pairs can be given as,

( ) ( )φ = φπ ∫1

, , ,2

ikxx t k t e dx (8.68)

( ) ( ) −φ = φπ ∫1

, , .2

ikxk t x t e dx (8.69)

The Fourier transform of the wave equation now becomes,

∂φ

ρ + φ =∂

22 0k

kkt

(8.70)

which is now the equations of motion of the system, but containing an infinite number of degrees of freedom. These equations can be derived from the Lagrangian,

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( ) − −φ φ = ρ φ φ − µ φ φ∑ ∑& & & 21 12 2,k k k k k k

k k

kL . (8.71)

Using the relations given in Eq. (281) and Eq. (280) these transitions from the discrete formulation using φk to the continuous formulation

using ( )φ x can be made.

Considering the first term of the Lagrangian,

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

−

−

ρ φ φ = ρ φ φ −

= ρ π φ φ −

= ρ φ φ

= = φ φ

∑ ∫

∫ ∫∫

∫ ∫

& & & &

& &

& &

& &

1 12 2

12

12

12

,

2 ,

,

.

k kk

ikx

k k dk

x dx e k dk

x x dx

x dx x xT

(8.72)

which is the kinetic energy density. The second term,

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

−

−

−

−

µ φ φ = µ φ φ −

= µ π φ φ −

⎛ ⎞∂= µ π φ − φ −⎜ ⎟∂⎝ ⎠

⎛ ⎞∂= µ π − φ φ⎜ ⎟∂⎝ ⎠

⎛ ⎞∂= µ − φ φ⎜ ⎟∂⎝ ⎠

⎛ ⎞∂ φ= µ ⎜ ⎟∂⎝ ⎠

=

∑ ∫

∫ ∫

∫ ∫

∫ ∫

∫

∫

∫

2 21 12 2

212

212 2

212 2

212 2

212 2

,

2 ,

2 ,

2 ,

,

,

.

k kk

ikx

ikx

ikx

k k k k dk

x k e k dxdk

x e k dxdkx

x dx e k dkx

x dx xx

dxx

x dxV

(8.73)

which is the potential energy density.

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Frustra fit per plura, quod feiri potest per pauciora

OR Essentia non sunt multiplicanda praeter necessitatem

(It is vain to do with more what one can do with less) OR

(Entities are not to be multiplier beyond necessitity) [16]

— Occam’s Razor

16 Attributed to William of Occam, or Ockham, or probably Oakham in Surrey

(1300–1349), Oxford scholar in the Order of Franciscan Friars. Occam’s razor is widely used in scientific analysis with an interpretation akin to: One should always choose the simpler of two otherwise competing descriptions of physical phenomena. [Doug90].

Copyright 2000, 2001 9–1

§9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD

In the previous section the Lagrangian and Hamiltonian of an ensemble of point particles was developed. This approach is based on a discrete set of coordinates ( )iq t . This discrete formulation can be extended to a continuous formulation. Each point in a region of space – either finite or infinite will be associated with a continuous variable,

( ),x tφ . (9.1)

The set of variables constitute a system with an infinite number of degrees of freedom – a field. In this formulation, the Lagrangian of the field variable is functional of the field. A functional is a mapping from a space of functions, the Lagrangians, to a set of real numbers. This mapping is given by,

( ) ( ) ( ), , ,L t L x t x t = φ φ & . (9.2)

The functional ( )L t , depends on the value of φ and φ& at all points in space at simultaneous time. Hamilton’s principal can now be applied to this functional by defining a variation of the functional as,

[ ] [ ] [ ]

[ ]( ) ( ) 3

,

.

F F F

Fx d x

x

δ φ = φ+δφ − φ

δ φ≡ δφ

δφ∫ (9.3)

The term in Eq. (9.3) [ ] ( )F xδ φ δφ is called the functional derivative of the functional [ ]F φ with respect to the functional φ at a spatial point x. This derivative decribes how the functional is changing when the values of the function φ is varied at the point x.

The functional derivative is formally defines as,

( )

( ) ( ) ( )0

limF g x x y F g xF

g y ε→

+εδ − − δ =δ ε

(9.4)

where ( )x yδ − is the Dirac delta function. The derivative in Eq. (9.4)

displays all the properties of a standard derivative. [1]

1 If F and G are two functionals, than their product is a functional derivative defined

as, [ ] ( ) ( )( ) ( )( )FG g x G F g x F G xδ δ = δ δ + δ δ , which is the Leibniz property. If F g is

a functional that is well behaved in the interval in the function space around g = 0 , than

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The functional derivative can now be applied to the Lagrangian in Eq. (9.3) to give,

( )

( )( )

( ) 3, L LL x x d xx x

δ δ δ φ φ = δφ + δφ δφ δφ ∫& && (9.5)

The Lagrangian can now be integrated to produce the action function, usually denoted by ,W φ φ

& , which is a functional of φ and φ& . By

integrating over the time interval 1t = −∞ to 2t = +∞ to give,

( ) ( ) ( ) ( )

( ) ( ) ( )

2

1

2

1

2

1

3

3

, ,

, , , ,, ,

, ., ,

t

t

t

t

t

t

W L dt

L Lx t x t dt dxx t x t

L L x t dtdxx t t x t

δ = δ φ φ

δ δ= δφ + δφ

δφ δφ

δ ∂ δ= − δφ δφ ∂ δφ

∫

∫

∫

&

&&

&

(9.6)

Since Hamilton’s stationary function is,

( )2

1

, , 0t

t

W L dt δ φ φ = δ φ φ = ∫& & , (9.7)

the Euler–Lagrangian can be generalized to Classical Field theory as,

0L Lt

∂ ∂ δ− =

∂φ ∂ δφ& . (9.8)

Applying the principals developed in §8 to Maxwell’s equations began as early as the 1870’s. Maxwell made use of Hamilton’s principal in his Treatise [Buch85]. By formulating the electromagnetic field as a continuous dynamic medium, Maxwell built on the ideas put forth by William Thomson and Peter Tait in [Thom62]. Thomson and Tait considered Hamilton’s principal as a fundamental formulation of physics…

the functional F can be expanded in a Taylor series

[ ] [ ]( ) ( ) ( )( )1 2 11 ! , , ,0

n

n nF g n dx dx dx F g g x g xg

= δ δ δ=

∑ ∫ … L .

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…(the) principal of Least Action is a useful guide to kinetic investigations. We are strongly impressed with the conviction that a much more profound importance will be attached to it … in the theory of several branches of physical science now beginning to receive dynamical explanation … his method of Varying Action which undoubtedly become a most valuable aid in the further generalization [Thom62] Vol2. §326.

Restating Hamilton’s principal –– requires that the path taken by a physical system between two states at specified times and with fixed values of the variables at these times such that the value of the function

( )T V dt−∫ , where T is the kinetic energy V is the potential energy, must

be an extermimum such that,

( )1

0

0t

t

T V dtδ − =∫ .

In this form Hamilton’s principal is capable of generating both the equations of motion of the electrodynamic system and the boundary conditions for any continuous field with localized forms of energy. By adopting Hamilton’s principal as the fundamental formulation of electromagnetics, then every problem in field theory reduces to finding an appropriate expression for the field’s potential and kinetic energies. [2] The essence of the Lagrangian or Hamiltonian approach consists of two parts: (i) the specification of the generalized coordinates which fix the state of the field and (ii) a choice of expressions, in terms of these coordinates and their spatial and temporal derivatives.

To make use of the results developed in §8, the Lagrangian, L, for the electromagnetic field in vacuum will be constructed. The rf 's are given by the vector and scalar potentials and the field equations take the form,

22

2

22

2

0

0

t

t

∂∇ − = ∂

∂ φ ∇ φ − = ∂

AA

.(9.9)

The potentials have to be subjected to the further constraint of the Lorentz condition,

2 As is always the case this simplification of the situation is not universally true.

Independent conditions on the boundary values or generalized coordinates must be imposed where the fields energies are functions.

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0t

∂φ∇ ⋅ + =

∂A . (9.10)

The Lagrangian which provides the equations in Eq. (8.X) is given by,

( )2 218

= +π

E BL . [3] (9.11)

The BB and EE fields can now be expressed in terms of the potentials, AA and φ as,

( )2

218 t

∂ = −∇φ− − ∇× π ∂

A AL . (9.12)

Expanding Eq. (9.11) in Cartesian coordinates,

22 2

31 2

1 2 3

2 2 2

3 3 1 1 2 1

2 3 3 1 1 2

18

.

AA Aq t q t q t

A A A A A Aq q q q q q

∂∂φ ∂ ∂φ ∂ ∂φ= − − + − − + − − − π ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ − − − − − − ∂ ∂ ∂ ∂ ∂ ∂

L

L

L

(9.13)

Using the Lagrange relationship,

1 1 1r rA At q At t

∂ ∂ ∂ ∂ ∂ + − ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

∑L L L, (9.14)

gives for the 1A component of the vector potential,

3 The factor 1 8π is completely arbitrary and is used so in the integration of the

electromagnetic field for charged bodies. There are other changes in the notation used in this section, including the absence of the absolute value signs around the EE and BB field variables — it will be assumed that the real nature of these variables is understood by the reader. In all instances the field variables are assumed to be functions of time and space — although no explicit parameters are shown in the equations.

Where there is a explicit dependence on a position variable a subscript will be used to indicate the dependent variable. In general the variable qr will be used for a generalized coordinate value in place of rr or x.

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Copyright 2000, 2001 9–5

2

31 2 1 1

1 2 1 2 3 3 1

22 1

1 21

22 1

1 2

1 ,4

1 ,4

1 .4

AA A A At q t q q q q q q

AAt q t

AAt

∂∂ ∂φ ∂ ∂ ∂ ∂ ∂ ∂= + + − − − π ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂φ = −∇ + + ∇ ⋅ + π ∂ ∂ ∂

∂= − ∇ − π ∂

A

L

(9.15)

with similar equations for 2A and 3A components.

A similar expansion can be developed for the scalar potential, such that,

2

22

1.

4 t ∂ φ

= − ∇ φ− π ∂ L (9.16)

A Lagrangian which includes the Lorentz condition and represents the electromagnetic field independent of any constraints is given by,

( )2 2

218 t t

∂ ∂φ = −∇φ− − ∇× − ∇ ⋅ + π ∂ ∂

A A AL . (9.17)

Considering the Lagrangian for a electromagnetic field, with charge sources, is the next step in the development of the complete Hamiltonian.

The potential equations as originally developed in Eq. (4.11) are,

22

2

22

2

t

t

∂∇ − = − ∂

∂ φ ∇ φ − =−ρ∂

AA j

. (9.18)

The Lagrangian that will result in these equations is Eq. (9.17) plus an additional term ( )vρ ⋅ − φA , where ρ is the charge density and v is the velocity of the charge, so that the radiation and interaction Lagrangian is given by,

( ) ( )2 2

218

vt t

∂ ∂φ = −∇φ− − ∇× − ∇ ⋅ + + ρ ⋅ − φ π ∂ ∂

A A A AL . (9.19)

Eq. (9.19) now defines the Lagrangian for a charged particle moving in a electromagnetic field produced by the charge ρ and current jj . The Lagrangian for the entire field, including the particles themselves, is

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given by integrating each individual charge with the volume using dV= ∫L L to give the field Lagrangian as,

( ) ( )2 2

218

T vt t

∂ ∂φ = + −∇φ− − ∇× − ∇⋅ + + ρ ⋅ − φ π ∂ ∂ ∫

A A A AL , (9.20)

which results in a description of both the equations of motion for charged particles embedded in the field and the equations for the electromagnetic field itself.

Eq. (9.20) represents the total Lagrangian of a set of particles interacting with the electromagnetic field. The dynamical variables of the particles form a discrete set involving the components of the position rr and the velocity 2 2 2d dtr . For the electromagnetic field, it is the field potentials and the fields which represent the generalized coordinates of the Lagrangian.

The total Lagrangian has three terms, the Lagrangian for the particles, objL , the Lagrangian for the radiated field, radL and the

Lagrangian for the interaction between the electromagnetic field and the particles, intL .

The total Lagrangian is then given by,

obj rad int+ +L = L L L (9.21)

where,

2obj

12 r r

r

m q= ∑ &L , (9.22)

and

2 2rad

12

dv= +∫E BL , (9.23)

and

[ ]int r r r rr

= ⋅ − ρ φ∑ j AL . (9.24)

Regrouping the radiation and interaction Lagrangian terms allows the re–introduction of the Lagrangian Density.

[ ]2 212

= + + ⋅ −ρφ E B j AL , (9.25)

and the following form for the Standard Lagrangian,

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2i r

r

mq dv= +∑ ∫&L L . (9.26)

It should be noted that the interaction term is local with the current density at the point multiplied by the vector potential. In the radiated field Lagrangian the spatial derivatives of the potentials come about from EE and BB , which describes the coupling between the fields at each point. This coupling is the origin of the propagation of the free field.

§9.1. FIELD ENERGY DENSITY

Using the Hamiltonian expressions given in §8, Eq. (9.26) serves as the starting point for computing the Hamiltonian of the electromagnetic field in the absence of charged particles and field sources, that is the Hamiltonian of a freely propagating electromagnetic wave. The contribution of the first and last terms in Eq. (9.26) is determined by Eq. (8.44).

The remaining contribution will be from the terms in the field quantities alone,

dV= ∫H H , (9.27)

where,

( )

22 2

2 2 2

222 2

1 2 2 ,8

1 2 2 ,8

1 2 .8

t t t

t t

t

∂φ ∂φ ∂φ = − ∇ ⋅ + − + + ∇ ⋅ + π ∂ ∂ ∂ ∂φ ∂φ = + ⋅∇φ+ ∇ ⋅ + ∇⋅ − − + π ∂ ∂

∂φ = + + ⋅∇φ+ ∇⋅ − π ∂

E A E B A

E E A A E B

E B E A

H

(9.28)

The total Hamiltonian given by Eq. (8.46) is now,

( )2

22 2

,

1 2 .8

sS r r s

s r

vq p dVq

dvt

∂= − ρ + ∂

∂φ + ρφ + + ⋅∇φ+ ∇ ⋅ − π ∂

∑∫

∫

A

E B E A

L&

L

H H

(9.29)

Where SH is the Hamiltonian function of the dynamical system. When the field variables are taken as discrete, the ordinary Hamiltonian canonical equations can be used for the dynamical as well as for the electromagnetic variables. This first term of this Hamiltonian is the

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kinetic energy of the dynamically system. The sum of the last two terms under the integral sign is zero, due the Lorentz condition.

The first and fourth terms under the integral sign cancel since,

( ) 1 1 1 ,4 4 4

1 ,4

0.

dv dv

dv

ρφ+ ⋅∇φ = φ ∇ ⋅ + ⋅∇φ π π π

= ∇⋅ φπ

=

∫ ∫

∫

E E E

E (9.30)

If the field potential vanish sufficiently rapidly at infinity what remains is the radiation field energy density,

( )2 218

dv+π∫ E B . (9.31)

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§10. A PREVIEW OF QUANTUM MECHANICS

At the end of the 19th century, theories of matter and energy based on the classical concept of continuity were being replaced by theories based on discontinuities. These theories were being put forth in several fields of science including: physics, biology and geology.

In the field of physics, this transitions to discontinuity can be seen in the writings of Helmholtz and his comments on Faraday’s Law of Electrodynamics:

Now the most startling result of Faraday’s Laws is perhaps this: if we accept the hypothesis that the elementary substances are composed of atoms, we cannot avoid concluding that electricity also, positive as well as negative, is divided into elementary portions which behave like atoms of electricity.

Although it would be a great leap to conclude that Helmholtz foresaw the quantum nature of the electric charge — the electron — the concept of an indivisible unit of mater has been part of physics since Newton.

The theories of physics put forth at the end of the 19th century did not however provide a uniform and constant picture of nature. Although the common popular science impression is that scientist believed that a combination of thermodynamics, electromagnetism and classical mechanics could account for the known physical phenomena — such was not the case. Clausis, Maxwell and Boltzman presented an ambiguous description of the kinetic theory of gases. The theories of the structure of matter and atoms cam under attack for several reasons. Both technical and philosophical changes were taking place. Movement away from mechanistic models of the atom toward an empirical or phenomenological theories were lead by Boltzman and his statistical theory of thermodynamics.

§10.1. DOMINATION OF MODERN QUANTUM MECHANICS

Modern physics is dominated by Quantum Mechanics and the application of Quantum Field Theory. The majority of the experimental activities take place in the fields of High Energy Physics (HEP) and related fields. The research funding and the academic programs are focused on the frontiers of size and energy. However, in the beginning of the 20th century the physical world was described by the laws of classical (or Newtonian) mechanics — the physics of every day life. Experiments

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were conducted in which the results could be comprehended on a human scale, in size, energy and funding.

Before proceeding with a more detailed description of quantum mechanics and its origins, the physical aspects the quantum theory needs some explanation. At the macroscopic scale of matter there are two broad types of classical physics: Waves and Particles. Particles are considered localized phenomena which transport mass and energy as they move. Waves are de–localized phenomena which carry energy, but not mass, as they move [Auya95], [Kim91].

In quantum mechanics the distinction between particles and waves is less clear. Particles can exhibit wave–like behavior and waves can exhibit particle–like behavior. It is important to understand the –likeness aspects of these behavior’s. Whether an object is a particle or a wave can not be clearly defined — only its behavior in the presence of some measuring device or interaction with other objects can be described. The blurring of the distinction between particle and wave causes serious problems when a visualization of the behavior is desired. Electrons can create wave–like diffraction patterns, while the electromagnetic radiation of light can interaction with matter in a particle–like manner.

At the turn of the century the axioms of Newtonian physics provided a description of the dynamics of point masses — planetary motion, the motion of rigid bodies, the elastic properties of solids, hydrodynamics and acoustics [Schw92]. In classical mechanics the position and momentum of a particle can be measured with arbitrary accuracy. With the additional of Maxwell’s electrodynamics, classical physics was nearly complete.

Specific quantum phenomena began to be discovered which challenged to classical theories. In chronological order, these major phenomena are: blackbody radiation (1900), the photoelectric effect (1905), line spectra of hydrogen (1913), the Compton effect (1923), the diffraction of particle beams (1927), the Ramsauer effect (1927) and the tunnel effect (1928) [Spey94]. [1] As each of these phenomena was explored further,

1 Each of these effects and the experiments that confirmed them has been performed

numerous times and have become the stock in trade of University physics. The following is a brief summary of each selected effect:

(1) Blackbody radiation — is described in more details in §10.2.1. This quantum effect occurs when the intensity of electromagnetic radiation is plotted as a function of frequency. The plot does not adhere to the principals of thermodynamics. As shorter wavelengths are examined, the predicted spectral intensity should increase without limit. The thermodynamic equation

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( ) 48I kT= π λ (the Rayleigh–Jeans equation) fails to predict the measured

results.. Max Planck solved the problem with the equation

( ) ( )( )58 1ch kTI ch e λ= π λ − . The result is the radiated energy is quantized — see

§10.21. for further details.

(2) Photoelectric Effect — after Planck’s equation was accepted by the community, Albert Einstein publishes a paper describing the photoelectric effect. When weak red light is shone onto a surface of cesium, electrons are emitted. The brighter the light the more electrons are emitted. These electrons travel at a velocity slower than normal electrons. If a blue light (shorter wavelength) is used, electrons are emitted from the cesium surface — but they are traveling at a higher velocity. The brighter the blue light, the more electrons are emitted at this higher velocity. If a very long wave length light is used (infrared) no electrons are emitted no matter what the intensity. The classical description of this effect states that the incident light would be absorbed over the entire surface of the cesium and require some amount of time to raise the energy level of all the cesium atoms in order to cause electrons to be emitted — contrary to the observed results. By using Planck’s theory that the emitted energy of a black body is quantized, Einstein made the reverse conjecture — the incident electromagnetic field is quantized — behaving as a group of particles colliding with atoms of cesium [Eisn05b]. Einstein’s equations of the interaction is the light with the atoms of cesium is E h= λ . Using this model, only the cesium atoms struck by the particles of light

(which is called the photon but had not been named yet), emitted electrons. The energy of the incident photon gave rise to a proportional energy electron. The higher the energy of the photon — the shorter its wavelength — the higher the energy of the emitted electron. The photoelectric effect laid the groundwork for the complementarity principal in which the particle nature of light complements the wave nature. Einstein’s description of the photoelectric effect was heuristic, simple in its mathematical formulation and fit the known experimental data. The theoretical formulation was verified in 1916 by Robert Millikan [Mill16].

(3) Line Series — the description of the emission lines of hydrogen can be described by

the Ritz formula ( )21 4 1f R n= − . Other spectral lines shapes are given in

§10.2.

(4) Compton Effect — the Compton effect occurs when X–rays are scattered by electrons. In the classical description of the effect using Maxwell’ equations, the wavelength of the scattered light should be the same as the incident light. Arthur Holly Compton (1892–1962) observed that X–rays were shifted to a lower frequency when scattered from a collection of electrons. If the incident X–rays are considered as quanta of the electromagnetic field, with an energy of E h= λ , then

the collision between the electron and the X–ray photon can transfer energy to the electron. Since momentum must be conserved, the incident photon gives up energy during the collision and is scattered at a lower energy, thus a longer wavelength.

(5) Diffraction of Particle Beams — if the complimentary principal is applied to beams of particles, then wave–like behaviors can be observed. This symmetry between particle and wave is predicted by Louis–Voctor Pierre Raymond de Broglie (1892–1987). De Broglie stated that the wavelength is a particle is λ = h mv . This idea

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quantum theory and the experimental techniques used to verify the theory grew. With this growth came the understanding that the Newtonian view of nature was seriously flawed.

Three principals of quantum theory of nature were developed:

(1) The Correspondence Principal — the quantum mechanical description and the classical description of a phenomena are equivalent when Planck’s constant becomes zero, that is 0→h .

(2) The Complementarily Principal [Holt70] — in which both the wave and particle view of an effect produced similar results.

(3) The Uncertainty Principal — which describes the hidden interdependence between pairs of measurements.

§10.2. EARLY QUANTUM THEORY

In 1900 the wave theory of light, built on Maxwell’s electromagnetic theory and the description of the motion of particles, based on Newton’s laws of motion went unchallenged. Two developments in the early 1900’s served to upset forever the certainty of these theories.

On December 14th, 1900, Max Planck (1858–1947) presented a paper [Plan01], [Segr80] to the Berlin Academy of Science which stated that under certain ideal conditions energy radiated from a hot body was distributed in a characteristic manner, which could only be explained by assuming the electromagnetic radiation was emitted by the body in discrete amounts called quanta (Latin for how much) [Crea86], [Waer68].

Prior to this momentous announcement, the 19th century produced many discoveries which lead to the formulation of quantum mechanics.

was developed in de Broglie’s Ph.D. thesis and was verified when a beam of electrons was defracted by gratings — as if they were waves — electron waves [d’Broh24], [d’Brog46], [d’Brog53].

(6) Ramsauer Effect — when a beam of electrons travels through a gas, this gas becomes transparent. If the electrons are considered as waves and are projected through the gas at just the right velocity (slow electrons), the incident electrons resonant with the atoms of gas and are not scattered. Rather they tunnel through unimpeded.

(7) Tunnel Effect — the tunneling observed in the Ramsauer effect can take place in other instances. If the position and momentum of a particle are described by a wave equation, then the particles can have a finite probability of passing through a potential barrier that would be forbidden in classical physics. The description of this behavior involves the use of Schrödinger’s equation in which both the particle and wave nature of matter are required to describe the effect.

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The achievements that lead to the joining of electricity and magnetism has been presented in the previous sections. The development of the mathematical foundation of theoretical mechanics and the confirmation of the laws of the conservation and transformation of energy lead to the belief that the description of the laws of nature were in their final stages. This situation was illustrated by the experience of Max Planck. After completing the defense of doctoral thesis, he wrote to his teacher and mentor Philip Jolly asking his advise on a career in theoretical physics [Heil86]. Jolly replied that

... theoretical physics is practically finished, the differential equations have all been solved. All that is left is to consider individual special cases...

In August of the same year, David Hilbert (1862–1943) presented his famous 23 problems before the Second International Congress of Mathematicians. [2] The sixth problem pertained to axiomation in physics.

2 Hilbert presented a talk titled “Mathematical Problems,” in which he outlines the future of mathematics [Reid69]. The significance of mathematics and the role it played in the mind of Hilbert was presented through a set of problems. Only 10 of the 23 problems were given during the talk. The full list of problems appeared in L’enseignement mathematique, Volume 2, 1900, pp. 349–355. They are:

(1) Cantor’s problem of the cardinal number of the continuum.

(2) The compatibility of the arithmetical axioms.

(3) The equality of the volumes of two terrahedra of equal bases and equal altitudes.

(4) The problem of the straight line as the shortest distance between two points.

(5) Lie’s concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.

(6) The mathematical treatment of the axioms of physics.

(7) The irrationality and the transcendence of certain numbers.

(8) Problems of prime numbers (including the Riemann hypothesis).

(9) The proof of the most general law of the reciprocity in any number field.

(10) The determination of the solvability of a Diophantine equation.

(11) The problem of quadratic forms with any algebraic numerical coefficients.

(12) The extension of Kronecker’s theorem of Abelian fields to any algebraic realm of rationality.

(13) The proof of the impossibility of the solution of the general equation of the 7th degree by means of functions of only two arguments.

(14) The proof of the faintness of certain complete systems of functions.

(15) A rigorous foundation of Schubert’s enumerative calculus.

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Hilbert proposed that a finite number of initial axioms should be formulated so that all the results needed for a complete description of the physical picture of nature could be obtained by purely logical means [Wigh76].

Events would soon follow to dispel these illusions. At the end of the 19th century a number of discoveries were made which could not be described within the framework of existing theories. These included: X–rays, the dependence of the mass of an electron on its velocity, the photoelectric effect and radioactivity.

§10.3. EXPERIMENTAL NECESSITY FOR THE QUANTUM THEORY OF RADIATION

By the end of the 19th century, experiments that established that the radiation spectra of free atoms consisted of sets of discrete lines, or line spectra, which formed order groups or series. In 1885 Johann Jakob Balmer (1825–1898) [3] discovered that atomic hydrogen emits radiation frequencies which can be described by,

2

1 124n cR

n ω = π − ,

(10.1)

(16) The problem of the topology of algebraic curves and surfaces.

(17) The expression of definite forms by squares.

(18) The building up of space form congruent polyhedra.

(19) The determination of whether the solutions of “regular” problems in the calculus of variations are necessary analytic.

(20) The general problem of boundary values.

(21) The proof of the existence of linear differential equations having a prescribed mondromic group.

(22) Uniformization of analytic relations by means of automorphic functions.

(23) The further development of the methods of the calculus of variations.

3 Balmer was a Swiss mathematician and physicist who developed the first formulation of spectroscopic data. Other workers in the field had attempted to establish a mechanical acoustical relationships between the spectral lines of an element. Balmer found his formula for hydrogen by empirical means and used it to predict other spectral lines series that were subsequently confirmed experimentally.

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where n are integral numbers 3,4,5,… , c is the velocity of light in a

vacuum and R is the Ryberg constant, where 7 11.097 10R m−= × . [4] This formula was derived by Balmer and is called the Balmer series whose frequencies fall in the visible region of the electromagnetic spectrum, with

the first Balmer lines having a wavelength of 6563Ao

, with the next two lines falling in the blue and violet regions of the visible spectrum. Additional series were discovered for hydrogen, atoms which fell in the infrared region of the spectrum. The regularity of these new series, which were similar in structure to the Balmer series, which resulted in the generalized expression,

2 2

1 12n cRk n

ω = π −

, (10.2)

where k fixes the series such that 1k = gives the Lyman (Therdore Lyman (1874–1919)) series (ultraviolet), 2k = gives the Balmer series and 3k = gives the Paschne (Friedrich Paschne (1865–1947)) series (infrared), 4k = gives the Brackett series (F. S. Brackett) and 5k = gives the Pfund series. [5] The regularity of the structure was also observed in the spectrum of other atoms, which lead to more generalizations. One such theory was proposed by Ritz in 1908. Ritz’s combination principal stated that if the formula of a series was given and the constants of the formulas known, then newly discovered spectral lines could be obtained from known spectral lines.

§10.3.1. Black Body Radiation

The black body experiments performed during the same time produced results inconsistent with the thermodynamic description of the equilibrium of radiation with matter [Plan14]. If the radiation emitted by the heated body is completely enclosed by matter as a given temperature, it will be in a temperature equilibrium with its surroundings, but independent of the actual surrounding material. This radiation is called black body, since the frequency distribution of the radiation is same as

4 Janne (John) Robert Ryberg (1854–1919) described the spectra of periodic elements in a paper submitted to the Swedish Academy of Sciences in 1889, titled “Recherches sur la consitution des spectres d’émission des éléments.”

5 Recently, hydrogen–like atoms in highly excited states up to k = 100 have been observed. They are called Ryberg atoms and their diameter is approximately 105 times larger than the diameter of a ground state hydrogen atom [Litt79].

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that radiated or absorbed by a completely black or completely absorbing material body at that temperature.

The reason that the spectral distribution of the radiated energy is independent of the radiating material can be deduced from classical thermodynamic theory, but the distribution of the radiated frequencies must be measured to confirm the theory. The original measurements were expected to produce results based on the electromagnetic theory of light. If the electromagnetic field inside the enclosed volume can be described by an infinite number of normal coordinates, then the statistics of such a system can express the average energy as a function of the temperature.

During the experimental observation of thermal radiation, the intensity of the radiation, per unit frequency interval, rises from zero at very low frequencies to a maximum value whose position and magnitude depend on the temperature of the body, then falls again, approaching zero at very high frequencies. The drop in intensity in the high frequency region was in conflict with the theoretical results obtained by Lord Rayleigh. This conflict was based on the equipartition theorem derived from classical mechanics and of the wave theory of light. In the theory of the atom developed by Lenard and Rutherford, the atom was composed of a positive nucleus surrounded by negative electrons. The hydrogen atom is the simplest consisting of one electron and one proton in its nucleus. The electron is held by the nucleus by the electrical force of the proton. If this binding force were purely electrical, there was a theorem that states that a system of electrical charges cannot be in a stable equilibrium. This forced Rutherford to propose that the electron moved around the nucleus in such a way that the centrifugal force balanced the electrical force.

At room temperature the atoms or molecules of a gas undergo 100,000,000 collisions per second. If the ordinary laws of mechanics are applied to this gas there would be a slight change in the orbit of the electron with each collision. These changes would accumulate over time and the system would collapse in a few seconds. Since this is not the case, there must be some other principal at work other than the laws of classical mechanics [Born26].

In order to illustrate the black body nature of radiation, the electromagnetic waves emitted by an oscillating electron can be examined. If this electron were placed in free space, the radiated energy would escape to infinity. The electron would continually loose energy and eventually stop radiating. If however, the electron were placed inside a container whose walls were perfect reflectors, a different behavior would result.

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As the radiation is reflected from the wall it would strike the electron causing it to be re–accelerated. In this way, the electron is bathed in its own radiation. This radiation is absorbed and re–emitted through the process of scattering. Since the electron obeys Kirchoff’s Law, it acts as both a radiator and an absorber with equal results. Once equilibrium has been reached between the emission and absorption of radiation, the cavity attains a constant temperature, with the energy density uniformly filling the cavity. If a small hole is placed in the cavity, the radiating energy has the characteristic of a black body.

In such a system, the energy per degree of freedom (per mode of oscillation) is given by,

12

U kT= , (10.3)

where 231.381 10k −= × joules is Boltzman’s constant and T is the temperature in degrees Kelvin. The modes of oscillation in the cavity are simply standing waves. The total electromagnetic energy can be determined by calculating the total number of modes of oscillation. The electric field of the thn mode is given by,

( ) ( ), sin coszn n n

nE z t A tLπ = ω

, (10.4)

whose frequency is given by,

, 1, 2,3,nn c nLπ

ω = = … (10.5)

where L is the dimension of the cavity. By rewriting Eq. (10.5) I terms of a wavelength, 2n L nλ = , it can be seen that in order to fit a standing wave the dimension L must equal an integral number of half waves.

Planck attributed the discrepancy to the breakdown of the equipartition theorem when applied to high frequency oscillations and made the suggestion that, if the vibrating matter particles which emit radiation have motions restricted to certain discrete energy values there would be a departure from the laws of the classical statistical mechanics, that was required to describe to experimental facts. [6] That is, the emission

6 The German physicist Max Karl Ernst Ludwig Planck (1858–1947) found a empirical formula that described the dependency of the intensity of radiation on temperature and wavelength. Planck assumed that the radiation escaping from a black body (which may be realized in practice as the radiation escaping a heated cavity through a

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and absorption of radiation always takes place in discrete portions of energy quanta, νh , where ν is the frequency of the emitted or absorbed radiation [Born26].

This remarkable development is best demonstrated by the photoelectric effect, which had been discovered by Heinrich Hertz in 1886–87, in which the kinetic energy of the photon emitted is given by

2 2mv h= ν . This equation, proposed by Einstein, was proved experimentally by Robert A. Millikan (1868–1953) provided the first evidence of the existence of the quantum. [7]

The acceptance of Plank’s theory meant a revolution in physics since it was incompatible with both Newtonian mechanics and the electromagnetic theory of light. This proposal was verified in short order by Heinrich Reubens, when he compared his experimental results with Planck's formula [Bagg92]. The explanation of this behavior however was unknown to Planck and Reubens.

In 1905 Albert Einstein observed that Planck's assumption regarding the energy radiated by a black body ε = νh , followed closely his predictions of the behavior of light...

small hole) originated in oscillations in the wall of the cavity. According to classical physics an oscillator may have any value for its total energy. Planck made the assumption that the oscillations occur only at discrete values of the total energy. By using statistical arguments, Planck derived the radiation density law, as a function of temperature,

( ) ( )3 3 /( ) 8 ( 1)h KtT h c e ν

νυ = π ν − . This derivation came about by dividing the total energy of

each oscillator into equal values, then letting these values become infinitely small. Using a standard calculus technique, these values were integrated to restore the original energy calculation. In Planck's calculation this did not work. He found that in order to make the integration fit the experimental data the infinitesimal values of the oscillator energy could not be vanishingly small, but had to have a finite value — meaning their sum, or the total energy of the oscillator, could only have particular energy values [Bagg92], [Serg80], [Kuhn78], [Klei66].

An important point should be made here. The units of Planck's constant are the same as the units of action, energy–seconds. The idea of action will be developed in later sections with Lagrangian and Hamiltonian dynamics. Hamilton’s are a method of finding the minimum value of a given equation. Neils Bohr won the Nobel Prize for the insight that when calculating the orbits of electrons, Planck's constant should be used in place of the Hamiltonian action variables.

7 R. A. Millikan developed a method to determine the value of the fundamental unit of electrical charge. In 1923 Millikan received the Nobel Prize for t his work as well as the measurement of Planck’s constant to an accuracy greater than previously possible [Nobe65]. Millikan’s paper, with G. Winchester [Mill07] describes the photo–electric effect that was theorized by Einstein and Planck, which Millikan called the Planck–Einstein light unit theory [Mill50], [Mill68].

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... monochromatic radiation ... behaves ... as if it consists of mutually independent energy quanta of magnitude [ ]νh . ... then

this suggests an inquiry as to whether the laws of the generation and Conservation of light are also constituted as if light were to consist of energy quanta of this kind. [Eins05]. [8]

Planck was unwilling to accept so radical a theory and spent much of his time in a fruitless attempt to save the wave theory of light by a modification of his original energy–level assumptions.

In 1909 Einstein showed how the fluctuations of the blackbody radiation confined in a volume V could be written as the sum of two terms – one describing the quantum properties of the radiation and the other describing the wave properties of the radiation [Eisn09]. The mean square fluctuations of the black body energy is given by,

( )23

2

28c E

E E E hvd V

− = +πν ν

. (10.6)

In another paper [Eisn19], Einstein described the probability transition rates of atoms which experience induced emission and absorption as well as spontaneous emission. The emission transition rate

( )mn m mn nmdW dt N B A= ρ + , the absorption rate nm n nmdW dt N B= ρ and spontaneous rate mnA , where nN is the number of oscillators on the energy level nE and ρ is the radiation energy per unit volume in the frequency interval dν → ν + ν .

This two expressions can be used to derive Planck’s Law. The A and B coefficients are related by,

3

3

8mn nm

hA Bc

π ν= . (10.7)

8 The concept of photons as the quanta of light was proposed by Gilbert. N. Lewis

(1875–1946), a physical chemist at the University of California, Berkeley, in 1926 [Lewi26], [Sutt92]. Lewis speculated the light consists of...

...a new kind of atom ... uncreatable and indestructible (for which) I ... propose the name photon [Pas82].

Nearly 90 years after Einstein's observations, it is difficult to imagine just how revolutionary his ideas were. The ideas of the quanta were not readily accepted by most physicists of the time, but the mounting experimental evidence eventually convinced the world of the need for the quantum hypothesis.

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It was not until Neils Bohr (1885–1962) united the Rutherford nuclear model of the atom with the energy–level hypothesis to formulate his theory of the structure and spectrum of hydrogen, that the quantum theory of the atom was accepted as a fact of nature. [9]

Bohr showed in 1918 that the transition probabilities were associated with the electric dipole moments of the radiation oscillators given,

( )4 3

2

3

23nm mnA Phc

π ν= , (10.8)

where mnP is the Fourier coefficients of the dipole moment. This result was valid only for large quantum numbers – large numbers of oscillators. In the summer of 1925 Jordan made use of Heisenberg’s matrix mechanics to construct a theory of the radiated electromagnetic field [Born25]. Jordan showed that the black body radiation could be properly described if the oscillators had an energy of,

( )12E n h= + ν . (10.9)

§10.4. STATES OF A MECHANICAL SYSTEM

The underlying structure of the theory of quantum mechanics is based on the representation of the states of a system by a one–to–one correspondence with vectors in a suitably chosen linear vector space. [10]

9 Bohr was 28 years old when he published his first paper on the theory of the

hydrogen spectrum. Einstein had already written his initial paper on the corpuscular theory of light at the age of 26. Heisenberg was 24 years old when he laid the foundation of matrix mechanics. Dirac and Jordan write their papers at 24 and 23. All of these efforts as well as many others in the field of quantum mechanics focus attention on youth as one of the ingredients of scientific discovery.

When Bohr went from Cambridge to Manchester University in march of 1912 on a visit to Rutherford’s laboratory, he was unaware of the regularities of the hydrogen spectrum lines. He did not learn of the Balmer series until the beginning of 1913, a few weeks before submitting his paper to Philosophical Magazine [Bohr13]. The starting point for Bohr’s atomic model was instead Rutherford’s experimental discovery of the existence of the nucleus and the contradiction of the observed stability of atoms that cam about if the laws of classical electrodynamics were applied to the Rutherford model of the atom.

10 The concept of state is one of the most subtle and controversial concepts in quantum mechanics. In classical mechanics the word state is used to refer to the coordinates and momenta of an individual system, and early on it was supposed that the quantum state description would also refer to attributes of an individual system. However,

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The measurement of the attributes of these states are described in terms of operations on these vectors. These operations are assumed to be linear. The operation of an operator on a vector is intended to describe a physical operation on the system. The result of the operation (measurement) of an attribute (dynamical variable) is an eigenvalue of the linear operator representing the dynamical variable. The state in which the dynamical variable has that value is represented by the corresponding eigenvector.

In quantum theory, quantities like position and momentum cannot have definite numerical values. Heisenberg's uncertainty principle states that the product of the uncertainty in coordinate position and the uncertainty in momentum is given by: xx p∆ ∆ ≈ h , where

34 11.054572 10 kgms− −= ×h is Planck's constant [Heis30]. A similar

the assumption that a quantum state is a property of an individual physical system leads to contradictions and must be abandoned [Wign73], [Heal79], [Shim74], [Redh87].

The quantum state description may be taken to refer to an ensemble of similarly prepared systems. One of the earliest advocates of the ensemble interpretations was Einstein. His view is concisely expressed as follows [Eins49]:

The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not individual systems.

Criticisms of the ensemble interpretation have often resulted from a confusion of the ensemble, which is the virtual infinite set of similarly prepared systems, with a concrete sequence or assembly of similar systems. These criticisms may be alleviated by a slightly more abstract interpretation in which a state of identified with the preparation procedure itself. State is then an abbreviation for state preparation procedure.

In quantum mechanics the state of the system is usually taken to mean a mathematical object such as a wave function, state vector or density operator. Due to the underlying complexities involved in the description of the quantum mechanical process it is important to further distinguish between the dynamical state of the system and its quantum state [Heal89]. The role of the quantum state is to generate probabilities concerning the possible outcomes of a measurement performed on the system. The dynamical state is usually identified with a quantum variable, position, momentum, spin at some instant. Using these weak definitions a quantum system always has a dynamical state. There is however no general connection between a system quantum state and dynamical state [d'Esp71].

By identifying the state concept directly with a set of probability distributions, this approach makes clear the fact that the interpretation of quantum mechanics is dependent on choosing a suitable interpretation of probability and the probability density function.

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uncertainty can occur for the product of time and energy t E∆ ∆ ≈ h . [11] Planck's constant is so minute that its effect on the macroscopic world can be ignored. In the microscopic world of atoms h places a significant role in the description of the nature.

The uncertainty principle, although quite foreign in the realm of classical physics results from the Correspondence Principle [12] which states that any quantum mechanical theory must provide proper classical results when the action per cycle of the system is large compared to Planck's constant [Dirac25].

It is agayans the process of nature.

— Geoffrey Chaucer (1343–1400), The Frankeleyn's Tale [Chau94]

11 Although the time–energy uncertainty given by ∆ ∆t E ≈ h is given as an extension

of the position–momentum uncertainty in many quantum mechanics texts. This relation can not be directly derived from the original position–momentum relation ∆ ∆x p ≈ h [Eber73]. The first problem is that the time variable t is not an operator associated with an observable characterizing the particle, but is a parameter. Secondly the energy E is not a generalized canonical conjugate of time t. These difficulties can be overcome by considering the time–energy uncertainty relation in light of the wave nature of matter. In this description, the dispersion of the wave packet represents the quantum mechanical particle traveling through space. Consider a one–dimensional matter wave described by the

function ( )f x whose uncertainty (dispersion) of x is given by ( )[ ]( ) 122

, ,x f x f xf∆ = −

where ( ),f Af f Afdx∗= ∫ . Using the Fourier transform ( ) ( )f x g k→ the theorem

provided by Heisenberg in [Heis30] states that the minimum product of the dispersions

∆ ∆x k⋅ can be obtained if ( ) ( )22

1 2 x xf x e − ∆= π is a Gaussian distribution, which results

in ( ) ( ) 22

1 2 k kg k e − ∆= π also being a Gaussian distribution, where ∆x and ∆k are identical

with the dispersions defined by the original expression. Their product is ∆ ∆x k⋅ = 1 2 which is independent of the meaning of the two values x and k. By identifying x with a coordinate and k with the inverse of the de Broglie wave length so that k = 2π λ the usual uncertainty relation between position and momentum can be derived. Using the standard parameters for the Fourier transform t and ω a similar expression can be formed by substituting x by t and k by ω to give ∆ ∆t ⋅ =ω 1 2 or ∆ ∆t E⋅ = h 2 [Rays77].

12 The traditional correspondence principle implies the Bohr Correspondence Principle as a statement that for large quantum numbers, quantum mechanical laws must reduce to their classical counterparts. In this monograph the correspondence principle will imply additionally that operators are assigned to physical quantities. The details of this statement will be developed later, but for now this implies momentum is given by ( )p i→ ∇h and

energy is given by ( )E i t→ ∂ ∂h [Jord75].

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Instead of certain values of a measurable component, quantum theory provides probabilities for the measurable value, derived from the Wave Equation ψ which describes the measurable component. [13] The wave equation states the probability of finding a particle at a certain position, given its momentum. In this way ψ is something that varies from point to point in the same manner as a field varies from point to point. The result of this approach is that particles become interchangeable with waves and fields become interchangeable with propagating particles.

§10.5. QUANTUM MECHANICS OF ELECTROMAGNETIC FIELDS

In quantum mechanics, the fields generated by the electric charge are subjected to the uncertainties of Werner Karl H. Heisenberg's (1906–1976) principle. In classical physics, fields are associated with forces. In quantum physics, fields are associated with particles. [14] For particle fields and their associated force, the range of the force is related to the mass of the particle carrying the force. In order for the particle to have an effect, it has to travel over a distance 0m ch . Because the electromagnetic force is long

range or possibly infinite, a particle of zero mass is need to carry the electromagnetic force. The Heisenberg uncertainty principle provides for the suspension of the conservation of energy for the duration t∆ , by an

13 The concept of a wave equation originated with Einstein's 1905 theory of the

photoelectric effect in which particles — photons — of energy hν and momentum h λ are associated with electromagnetic radiation of frequency ν and wavelength λ . In 1923 Louis de Broglie suggested a particle of momentum p should be associated with a wave of wavelength λ = h p [D’brog24]. It was William Hamilton in the early nineteenth century (1828–1837) that noted a formal similarity between the description of a light ray in optics and the motion of a particle in classical mechanics.

14 The term particle here refers to elementary particles. In the standard model all known particles belong to three classes:

(1) Leptons — are point like particles with no detectable spatial extent. The six known leptons are the electron, muon, tauon and their corresponding neutrinos.

(2) Hadrons — are particles believed to be composed of quarks. Hundreds of hadrons have been observed, with the most important being the proton and the neutron.

(3) Bosons — are exchange or gauge particles. These particles are the carriers of force, with the photon carrying the electromagnetic force, the intermediate vector bosons carrying the weak force, the gluon carrying the string force and the graviton carrying the gravitational force. The electromagnetic and weak forces were united in the late 1970's into the electroweak, which predicts a massive boson which has yet to be detected – the Higgs boson.

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amount E∆ , where t E∆ ⋅∆ ≅ h . This value is the Compton (Arthur Holly Compton (1892–1962)) wavelength of a particle representing the electromagnetic force — such a particle is the photon.

In quantum mechanical terms charge — the electrostatic unit — is available only in units of ch . Coulomb's law given in Eq. (2.2), is then given as,

1 22 .e ec= αFr

h (10.10)

The quantized charge and the resulting quantized field are represented by a quantum particle of spin 1 — the photon. Particles that correspond to the constitutes of matter — particles of spin ½ — and particles that correspond to force — particles of spin 1 can both be represented by quantized field equations. [15]

The electromagnetic field equations are introduced into the realm of quantum mechanics through the potential field equation Eq. (4.22) and Eq. (4.23). The electromagnetic force was derived from the vector potential AA , where,

= ∇ ×B A . (10.11)

With the scalar potential 0φ = and gauge condition 0∇ ⋅ =A , the EE and BB fields are given as,

,t

∂=−∇φ−

∂AE (10.12)

and

,= ∇ ×B A (10.13)

resulting in,

15 The concept of intrinsic spin was introduced in 1925 by the Dutch physicists Samuel Abraham Goudsmit (1902–1978) and George Eugene Uhlenbeck (1900–1988) [Gard90], [Pais86] and was confirmed experimentally, even though the property was not fully understood at the time. In 1928 P. A. M. Dirac made progress in the explanation of spin when he combined special relativity with the understanding of angular momentum. Twenty years later Wolfgang Pauli earned a Nobel Prize, in 1989, for exclusion principle. This principle forbids two particles with spin ½ (fermions) from being too close to each other when they are in the same state — that is they both have the same spin orientation. Pauli's exclusion principle explains the patterns in the periodic table of elements which in turn explains most of chemistry. It explains why electrons in atoms maintain their orbital positions, which explains the stability of matter.

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2

22

1 0,c t

∂∇ − =

∂AA (10.14)

which is the traveling wave equation given in terms of the vector potential A, A, where AA satisfies the transverse condition 0∇ ⋅ =A . According to Eq. (10.14) if AA and t∂ ∂A are given as functions of position at 0t t= the electromagnetic field is completely determined as a function of time. With no charges or currents present in the volume, a single vector potential A A is sufficient to describe the electromagnetic radiation within the volume. Although this expression has been stated elsewhere in this monograph, its meaning is now important in the quantum mechanical behavior of the radiation field.

§10.6. PRELIMINARIES TO QUANTIZING THE RADIATION FIELD

Quantizing the electromagnetic field is the central problem addressed in this section. In performing this quantization care must be taken to deal with the gauge arbitrariness and the redundancy of the vector scalar potentials. [16] The approach used here is heuristic in nature and is developed with hind sight. This heuristic is augmented with the Lagrangian formalisms by defining the conjugate dynamical variables of the system and an expression for the energy of the system in terms of the Hamiltonian in the Coulomb gauge, 0∇ ⋅ =A .

This approach consists of showing that the equations of classical electrodynamics, the Maxwell–Lorentz equations, can be thought of as Lagrange's equations of motion. Canonical quantization of the system can then be achieved by associating each pair of generalized coordinates and canonical conjugate momentum variables with the annihilation and creation operators used in the quantum electrodynamics description of radiation [Heit54].

16 Several approaches are available to quantize the electromagnetic field. By showing

that the electromagnetic field and a set of particles is formally equivalent to a set of mutually interacting particles and oscillators the Lagrangian method of canonical variables can be used. This approach will quantize the system by associating the momentum and position of particles with commuting operators and replacing the normal mode variables of the field oscillators with annihilation and creation operators. All physical quantities of the classical electromagnetic theory become operators acting on the quantum states of the system.

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§10.6.1. Vector Potential Expanded as a Fourier Series

The vector potential found in Eq. (10.14) is a function defined at all points in space and time. If this vector potential field equation is to be described in quantum mechanical terms, the number of variables needed becomes infinite. However, it is possible to choose an innumerable set of variables if the radiation field is enclosed in a volume of size 3K d k= . The exact form the boundaries of the enclosing volume are not needed, since the radiation equations can be described properly if the fields on one side of the volume are the same as they are on the other.

Each component of the vector potential can then be expressed as a component of a Fourier series. This new set of variables will represent the vector potential in Fourier space rather than xyz coordinate space. In this new volume some boundary conditions on the surface of the volume can be satisfied such that AA and its derivatives have the same values on two opposite planes of the volume — AA is periodic on the volume surface, and

3d k is considered large compared to the material system under analysis. For convenience 3 1d k = will be used in this monograph.

Fourier's theorem [Korn92] states that, any function of x real or complex, defined within the limits x−π≤ ≤ π , that has only a finite number of discontinuities, can be expanded in the Fourier series,

0 0

( ) cos( ) sin( ),k kk k

f x a kx b kx∞ ∞

= =

= +∑ ∑ (10.15)

by using,

cos sin ,ixe x i x≡ + (10.16)

so that,

( ) ,inKxn

n

f x a e∞

=−∞

= ∑ (10.17)

with the Fourier coefficients,

1 ( ) ,2

ikxka f x e dx

+π−

−π

=π ∫ (10.18)

and,

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1 ( ) .2

ikxk ka f x e dx a

+π∗

−−π

= =π ∫ [17] (10.19)

The vector potential field can be converted from a continuous expression to a discrete expression using the Fourier series expansion. This is done by assuming the potential field is a periodic non degenerate dynamical system of k degrees of freedom, defined by equations connecting the coordinates and their time differential coefficients. Each coordinate can be expanded in the form of a multiple Fourier series in the time t, giving,

( )i ie e⋅ ∗ − ⋅= +∑ k r k rk k

k

A a a (10.20)

explicitly indicating that AA is real and where the coefficients ka are vector

functions of time such that i te− ω≈ka with kω ≡ ω = k . [18] Since the vectors

17 The term 1 2π is inserted for convenience of the Fourier theory and has no

particular purpose for this development.

18 Jean Baptiste Joseph (Baron: conferred by Napoleon in 1808) Fourier (1768–1830) formulated his theorem for complex variations in 1807. The development of Fourier's trigonometric series expansion technique was motivated by the description of the heat diffusion between disjoint masses and in special continuous bodies based on the diffusion equation ∇ =2v k v t∂ ∂ [Gill72], [Ligh60], [Heri75]. Fourier found that any periodic function could be expressed as a sum of sine functions. The harmonic analysis makes the study of sound, heat, light and all periodic phenomena amenable to mathematical treatment.

Fourier's contribution to mathematical physics is best illustrated by his 1822 publication Théorie Analytique de la Chaleus (Analytical Theory of Heat) [Four22]. In this work Fourier described his theory of the conduction of heat. In previous theories, heat was seen as the flow of an imponderable fluid the caloric, which was responsible for the repulsive force in matter. Fourier formulated his theory on rational mechanics using differential equations to characterize the conduction of heat. Fourier's contribution to physics was to describe heat conduction in terms of a mathematical theory independent of any physical behavior.

His description was based on the effects of heat on the temperature distribution in materials, not on the way in which the repulsive power of heat determined the physical state of the material. Fourier's analytical approach to the study of heat had an important influence on later mathematical descriptions of electromagnetic phenomenon. Fourier stressed the distinction between a mathematical theory and a physical interpretation. This approach allowed models of nature to be built using mathematical tools. These models could then be compared with physical reality using experiment — but it was the formulation of the model that moved the study of electromagnetism forward.

The Fourier expansion and subsequent Fourier transform allow time dependent descriptions of Maxwell's equations to be transformed into frequency based descriptions. According to Fourier every wave of the form f x ct( )− can be decomposed into an integral

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are functions of time they behave like periodic functions with frequency cω = k and the expansion of the field appears as an expansion of

propagating plane waves.

The divergence condition 0∇ ⋅ =A (transversallity condition) states that the complex vectors ka , represented by the Fourier coefficients, are orthogonal to the corresponding wave vectors kk , such that ( ) 0t ⋅ =ka k and

( ) 0t∗ ⋅ =ka k , which can be verified by taking the divergence of each side of Eq. (10.20) using,

i ie i e⋅ ⋅∇ =k r k rk (10.21)

If the vectors ka are specified, the field in a volume 3x y zd k dk dk dk= is

completely determined. These quantities may be regarded as a discrete set of classical field variables.

It is convenient to include the polarization vectors in the notation for the wave vector, through the following notation. The polarization vectors in Eq. (5.37) can be used to define the wave vector of,

1 2 0ε ⋅ = ε ⋅ =k k , (10.22)

,i j i jε ⋅ ε = δ (10.23)

The Fourier coefficients can now be expanded as,

( ) ( )1,2

t a tλ λλ=

= ε∑k k ka . (10.24)

For the electric field,

( )i ie et

⋅ ∗ − ⋅∂= − = − +

∂ ∑ k r k rk k

k

AE a a& & (10.25)

or simply,

( )i ii e e⋅ ∗ − ⋅= +∑ k r k rk k

k

E k a a (10.26)

For the magnetic field,

over such waves: f x ct A e di t x c( ) ( ) .( / )− = −

−∞

+∞z ω ωω The coefficient A( )ω is obtained from

f x ct( )− by the Fourier transformation: A f x ct e dti t x c( ) ( ) .( / )ωπ

ω= − −

−∞

+∞z12

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( )i ii e e⋅ ∗ − ⋅= ∇ × = × − ×∑ k r k rk k

k

B A k a k a (10.27)

The Fourier expansion expresses the field in terms of a series of discrete parameters — the vector ka in place of the continuous parameters given in Eq. (10.20) Because of the orthognality of the trigonometric functions in the Fourier expansion of the vector potential, the following equation is required:

2

22 0d k

dt+ =k

ka a , (10.28)

where 2k is the square of the length of the vector kk . Each of these variables will be used later in the formulation of the Hamiltonian representation of the radiation field. [19]

§10.6.2. Planck’s Conclusions Using the Vector Potential

With the vector potential written in terms of ka , it is possible to transform the expression for the energy stored in the field from an integral over the volume to a sum over the various ka ’s and their time derivatives, giving:

( ) ( )2 2 2 2 212 16

VW E B dv K= + = +∑∫ k kk

a a& (10.29)

Because of the orthognality of the trigonometric functions most of the terms in the integral vanish, and the remaining integrate to 8V . Since this term contains only the squares of their derivatives, the ka ’s are normal coordinates. Eq. (10.29) is an exact expression for the energy in the field. It contains no assumptions regarding the nature of the field except for the periodicity of the boundary conditions. If there were only one field present, the distribution of energy among the different normal coordinates would remain constant and there would be no approach to an equilibrium distribution.

19 With the introduction of the Fourier representation of the radiated field, the

groundwork has been laid for the quantum mechanical description of electromagnetism that will take place in later sections. An important linguistic change will also take place. The classical vocabulary of describing a physical phenomenon as is must now be replaced with the provisional statement acts as though it were [Baey92]. This change results from the mathematical descriptions of the physical processes that are counter intuitive in nature.

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However, black body radiation is radiation in equilibrium with matter, and the presence of the matter provides the means by which energy can be transferred from one normal coordinate to another, maintaining the state of equilibrium. This state can be described in statistical terms without reference to the process which created it. According to the equipartition theorem of classical statistical mechanics, when equilibrium is obtained, the average energy associated with each normal vibration is E kT= .

Eq. (10.29) demands that each normal coordinate vibrate as a simple harmonic oscillator of frequency 2Kν = π . In order to determine the energy of the electromagnetic field as a function of frequency, it is necessary to find how many normal vibrations have frequencies in the range ν to dν + ν . This is equivalent to finding the number of vectors kk whose lengths are in the range 2πν and ( )2 dπ ν + ν . The components of the

kk ’s are restricted by the relation ( )1 2 32 L k k k= π + +k i j k . The length of such

a vector is ( )2 2 21 2 3k k k L+ + = ν . The number of these vectors in a solid angle

of 2π and in the range corresponding to dν is 3 22 L dπ ν ν . The total number of coordinates whose frequencies lie between ν and dν + ν is given by:

28dN V d= π ν ν . (10.30)

According to classical statistics, the energy distribution in a black body is then given by:

8E d VkT dν νν = π ν . (10.31)

This expression is contradicted by experiment in many ways. [20] Planck found that in order to correct these problems a change in the

20 There are several problems that arise from using classical electromagnetic theory

to predict the energy distribution of a black body. If a cavity is heated and the frequency of the radiation being emitted by the cavity through a small hole is measured several problems arise with the predicted results:

(1) There is an infinite total energy in the radiating cavity. This cannot be immediately disproved experimentally because the total energy in not a directly observable quantity, but it is an undesirable result.

(2) The radiation in the cavity has an infinite specific heat. This is not possible, since the temperature of the cavity can be raised with a finite amount of energy.

(3) The specific heat is independent of the temperature and the total energy is proportional to the temperature. This is also not possible since it can be shown thermodynamically that the total energy in the cavity is proportional to the fourth power of the temperature T.

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statistical treatment of the radiation was needed. He assumed — quite arbitrarily — that each normal coordinate can vibrate with an amplitude which gives it the energy nhν , where n is an integer and ν is the frequency of the normal vibration. He also proposed that the probability of an oscillator has the energy nhν is proportional to nh kTe − ν , which is not really different from the assumptions based on classical mechanics. The critical point made by Planck was the assumption that energy came in discrete values and was proportional to the frequency. With these assumptions the average energy in one normal coordinate whose frequency is ν is given by:

0 0

0 0

1

nh kT nx

n nnh kT

nh kT nx

n n

dnh e h ehdx

ee e

∞ ∞− ν −

= =∞ ∞ − ν

− ν −

= =

ν − νν

= =−

∑ ∑

∑ ∑ (10.32)

From Eq. (10.31) and Eq. (10.32) the energy distribution law is given as:

3

81h kT

hE Veν ν

ν= π

−. (10.33)

This expression is Planck’s Law and agrees closely with experimental measurements. Through this development Planck’s constant h became a famous icon for modern physics and the concept of quanta of radiated energy was formed.

§10.7. RADIATION FIELD EXPANSION USING CANONICAL VARIABLES

By adopting the Hamiltonian description of the electromagnetic field, the formulation of the field equations is reduced to finding an appropriate expression for the potential and kinetic energies of the field.

In order to transform the quantities from classical form, stated in terms of the Fourier series expansion of the vector potential, to quantum form, a canonical set of Hamilton equations are generated [Dick60]. Eq. (10.32) expresses the field in terms of a series of discrete variables in place of a series continuous variables.

(4) The energy is concentrated in the high frequencies and the energy per unit frequency range is proportional to the square of the frequency. Experiments show that the energy as a function of frequency has a definite maximum at a frequency which is a function of the temperature.

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The canonical field variables of the classical field potential developed in the previous section are,

( )14n n nQ ∗= +π

a a , (10.34)

and,

( )4n n niP Q∗− ω

= + =π ka a & , (10.35)

where the canonical variables Qk are generalized coordinates and Pk are

generalized momentum(s). [21] The field vector potential is now expressed in terms of the canonical variables as,

14 cos sinn n

n

Q P = π ⋅ − ⋅ ω ∑A k r k r . (10.36)

The Hamiltonian H can be expressed in terms of the canonical variables Qk and Pk by calculating the total energy of the electromagnetic field as,

( )

2 2

22

1 ( ) ,21 1 .2

dv

c dvt

= +

∂ = ∇× + ∂

∫

∫

E B

AA

H

(10.37)

The derivation of the Hamiltonian of the electromagnetic field, shown in Eq. (10.37) is shown in detail in the following section. The method of this derivation is important in that it lays the groundwork for the quantum mechanical representation of the radiation field.

When the field potential AA is expanded in terms of the canonical field variables, the Hamiltonian becomes,

21 The operators for the electric and magnetic fields can be obtained by analogy with

those for a mechanical oscillator. The identification of these operators with the electromagnetic fields should not be regarded as established principles but rather a hypothesis necessary to make the transition from classical mechanics to quantum mechanics.

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( )2 2 21

2

( ) ( )2 ,2 2

.

c Q iP c Q iPc

P Q

ω − ω +ω = ω ω

= + ω

∑

∑

k k k k

k

k kk

H (10.38)

This expression is referred to as an oscillator expansion of the field. [22] Each vector nP and nQ is perpendicular to the wave vector kk and is denoted by ,nQk , ,P αk giving the Hamiltonian as,

( )2 2 21, ,2

1,2

.P Qα αα =

= + ω∑ ∑ k kk

H (10.39)

Each term in the above expression corresponds to a traveling wave with a definite wave vector and polarization and has the form of the Hamiltonian for a one-dimensional oscillator. Thus the Hamiltonian is the sum of independent terms, each of which contains only one pairs of quantities ,Q αk and ,P αk .

§10.8. SCHRÖDINGER’S EQUATION

During the years 1828–1837, William Rowan Hamilton noted the formal similarity between the decryption of a light ray in the field of optics and the motion of a particle in the presence of the potential field. The basis of the that waves are related to particles started with Fermat in 1657 when he discovered a principle describing the path of a light ray. If a light ray moves between two points A and B in a medium where the index of refraction is a function of the spatial coordinates ( ), ,n n x y z= , then the path the ray takes is determined by the condition that the time of

22 The original concept for the oscillator expansion came in the 1920's when Born,

Heisenberg, Jordan and Paulie formulated a quantum description of Einstein's energy description of a dynamical system, E nhn = ν [Pais86]. Born, Heisenberg and Jordan [Born25] utilized a one–dimensional model — the energy fluctuations in a segment of a vibrating string with length l and fixed endpoints. The displacement of the string ( ),u x t

at point x and time t was, ( ) ( )1

, sink kk

u x t q t x∞

=

= ν∑ where ν πk k

l= , k = 1 2, ,…, which

reduces to an infinite set of uncoupled oscillators with coordinates ( )kq t . The total string

energy is then ( )2 2 212 k k

k

p q= + ω∑H , ω πν= 2 which can be rewritten

as, ( )12k

k

n h= + ν∑H . This Hamiltonian forms the basis of the quantum mechanical

description to be developed later.

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traversal is a minimum. If the phase velocity of the ray is u c n= , the time of the traversal for the distance ds is given by ds u . Since the phase velocity can also be given as u =νλ , where λ is the wavelength. The time for traversal can now be given as,

1 B

A

dsI =ν λ∫ . (10.40)

The minimum condition becomes,

( )10 0, ,

B

A

I dsx y z

δ = ⇒ δ =λ∫ . (10.41)

A similar principal was developed in the section on Hamiltonian mechanics, where the kinetic energy of a particle, T, in motion has the least possible value, such that,

0B

A

Tdtδ =∫ , (10.42)

which can be restated in terms of the momentum of a particle by the use of 2Tdt mvds pds= = , which gives,

0B

A

pdsδ =∫ . (10.43)

This formal similarity, described by Hamilton, between the path of a ray of light and the motion of a particle, was the starting point for Prince Victor de Broglie (1892–1987) nearly one hundred years later. [23] In 1923 de Broglie [D’brog24] proposed that a particle’s momentum p and the energy E of a wave with wavelength λ and frequency ν , could be related as,

hp

λ = and Eh

ν = . (10.44)

23 Prince Louis–Victor de Broglie presented his doctorial thesis “Recherches sur la

Theorie des Quants” (“The Theory of Matter Waves”) to his committee at the Sorbonne in 1924. De Broglie’s concept of matter waves was beyond the thesis committee’s ability to make any critical review. They awarded de Broglie his degree after Einstein commented on his work. De Broglie was awarded the Nobel Prize in 1927 after Clinton Joseph Davidson (1881–1958) and Lester Halbert Germer (1896–1971) verified (in 1927) that electrons can be diffracted by a crystal.

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This relationship implies that a particle with a well defined momentum p would experience diffraction, as if it were a wave with a wavelength of h pλ = . The experimental verification of de Broglie’s ideas came in 1927 when Clinton Joseph Davisson (1881–1958) and Lester Halbert Germer (1896–1971) at Bell Laboratories and Sir George Paget Thompson (1892–1975) in England observed that an electron beam diffracted from the surface of a nickel crystal, just as light is diffracted by a ruled grating. [24] de Broglie compared the energy and momentum of a particle, 2 2 4 2 2E m c p c= + with the form of a plane wave, ( )i te ⋅ −ωk r . This concept of a matter wave became the basis of Erwin Schrödinger’s (1887–1961) formulation of wave mechanics. [25] Schrödinger first published his results in a series of papers starting March of 1926 [Darr86].

§10.8.1. Development of Schrödinger’s Equation

There are actually two Schrödinger equations, one time–dependent and one time–independent. The approach suggested by Schrödinger was to postulate an equation that could vary in both space and time — in a wave–like manner, thus the wave equation. There are actually two Schrödinger equations. The time–dependent equation provides a description of the wave function as it evolves in time, once the initial conditions are determined. These initial conditions are contained in the potential that would be experienced by a particle. All the solutions to the time–dependent wave equation vary over time in a wave–like manner, these solutions may be converted into eigenfunctions of a time–independent wave equation multiplied by a time–dependent factor related to the energy of the wave. The simplified (time–independent) solution to the wave

24 This observation won Davisson the Nobel Prize in 1937, [Davi27].

25 Schrödinger came upon the wave equation while attending a seminar held jointly by the Eidgenössische Technische Hochschule in Zurich and the University of Zurich. The seminar was lead by a professor at the University of Zurich, Peter Joseph Wiheim Debye, who suggested that Schrödinger present a report on de Broglie’s concept of particle waves. During Schrödinger’s talk Debye remarked that the proper description of the particle wave required a wave equation [Bloc76].

Several weeks later Schrödinger had formulated his wave equation and produced four papers in Annalen der Physik titled “Quantiziering als Eigenwertproblem,” [Blin74], [Schr26]. In three papers Schrödinger derived both the time independent and the time dependent equations and formulated their solutions for the harmonic oscillator. Schrödinger arrived at his equation by using an analog between classical geometric optics and classical particle dynamics.

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equation in which the time–independent wave equation is solved, them multiplied by a sinusoidal factor related to the energy. The solutions to the time–independent wave equation are simply the amplitudes of the full time–dependent equation. In both cases the wave equation has no physical interpretation. It simply contains information regarding the system to which it refers. The most important characteristic of the wave equation is that the square of its magnitude is a measure of the probability of finding a particle described by the wave function at a given point in space.

The development of Schrödinger’s wave equation can be approached using a classical wave equation example. A wave equation can be formulated for a vibrating string can be constructed which can then be extended to the Schrödinger wave equation. [26] Assume a transverse displacement of a string from its undisplaced position be represented by the function ( ),x tψ , where x is the position along the string and t is time. If Newton’s equation F ma= is applied to an element of the string, the force arising from the tension of the string, T, will be acting in different directions on the two ends of the string element, dx. The transverse component of the tension force can be approximated by T x∂ψ ∂ as shown in Figure 1.0. At the point x dx+ , this tension force can be approximated by T x∂ψ ∂ , but there is a force at the point x of the opposite sign, T x− ∂ψ ∂ , where the derivative is computed at the point x. The sum of

these forces gives the total transverse force on the string and can be approximately given as, ( )2 2T x t dx∂ ∂ . This force must equal mass times

the acceleration. If µ is the mass per unit length, the mass of an element of the string is then dxµ . The acceleration of the string is given by, 2 2t∂ ψ ∂ , which results in Newton’s equation of motion for the string as,

2 2

2 2Tx t

∂ ψ ∂ ψ= µ

∂ ∂, (10.45)

which is the wave equation for a vibrating string [Gold51], [Arfk85], [Brau93], [Resn60], [Wall72], [Whit37].

26 This example of a vibrating string re–occurs throughout the history of physics.

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T

x d x+x

( )T xx

∂ψ−

∂( )xψ

( )T x dxx

∂ψ+

∂

( )x dxψ +

Figure F igure §10§10 .. 11 — Tensions in an element of a string.

Ordinarily the solutions to this wave equation vary sinusoidally with time so that ( )sinu x tψ = ω , where ( )u x provides a separation of variables

of the differential wave equation. In this case 2 2 2t∂ ψ ∂ = − ω ψ , which allows the wave equation to be rewritten in the form which cancels the time dependence,

2

22 0d u u

dx Tµ

+ ω = , (10.46)

which has the solution,

sin2 xu = πλ

, (10.47)

where λ is the wave length of the vibration given by, 2 Tπ λ = ω µ , which

allows the wave equation to be simplified to the form,

22

2

2 0d u udx

π + = λ . (10.48)

This is the standard form of a wave equation with the time parameter eliminated. This equation assumes that properties of the string, or in the case of the generalized three dimensional problem, the properties of the vibrating medium do not vary from point to point. In the case where the properties do vary with position, the wave length of the solution to the wave equation also vary with position. It is this property that can be used to derive the wave equation for de Broglie waves. Since the wavelength

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can be a function of position, it becomes a differential equation whose solution becomes very difficult, which can only be solved in certain special simple cases.

§10.9. FORMULATING SCHRÖDINGER’S WAVE EQUATION

The path taken by Schrödinger in the formulation of the wave equation was quite simple — now that hind sight provides a clear view of the events of the 1920’s and 30’ [Sopk80]. Given a particle of mass m moving along the x axis subject to a conservative force, ( )V x . Using the mathematics developed for the vibrating string, the momentum and hence the wave length at every point along the particles path can be described.

The conservation of energy states that the total energy, E, is given by 2 2E mv V= + or using the momentum representation 2 2E p m V= + . The

momentum of the traveling particle is then given by,

( )2p m E V= − . (10.49)

In de Broglie’s postulate the momentum is related to the wavelength of the particle by, p h= λ or h pλ = . Replacing the momentum gives,

( )2h m E Vλ = − . Substituting the expression the wavelength into the

wave equation Eq. (10.48)gives,

( )2 2

2 2

8 0d u m E V udx h

π+ − = . (10.50)

In order to generalize this expression for three dimensions, the wave equation becomes,

( )2

22

8 0mu E V uhπ

∇ + − = . (10.51)

§10.10. SCHRÖDINGER’S TIME DEPENDENT EQUATION

The time independent wave equation for the tension in the string developed above, involve the second time derivative. Only by assuming that the function varies sinusoidally with time can time be eliminated from the wave equation. By making use of the Hamilton description of the motion of particles, the bridge to quantum mechanics can be made through a simple mathematical transformation. As developed in the previous section, Hamiltonian function is simply the energy of the system

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expressed as a function of the coordinates and momenta. In rectangular coordinates, the Hamiltonian function for a particle of mass m moving in a potential field with energy V, is given by,

( ) ( )2 2 21 , ,2 x y zH p p p V x y zm

= + + + . (10.52)

The law of conservation energy requires that H E= , where E is a constant of the system. This relationship determines the momentum as a function of position. Using de Broglie’s postulate, the wave length of the particle can now be determined as a function of position.

It was observed by Schrödinger, that the wave equation can be formulated in the following manner. By replacing the momentum variables by differential operators the Hamiltonian can also be transformed into a differential operator. Thus,

2xhpi x

∂→

π ∂,

2yhpi y

∂→

π ∂,

2zhpi z

∂→

π ∂. (10.53)

By substituting these expressions into the Hamiltonian, it is changed into a quantity called the Hamiltonian operator,

2

228hH Vm

→ − ∇ +π

. (10.54)

Now taking the equation H E= and applying the differential operators to the wave function u, results in a wave equation Hu Eu= , which can be expanded to,

2

228h u Vu Eum

− ∇ + =π

. (10.55)

This expression is identical to the Schrödinger equation, which was derived in an intuitive manner. The last substitute for a differential operator can be made using,

2hEi t

∂→ −

π ∂. (10.56)

By substituting the energy differential operator into the wave equation and changing the symbol for the wave function to ψ , results in Schrödinger’s time dependent equation of,

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2

228 2h hVm i t

∂ψ− ∇ ψ + ψ = −

π π ∂, (10.57)

which is the form of Schrödinger’s equation involving time.

§10.10.1. The General Solution to Schrödinger’s Equation

By separating Schrödinger’s equation into a function of the spatial coordinates and a function of time, the solution to Schrödinger’s equation takes on the form,

( ) ( ), ,u x y z T tψ = , (10.58)

which results in,

2

22

1 18 2h h dTu Vu

u m T i dt − ∇ + = − π π

. (10.59)

Schrödinger’s equation is known as a homogeneous linear differential equation. The equation is linear because it does not contain ψ or its derivatives in powers higher than the first and it is homogeneous because it does not contain any terms dependent on ψ . Any such differential equation has the property that the sum of any two solutions to the equation is itself a solution. This allows for the general solution to be made up of other solutions, such that,

( ) ( )2 , ,iiE h ti i

i

c e u x y z− πψ = ⋅∑ , (10.60)

where ic is an arbitrary complex constant.

§10.10.2. Semi–Classical Theory of Radiation

In the Hamiltonian formulation of Maxwell's equations, the electromagnetic fields themselves become real entities, separate from the charges that generate them. The Hamiltonian given in Eq. (10.39) can now be used to describe the quantum mechanical motion of a charged particle. The Schrödinger equation is given in its simplest form as,

it

∂ψ = ψ

∂h H . (10.61)

Assuming a time–independent Hamiltonian, the following separate of the terms of the equation can be made,

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iEte −ψ ⇒ ψ h , (10.62)

where E is the total energy of the charged particle.

When this charged particle is placed in an electromagnetic field described by the vector potential AA , the Schrödinger equation becomes,

2

2

2iei V

t m mc ∂ψ

= − ∇ + ⋅ ∇ + ψ ∂ Ah hh . (10.63)

Although the motion of the particle is quantized in this description, the electromagnetic field is treated classically, which allows the potential to be specified with complete certainty in space–time by using Maxwell’s equations. This approach results in an correct description of the influence of the external radiation field on the charged particle — absorption and induced emission, but not of the influence of the charged particle on the radiation field — spontaneous emission [Beth64], [Schi68]. The reason for the correct result in the first case lies with the correspondence principal. When the radiation field is quantized it is regarded as a collection of quantized oscillators, with the thn excited state of the oscillator describing n photons in the electromagnetic field. The semi–classical description of the radiation field provides the correct results because the numbers of photons representing the field are very large. The details of this description will be developed in the next chapters.

Because the vector potential AA is linear, the results developed for intense radiation fields will also hold for weak fields, or low values of n. This consideration does not hold for spontaneous emission though. The spontaneous emission of a photon from a accelerated charged particle occurs regardless of the presence of an external electromagnetic field. At least one quantum of radiation must be emitted when the charged particle is accelerated. This effect is not linear in the field and the correspondence principal cannot be extrapolated in a simple way to the emission of one photon. For a satisfactory theory of both the electromagnetic field and the charged particle, the electromagnetic field must be quantized.

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We must accept the truth, even if it changes our point of view.

— George Sand

Copyright 2000, 2001 11–1

§11. GAUGE THEORY

In the previous sections gauge transformations and the expression ∇ ⋅A have been used without consideration to its motivation or importance. In 1919, only two fundamental forces were thought to exist — electromagnetism and gravity. In the same year the first experimental observations of starlight bending in the gravitational field were made during a total eclipse of the sun [Klub60]. The fundamental concept underlying Einstein's special relativity and general relativity is that there is no absolute frame of reference in the universe. The physical motion of any system must be described relative to some arbitrary coordinate frame specified by the observer, and the laws of physics must be independent of the choice of frame.

Nature uses only the longest threads to weave her patterns, so each small piece of the fabric reveals the organization of the entire tapestry.

— R. P. Feynman [Feyn94]

In special relativity an inertial reference frame is defined as one which is moving with uniform velocity [Mill81]. In general relativity, the description of relative motion is more complex for a coordinate system moving in a gravitational field. The essential difference between special and general relativity is that in General Relativity, a reference frame can only be defined locally or at a single point in a gravitational field. Since measurements can be made at different locations, the Lorentz (linear) transformations used in special relativity do not produce proper results. Einstein solved this problem be defining a new mathematical relation known as a connection in which the transformation from one reference frame to another is no longer assumed to be linear but consists of a transformation containing second derivatives of the space and time coordinates resulting from the curvilinear coordinate systems of general relativity [Eins55], [Misn73]. These second derivative terms arise from the curvilinear coefficients and represent the components of the connection between one reference frame and another. [1]

1 In General Relativity text books these connections are referred to as affine

connections or Christoffel symbols. Once again a simple example is used here. The gravitational connection is not simply the result of using curvilinear coordinates. The value of the connection at each point is space–time dependent on the properties of the gravitational field.

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This concept was generalized by Hermann Weyl in 1919 with the introduction of the idea of gauge invariance [Weyl19]. The basic concept of Gauge Invariance is that a physical system is invariant with respect to some rigid (space–time independent) group of continuous transformations, G, then it remains invariant when the group is made local (space–time dependent). That is ( )G G x→ where x xµ= . This transformation is valid if

the ordinary space–time derivatives µ∂ are changed to covariant

derivatives Dµ . These covariant derivatives take the form ( )D A xµ µ µ= ∂ +

where ( )A xµ are vector fields. This means that invariance with respect to

the local symmetry forces the introduction of the vector fields ( )A xµ and

determines the manner in which these fields interact with themselves and with matter.

Generalizing the concept that all physical measurements are relative, Weyl proposed that the magnitude or norm of a vector should not be an absolute quantity, but should depend on its location in space-time. A new connection is necessary in order to relate the lengths of vectors at different positions. This connection is known as scale or gauge invariance and provides a new property of local gauge symmetry. [2] The new concept here,

An example of the space–time dependency is an aircraft traveling along a great circle route from Peking to Vancouver. Early in the flight the aircraft is traveling in a northerly direction, but later in the flight it will travel in a southerly direction, although in the coordinate system of the earth's surface, the flight takes the straightest route between the two cities. The apparent change in direction indicates a turning, not of the aircraft's route, but in the coordinate system by which the aircraft's flight is described. The vector v describes the direction and speed (the velocity) of the aircraft remains constant throughout the flight. However the individual components of the vector, the latitude and longitude of the vector are not constant and are changing along the route. The changes to the individual vectors come about through turning coefficients that tell the navigator to turn the components of the vector components in order to keep the overall vector constant. These turning coefficient are used to describe the turning of the lines of latitude and longitude relative to the great circle route of the aircraft [Misn73].

2 The term gauge was first introduced in 1919 by Hermann Weyl in the context of a unified field theory of gravitation and electromagnetism. The basis of this theory was a symmetric tensor gµν of gravitation and the electromagnetic 4–vector AAµ . Weyl's theory

required that these two quantities be invariant under the transforms, ′ =g e gµνχ

µν and

′ = −AA AAµ µµ∂χ ∂x . The latter is now the familiar gauge transformation for the

electromagnetic potential. The gravitational transformation changes the length defined as, ds g dx dx2 = µν

µ ν into e dsχ 2 . Weyl choose the engineering expression change of gauge for

this transformation. In his first two papers on this subject he called his new invariance Masztabinvarianz, in the sequel paper he introduced the term Eichang — gauging.

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is that Maxwell's equations are invariant under a gauge change. This will become important when the quantum mechanical expansion of the radiation field in developed in this monograph.

Since electromagnetic theory was developed prior to Weyl’s gauge invariance, gauge symmetry groups and their descriptions of matter and fields did not play an essential role in defining electromagnetism. It is through modern particle physics and the study of nuclear forces that gauge theory has come to be applied to electromagnetism. With the discovery of the mediating particle and the charge independence of the nuclear force, gauge theories have been the cornerstone of the description of the string force. The Yang–Mills theory revived the idea that elementary particles have internal degrees of freedom which could be unified by using a geometric description — the afine connection of general relativity is similar to the phase of a wavefunction of the rotation in 3 dimensional space of the isotropic spin group SU(2). The symmetry space of the gauge group provides a local non–inertial coordinate frame for the internal degrees of freedom. The geometric nature of gauge theory can be represented as a fiber bundle [Hoft80], [Dres77].

§11.1. CLASSICAL MECHANICS EXAMPLE OF A GAUGE INVARIANCE

In order to understand the concept of gauge symmetry a simple classical mechanics example will be used which will lead to the development of local gauge invariance of the electromagnetic field and a new understanding of the field vector potential. [3] Consider a particle following a simple harmonic motion in a plane. In such an oscillator, both the x and y coordinates oscillate sinusoidally with the same frequency. The particle's motion will describe an elliptical path in the x, y – plane. The two motions along the x and the y axis can be combined into a single complex variable, z x iy= + . The following diagram describes the position of the particle in this complex plane,

Weyl's described these gauge transformation in 1928 [Weyl28] where he stated I now believe that ... gauge invariance does not tie together electricity and gravitation but electricity and matter.

3 Locality in the gauge theory refers to the idea that events can only influence other events in their immediate vicinity. A more restrictive meaning is that all physical events are assumed to propagate no faster than the speed of light, that is two spatially separate events that occur simultaneously cannot be causally connected.

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x

iy

z x iy= +r

θ

Figure 3.0 — Local Gauge Symmetry – defined for a mechanical model in the complex plane. The angular position of a particle, z, can be defined in an x and y coordinate system or in an angular coordinate system. If the angular coordinate system is used, the reference angle θ can be changed without changing the equation for the particle’s location iz x iy z ze − α′= + ⇒ = .

The magnitude z x iy= + is the distance of the particle form the origin and the polar angle θ is its angular position. The harmonic oscillators motion is described by,

, (11.1)

where 2π ω is the time period of the oscillator.

In this model, the angular position of the particle is measured relative to the x–axis. This measurement is simply a matter of convenience since the absolute axis of measurement is irrelevant. With this irrelevancy comes the freedom of choice — to rotate the x, y – axis counterclockwise by an angle α to the new axis x' and y'. This rotation results from replacing θ with θ − α , which results in regauging θ .

The regauged equation for z is then given by iz ze− α′ = . Since the change in the measurement α is constant in time, the irrelevance of absolute θ and the associated invariance of the equation for z' this transformation is called a global gauge, global because the shift of θ is a fixed α . If the shift is a function of time, ( )tα , the transformation becomes a local gauge invariance, i.e. local in time. With ( )tα varying in time, with the factor ie α in the gauge transformation can no longer be absorbed by the redefinition of z, because the time derivative of z will generate additional terms involving time derivatives of ( )tα , resulting in a loss of invariance.

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This invariance can be regained by introducing a term that will compensate for the additional time derivative terms in z. By replacing the time derivative d dt with ( )d dt i t− A , where ( )tA is the compensatory or gauge field. The time dependent shift ( )tα amounts to rotating the reference frame with an angular velocity d dtα . Such a rotation gives rise to fictitious forces — in this mechanical analog the centrifugal force and the Coriolis force. It is the existence of the gauge field ( )tA , that generates

the force. [4]

§11.2. ELECTROMAGNETIC FIELDS AND GAUGE TRANSFORMATIONS

Electromagnetic equations developed by Maxwell utilize the magnetic field BB and the electric field EE . These fields are related to the vector potential AA and scalar potential φ as shown in Eq. (4.7). In Weyl's original gauge theory gravitational fields were described by a connection which gives the relative orientation between local reference frames in space–time. By generalizing the concept that all physical measurements are relative, Weyl proposed that the absolute magnitude of norm of a physical vector should also not be an absolute quantity, but should depend on its location in space. A new connection [5] would then be necessary to relate the lengths of vectors at different positions in space–time.

4 The centrifugal force and the Coriolis force are examples of pseudo forces which

occur in coordinate systems that are rotating. In the case of the centrifugal force an observer in a rotating coordinate system will feel a force pressing him to outside walls of a carousel. This force is due merely to the fact that the observer does not have a Newtonian coordinate system in which to measure the real force.

Another pseudo force in a rotating system is the Coriolis force. Like the centrifugal force, it results from the rotating system and the angular momentum applied to the observer. For the observer to move radially in the rotating system a torque must be applied. In order to walk along the radius of the carousel, one has to lean over and push sideways.

5 In 1919 only two fundamental forces were thought to exist, gravity and electromagnetism. The confirmation by Albert Einstein of the gravitational bending of star light inspired Herman Weyl to propose his gauge invariance in 1919. The difference between special relativity and general relativity is that a reference frame in special relativity can be defined everywhere is space — in general relativity the reference frame can only be defined locally or at a single point in the gravitational field. This creates a problem in which measurements of the path of a test particle made at different locations do not follow the Lorentz transformation between the local reference frames. Einstein solved the problem of relating reference frames by a mathematical relation known as a connection. By using a curvilinear coordinate system Einstein constructed a curvilinear transformation between the two reference frames maintains the proper relationship

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Weyl's gauge invariance can be expressed using a physical model. A vector at position u has a norm ( )f u . If this vector is shifted to a new set of coordinates du, the norm becomes ( )f u du+ . Expanding this norm in the first order gives,

( ) ( )f u du f u fdu+ = + ∇ . (11.2)

In order to normalize to gauge units of measure, the multiplication by a scaling factor, ( )S u , is introduced. This factor can be seen as a change in the scale of a measuring device as a function of location as shown in Figure 4 .0Figure 4 .0 .

u

f u( )

du

f u du( )+S u du( )+

S u( )

Figure 4.0 — Gauge scaling as a function of position

Defining the scale factor as unity as u, the scaling factor at location u du+ is then given by,

( ) 1S u du Sdu+ = + ∇ . (11.3)

The norm of the vector at u du+ is then equal to the product of Eq. (11.2) and Eq. (11.3), giving,

( )Sf f S fdu fdu= + ∇ + ∇ . (11.4)

For the case of a constant vector, its norm is changed by,

between the measurements so that their results are equivalent for equivalent processes. The analogy with the curvilinear coordinates states that inertial reference frames and curvilinear coordinates are similar. Einstein's insight was to generalize this similarity and derive the revolutionary idea of replacing the Newtonian description of gravity with curvature of space–time in the General Theory of Relativity [Misn73], [Eins55].

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( )S fdu∇ + ∇ . (11.5)

The derivative of S∇ is the new mathematical connection associated with the gauge change.

Weyl made the connection between the gauge S∇ and the electromagnetic field potential AA . This could be done since the gauge connection transforms like the vector potential. For electromagnetism this transformation is,

S→ + ∇A A or → +∇χA A . (11.6)

The differential equations for AA and φ can be derived from Maxwell's equations and are given by,

22

2

2

,

( ) .

t t

t

∂φ ∂ ∇ ∇ ⋅ + − ∇ + = ∂ ∂ ∂ − ∇⋅ − ∇ φ = ρ ∂

AA A j

A (11.7)

Using the formulation developed above, the quantum mechanics of the electromagnetic field can now be developed using the potentials φ and AA rather than the fields EE and EE . Schrödinger’s equation for the motion of an electron can now be examined in light of this gauge transformation. Schrödinger’s equation for the electron is:

( )212

ie e im t

∂ψ − ∇ + − φ ψ = ∂ A . (11.8)

This equation is related to the equation of a free particle by making the substitutions:

,

.

iet t

ie

∂ ∂→ − φ

∂ ∂∇ → ∇ + A

(11.9)

This is called the principal of minimal coupling, which introduces the electromagnetic interaction into the free particle system. One of the first problems in this formulation is the fact the Eq. (11.8) is not invariant under the gauge transformation in Eq. (11.6). Eq. (11.6) can be augmented by a space – time dependent phase transformation of the wave function,

( ) ( ) ( ),, ,ie tt e t− φψ → ψxx x . (11.10)

so that the combinations of,

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iet

∂ − φ ψ ∂ , (11.11)

and

( )ie∇ + ψA , (11.12)

will have simple transformation properties and Eq. (11.8) will invariant.

Maxwell's electromagnetic theory contains local symmetry through the field potentials. In Maxwell's theory the value of the electric field is determined by the distribution of charges around a point. By defining the potential created by this charge distribution, the electric field is given by the voltage difference between the various charges. It is the symmetry of the voltage potential that makes Maxwell's theory a gauge theory.

For example, when a system of static charges is considered and the voltage potential between them is measured and then the global voltage potential in which the charges are embedded is raised, the relative voltage between the charges will remain unchanged. That is the relative potential between charges is unaffected by the global potential voltage of the external environment. Formally this can be restated as a gauge symmetry: the electric field is invariant with respect to the addition or subtraction of an overall potential. This is a global symmetry since the result of the voltage measurements remain unchanged only if the external potential remains constant everywhere.

When the charges in the example above are placed in motion, a magnetic field is produced in addition to the static electric field. The complete theory of electromagnetic fields must convert the global symmetry to a local symmetry. It is the presence of the magnetic field and its corresponding magnetic potential that results in a local symmetry. In the symmetry of the interacting electric and magnetic fields, local transformations can be carried out leaving the original electric and magnetic fields unaltered. Any local change in the electric field potential can be combined with a compensating change in the magnetic field in such a way that the electric and magnetic fields are invariant [Hoft80], [Tayl89], [Aitc82].

The solution to these equations are not unique; any new potential ′A and ′φ such that,

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,

.t

→ −∇χ

∂χ φ → φ + ∂

A A (11.13)

where χ is an arbitrary function of x and t, also satisfies these equations. The transform given in Eq. (11.13) is called the Coulomb Gauge transformation.

The significance of the gauge invariance can be seen in the formulation of electromagnetism as a classical Hamiltonian field theory — as developed above. By combining the vector potential and the scalar potential into a 4–vector,

( ),Aµ = φ A (11.14)

so that the transformations in Eq. (11.13) are now given by,

A A A µµ µ µ′→ = − ∂ χ (11.15)

In this Hamiltonian based theory the field momentum is replaced by the canonical momentum,

,p p eµ µ µ→ − A (11.16)

where the subscripts denote the 4–vector (general relativity) form of the momentum and field potential. The result of this transformation is that both Maxwell's equations and the equations of motion for charged particles form a single physical principle. As a result the Lagrangian density contains all the necessary information to describe the interaction of a charged particle with the electromagnetic potential µA . The derivation of

the Lagrangian is given in the 4–Vector Notation section as well as in [Jack62], resulting in,

21 1( ) ,2 4

p eA F F µνµ µ µν= − −L (11.17)

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where Fµν µ ν ν µ= ∂ − ∂A A is the Maxwell field stress tensor which describes

the energy densities of the electromagnetic field as a covariant tensor of

the form

3 2 1

3 1 2

2 1 3

1 2 3

00

00

B B iEB B iEAAFB B iEx xiE iE iE

µνµν µ ν

− − − −∂∂ = − = − −∂ ∂

.[6]

The Hamiltonian of the electromagnetic field interaction with a charged particle can now be derived using the Coulomb Gauge transformation:

xµ µ

µ

∂Λ→ +

∂A A , (11.18)

where Λ satisfies the D'Albertian equation 0 Λ =n . It is always possible to consider a gauge transformation of the type shown in Eq. (355) such that in the new gauge the tranversality condition 0∇ ⋅ =A is satisfied. To see this explicitly:

0 0

( , ) ( , ) ( , ),( , )( , ) ( , ) .

x t x t x tx tA x t A x tt

′ = +∇χ

∂χ = − ∂

A A (11.19)

where,

1 ( , )

( , )4

x tx t dv

x x′∇ ⋅

χ =′π −∫

A, (11.20)

resulting in,

( ) 0′∇ ⋅ = ∇ ⋅ +∇χ =A A , (11.21)

which shows the tranversality condition is satisfied for the vector potential.

The total Hamiltonian for the charged particles and the electromagnetic field is given by the spatial integral of:

6 Along with scalar and vector quantities another quantity enters into the

electrodynamics equations, a tensor. The Euler-Lagrange equations of motion given in relativistic form can be written in a covariant notation where b b b b b ibµ = =( , , , ) ( , )1 2 3 4 0b is a

four-vector. The four gradient is ( ) ( ) ( )x x x x a xµ υ µ µ µυ υ′ ′∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ .

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( )( )

( )

em

2 24

2 20 0

,

1 ,21 ( ).2

em

i A

A A

µ

µµ µ

∂∂= −

∂χ∂ ∂ ∂χ

= + − ⋅∇

= + − ρ +∇⋅

AA

B E E

B E E

LH L

(11.22)

where the last term vanishes when emdv∫H is evaluated.

The interaction portion of the Hamiltonian density is given by:

int ,cµ µ= −j AH (11.23)

where µj is the charge current density. Combining the interaction and

electromagnetic field portions gives:

( )2 2int em

1 ,2

dv+ = + − ⋅∫ ∫B E j AH H (11.24)

Hamilton's principle is formulated in terms of a path that a dynamical system follows between two points as described in Eq. (108). The Lagrange equation described in Eq. (115) is now given as,

( ) 0,qqλ

µλ µ

∂ ∂∂ − =

∂∂ ∂

L L (11.25)

where qµ is now the generalized 4–vector coordinate. By using Eq. (349)

and identifying the components qµ with spatial coordinates the Lorentz

force law for charged particles is derived. By equating qµ with the

electromagnetic potential µA , Maxwell's equations are obtained. By using

the Hamiltonian formalism, the electromagnetic potential becomes an integral part of the canonical momentum, and is treated as if it were a generalized coordinate in the Euler-Lagrange equations.

§11.3. LORENTZ AND COULOMB TRANSFORMATIONS

The relationship previously developed between AA and φ , 0t

∂φ∇ ⋅ + =

∂A

is the Lorentz condition or Lorentz Gauge. For potentials that satisfy this gauge, there is still arbitrariness in their solutions. The Lorentz gauge is commonly used because it leads to the wave equations,

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22

2

22

2

,

.

t

t

∂ φ∇ φ − =−ρ ∂

∂ ∇ − = − ∂ AA j

(11.26)

which treat AA and φ on equal terms.

Another useful gauge of the Coulomb Gauge

0∇ ⋅ =A . (11.27)

From Eq. (110b), the scalar potential φ satisfies the Poisson equation,

2∇ φ = − ρ , (11.28)

whose solution is,

( , )( , ) x tx t dvx x

′ρφ =

′−∫ . (11.29)

The scalar potential in the Coulomb gauge is just the instantaneous Coulomb potential due to the charge density, ( , )x tρ .

The vector potential in conjunction with the wave equation gives,

2

22t t

∂ ∂φ∇ − = − + ∇

∂ ∂AA j . (11.30)

The term involving the scalar potential, in principle, can be calculated from Eq. (361). Since it involves the gradient operator it is a term that is irrotational, that is it has vanishing curl. This allows the term to be cancel the corresponding term of the current density. The current density, or any vector field for that matter, may be written as the sum of two terms,

transverselongitudialj j= +j . (11.31)

The longitudinal or irrotational current has 0lj∇× = , while the transverse or solenoidal current has 0tj∇ ⋅ = . Starting with the vector

identity ( ) ( ) 2j j j∇× ∇× = ∇ ∇⋅ − ∇ together with ( ) ( )2 1 4x x x x′ ′∇ − = − πδ − ,

lj and tj can be constructed explicitly from jj as,

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1 ,41 .4

l

t

jj dvx x

j dvx x

′ ′∇ ⋅ ′= − ∇′π −

′= ∇×∇×′π −

∫

∫j

(11.32)

Using the now familiar continuity equation, 0t

∂φ∇ ⋅ + =

∂A and Eq. (361)

gives,

ljt∂φ

∇ =∂

. (11.33)

This allows the source for the wave equation for AA to be expressed entirely in terms of the transverse current tj ,,

2

22 tjt

∂∇ − = −

∂AA . (11.34)

This gives rise to the transverse gauge. The term radiation gauge is derived from the fact that transverse radiation fields are given by the vector potential alone, the instantaneous Coulomb potential contributing only to the near zone radiation fields, as described in the previous section.

The Coulomb Gauge is often used when no field sources are present. Then 0φ = and A A satisfies the homogeneous wave equation with the fields given by,

,

.t

∂= −

∂= ∇ ×

AE

B A (11.35)

§11.4. GAUGE SYMMETRIES AND POTENTIAL FIELDS

Eugene Wigner (1902– ) was among the first to apply the concept of symmetry to quantum mechanics. In his 1939 paper he showed that the mathematics of group theory could be used to classify quantum particles [Wign67]. [7]

7 In the context of quantum and particle physics a group can be defined as a set of

transformations that leave anything unchanged, whether a specific object or the laws of nature. The mathematics of these symmetry transformations is called group theory. each group can be characterized by the rules that describe the transformation. The transform

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It seems to me that the deliberate utilization of elementary symmetry properties is bound to correspond more closely to physical intuition than more computational treatments

— E. P Wigner [Pais86]

The concept of symmetry plays an important role in gauge theory. Symmetry becomes the glue that connects objects and the laws of physics that govern their dynamic behavior. The central theme is that symmetry is associated with a conservation law. A classical example can be seen in the Euler–Lagrange equation [Gold55], when the Lagrangian density satisfies,

0iq

∂=

∂L

. (11.36)

The action function is cyclic the coordinate iq . The canonical momentum then becomes,

0idpdt

= . (11.37)

and momentum is conserved. Eq. (368) states that the Lagrangian density is invariant in form and content under a translation of the coordinate iq by a time–independent constant ic , such that,

( ) ( ), ,i i i i iq q q c q= +& &L L . (11.38)

In a classical Newtonian system symmetries are understood as variations of transformations of the coordinates, that leave the action function invariant [Ityz80]. In this example continuous transformations are considered in which each transformation is connected smoothly to the identity transformation, which is defined as a transformation which leaves the coordinates unchanged. Because of this property only infinitesimal transformations will be considered since finite transformations may be built from repeated applications of infinitesimal transformations. An infinitesimal transformation of the coordinate iq will be given as

rules themselves do not depend on the objects being transformed, but rather on the mathematical structure of the symmetry group [Wein93].

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i i iq q q→ + δ , where iqδ is an arbitrary function of t satisfying ( )2 0iqδ ≈ . The variation of the action function S under this transformation gives,

,

.

i ii i

i ii i i

S q q dtq q

d dq q dtdt q q dt q

∂ ∂δ = δ + δ ∂ ∂

∂ ∂ ∂= δ + − δ ∂ ∂ ∂

∫

∫

&&

&

L L

L L L (11.39)

There is nothing new in Eq. (379) — it is a tautology. However if Sδ is evaluated for a set of classical trajectories cq which satisfy the original Euler–Lagrange equations of motion then,

b

a c

t

iit q q

S q dtq

=

∂δ = δ ∂

∫L

. (11.40)

If 0iqδ = at time at t= and bt t= , then 0Sδ = and these expressions have reproduced the Euler–Lagrange derivation. However if 0iqδ ≠ at the endpoint time and if 0Sδ = , under this transformation then Eq. (379) gives,

0dGdt

= , (11.41)

where.

ci q q

i

LG qq =

∂= δ

∂&& . (11.42)

If 0Sδ = under this variation, then this transformation is said to be a symmetry of the action function S, and an associated charge G, given by Eq. (379) is conserved by Eq. (373).

An important property of a classical mechanical system is that a time derivative of some function F of the canonical variables can be added to the Lagrangian, such that,

( ) ( )abdF S S F t F tdt

→ + ⇒ → + −L L , (11.43)

or

( ) ( ), ,

( ), , ( ), ,j i i ji x y z

d F t F tdt =

′ → + + ∂∑A r r A r rL L . (11.44)

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To calculate the new Lagrangian ′L it is necessary to integrate ′L over all spatial coordinates. The integral of F∇ ⋅ then transforms into a surface integral at infinity, which in turn vanishes. ′L then differs from L only by the time derivative of the function F.

Such a transformation does not affect the dynamics of the system during the time interval since F cannot affect the variations in that interval. Because of this property the transformation in Eq. (379) has been used as the definition of canonical transformation of the coordinates. Such a transformation preserves the Poisson bracket structure of the theory [Gold55]. [8]

§11.4.1. Gauge Invariance and the Lagrangian

In Maxwell's electromagnetic theory, gauge transformation are invariant, since only the electric and magnetic fields appear in the basic equations. This gauge invariance is less evident in the Lagrangian formulation of the electromagnetic theory. Restating the gauge transformation,

,

.t

′→ = −∇χ

∂χ ′φ → φ = φ + ∂

A A A (11.45)

where again χ is an explicit function of rr and t, but additionally the field variables in Maxwell's equations are also explicit functions of rr and t.

In the gauge transformations of Eq. (379) the Lagrangian of the charged particles is unmodified as well as the electromagnetic field Lagrangian, which depends only on the electric and magnetic fields. The only Lagrangian component that is modified by the transformation of the interaction term.

The gauge transformation of the Lagrangian is performed by adding the quantity,

FFt

∂⋅∇ + ρ

∂j , (11.46)

8 When considering such symmetries in quantum mechanics, complications appear

because of the underlying structure of the quantization [Mess66]. A quantum mechanical symmetry transformation may involve a unitary or antiunitary transformation of both the quantum state of the system and the quantum operators that operate of the quantum states.

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to the Lagrangian in Eq. (188) which can them be rewritten as,

( ) ( )212 r rm q dq F F F

t t∂ ∂ρ = + + ∇ + ρ − ∇⋅ + ∂ ∂

∑ ∫ j j&r

L L . (11.47)

The first two terms add to the Lagrangian density a divergence and a time derivative and according to Eq. (376), this transforms the Lagrangian into an equivalent Lagrangian. [9] The last terms added to the Lagrangian density is the expression for the conservation of charge, Eq. (26) and is zero.

§11.4.2. Symmetry and Conservation

It was Emmy Noether (1882–1935) who made this relation of symmetry to conservation laws mathematically precise [Noet18], [Hill51], [Kram82]. If the components of a multicomponent field — such as the electric and magnetic components of the electromagnetic field — can be transformed without alteration of the field interaction, then a symmetry is present along with the associated law of charge conservation. When fields are defined by properties that follow symmetry operations — mass, spin and charge — then the specification of these properties also fully specifies the field. [10]

In Weyl's original work it was postulated that space had a local symmetry. He argued that the behavior of space and time can vary randomly, but these variations are canceled by the action of the electromagnetic field [Weyl18]. The result of this local gauge symmetry is the conservation of charge. Upon reviewing Weyl's paper, Einstein pointed

9 There is not total equivalence between the changes in the Lagrangian and the gauge

transformations. All transformations which leave Maxwell's equations and the EE , BB , and AA fields invariant are gauge transformations. In the Lagrangian given above, the arbitrary function F can depend on the vector and scalar potential as well as their velocity and accelerations, which themselves are functions of rr and t. It is only when F does not depend on the velocities of the vector and scalar potentials that it also corresponds to a change in the Lagrangian, otherwise the acceleration terms of the vector and scalar potentials would appear in the Lagrangian.

10 This theorem is the foundation of gauge field theory. In more formal terms it states that if the Lagrangian L is invariant under a continuous one–parameter transformation, there will exist a 4–vector current which is differentially conserved, whose spatial integral of the zero–component will yield a conserved charge. [Aitc82]. In gauge field theory the currents are unique in that they play a dual role. They are the symmetry generating currents defines by Noether's theorem and they are the sources of the vector fields of electromagnetism.

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out that this postulate leads to illogical conclusions when one considered the motions of clocks traveling around a room [Crea86].

Weyl's theory was revised by Fritz London [Crea86] to state that it was not space and time that possessed the symmetry that conserved charge, but rather the phase of Schrödinger's wave equation. It is known that the electric charge of an electron never changes. London showed that the phase of the electron's wave equation can change and not effect the charge through a gauge symmetry which compensates for such changes by creating virtual photons — and their associated electromagnetic field — whose action ensures the conservation of electric charge.

In the language of gauge theory the purpose of the electromagnetic force is to maintain the symmetry of the electromagnetic interactions in which the phase of the interaction is altered.

§11.5. GAUGE PARTICLES AND THE CONVEYANCE OF FORCE

The gauge representation of Maxwell's unification of electricity and magnetism moves naturally into a quantum mechanical version, Quantum Electrodynamics (QED). This transition from the classical world to the quantum world seems clear in hindsight. In the late 1920's however gauge theory and QED were just being formulated. Modern gauge field theory was largely the creation of Chen Ning Yang and Robert Mill [Yang54]. In the late 1940's Yang started checking the calculations of the gauge invariant theories of electromagnetism. He realized that this invariance is a principle that can generate forces not only in electromagnetic theory but also nuclear forces. In Gauge Theory particles are the conveyers of force.

In the quantum theory of fields the particles are the result of quantizing the classical field. In this description the intensity of the field is equal to the probability of finding the associated quantum particle at some point in space. According to quantum field theory, it is through the interaction of these field particles that force is conveyed.

In order to deduce the nature of the particle carrying the electromagnetic force, some hindsight observations are made. From the long range nature of the inverse–square law, the exchange particle must be massless. It must be electrically neutral, otherwise the charge of the electron will be altered when the particle is emitted. It must be a boson in order to preserve the electrons identity after it absorbs the particle. Particles can be partitioned into two categories, boson's and fermion's.

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Boson's are particles that possess wave–like properties and integer spin [11] in that they exist in the quantum mechanical world rather than the classical world. Photons are boson's and have behaviors that are counterintuitive if they are thought to be tiny self-contained objects. These bosons do not exist as points of matter, but rather as delocalized probability densities. In this form boson's are indistinguishable from one another [Mart70], [Will71]. The Pauli exclusion principle applies to all quantum particles. [12] This principle states that particles with half integer (½) spin quantum numbers — fermion's — must have two–particle state vectors that are antisymmetric with respect to the pairwise interchange of particles, that is no two fermion’s can be in the same quantum state. Since all the physical constitutes of matter are fermion’s (electrons, protons, and neutrons), without Pauli's exclusion principle all matter would collapse on itself.

Particles with integer (1) spin quantum numbers — boson's — must have symmetric many–particle spin state vectors. [13] At the heart of this

11 The visualization of a spinning particle turning like a top, possessing angular momentum has limited usefulness here. The intrinsic spin of a particle has little resemblance to a spinning object with an angular momentum vector, which is the product of its moment of inertia and its angular spin [Roge60]. In the case of a spinning top its angular momentum can assume any value. For quantum particles the intrinsic angular momentum is quantized in multiples of h 2π. These quantum spin values are given integer and 1

2 integer values of 0, 12 , 1, 3

2 , and 2. Particles with integer spin (0, 1, 2) are bosons. Particles with half–integer spin (1

2 , 32 ) are fermions.

In terms of quantized spin, particles with spin 1 return to their original state after a full rotation of their spin axis, while particles with half–integer spin require two full rotations of their axis to restore them to their original state. If at the start of the rotation the particle has an orientation of up after one turn it will have an orientation of down. Another rotation will be required to restore its orientation to up. Particles with zero spin have no orientation, while particles with spin 2 return to their original orientation after and ½ rotation of their spin axis

12 Wolfgang Pauli put forward his exclusion principle as a hypothesis in 1925. The full quantum mechanical treatment of fermions was developed in 1926 by Enrico Fermi and Paul Dirac. The statistical behavior of fermions follows Fermi–Dirac Statistics which is the quantum mechanical version of Boltzmann statistics of distinguishable particles. The Bose–Einstein statistics was developed for the description of photons and electrons by Indian scientist Satyendra. N Bose and Albert Einstein in 1924.

13 In classical mechanics, identical particles do not lose their identity. These particles could be numbered in some way, with their motion followed and their individuality maintained throughout their course of motion. In quantum mechanics, the uncertainty principle prohibits the concept of the path from having any meaning. If the position of an electron is known exactly at a given time, its coordinates will have no definite value at the measurement time. Localizing and numbering quantum particles at some instant provides

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rather obtuse formalism lies the indistinguishability of all quantum particles. Fermion's cannot occupy the same energy state and boson's can. This indistinguishability is linked to the wave–particle duality of nature, just as the momentum–position relation of Heisenberg's uncertainty principle is linked. All of these various behaviors are in fact one underlying behavior.

Since the particle carrying the electromagnetic force is a boson, it will have integer spin and since it transmits both the electric and magnetic force it will carry non–zero spin. Such a non–zero integer spin particle with zero mass is of course the photon [Wu67].

The spin 1 nature of the photon is important in keeping its zero mass. A particle at rest with mass m has an energy given by 2E mc= . The value of this energy level shifts when the particle interacts with other matter, leading to a change in its mass. At first this may seem contradictory because a particles mass is an intrinsic and unchangeable quantity. This dilemma can be explained in the following manner. A particle at rest with spin s has 2 1s + spin states. These spin states can be realized through suitable rotations. A massive particle in motion has all of the 2 1s + states, because this particle appears to be at rest to an observer traveling with the same velocity.

The argument of having 2 1s + spin states is no longer valid if the particle is massless. In this case it travels at the speed of light and no (massive) observer can travel along with it. A massless particle therefore is allowed to have less than 2 1s + realizable spin states. By keeping the

no information about the identity of the particles later. This principle of indistinguishable particles plays a fundamental role in the quantum theory of system composed of identical particles.

The interchanging of two particles is then equivalent to interchanging their states described by the wave function ψ ξ ξ ψ ξ ξα( , ) ( , )1 2 2 1⇔ ei . The phase factor eiα becomes the

unimportant distinction between the particles, where eiα = ±1 , thus ψ ξ ξ ψ ξ ξ( , ) ( , )1 2 2 1⇔ ± [Land77], [Schw92].

The result of the interchange has two possibilities: the wave function is symmetrical, i.e. it is unchanged when the particles are interchanged or it is antisymmetrical, i.e. it changes sign when the particles are interchanged. The property of symmetrical or antisymmetrical wave functions depends on the nature of the particles. Particles described by antisymmetric functions obey Fermi–Dirac statistics and are called fermions, while particles described by symmetrical functions obey Bose–Einstein statistics and are called bosons. It has been shown that the statistics obeyed by particles is uniquely related to their spin: with fermions possessing ½ integral spin and bosons possessing integral spin [Land80].

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number of spin states less than 2 1s + , the particle therefore must be massless. For this to work for a boson, its spin must be 1.

The photon has spin 1 and has 2 1 3s + = spin states. A way must be found to restrict it to only 2 spin states. The best way to accomplish this is to propose a theory in which the third spin state is not suppressed, but is written in such a way that the third spin state becomes invariant under addition and subtraction — it has become physically decoupled and irrelevant.

Such a theory is a gauge theory with an exchange particle be a gauge particle or gauge boson and the invariance is a gauge invariance. The gauge invariance is a local invariance, because the invariance of the third spin state must occur at all times whenever the photon is in space, e.g. space–time. Such a gauge theory has an additional important feature. The photon couples only to electrically conserved charges (currents). Since the interaction strength is proportional to the charge, there are no interactions with neutral particles.

The gauge theory of electromagnetism derived in this way turns out to be Maxwell's theory. The two spin states (orientations) of the photon correspond to the left and right hand circular polarization's of the electromagnetic wave. The derivation of this theory contains many over–simplifications and large amounts of hindsight and should be taken as a layman's view of gauge theory. However it does illuminate the beauty and elegance of the mathematical process [Frie83].

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The confusion of the past is now replaced by a simple and elegant synthesis. This standard theory may survive as a part of the ultimate theory, or it may turn out to be fundamentally wrong. In either case, it will have been an important way–station and the next theory will have to be better.

— Sheldon Lee Glashow Nobel Lectures, 1979 [Adai87]

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§12. MATHEMATICS OF QUANTUM MECHANICS

In order for the science of physics to progress, physical theory needs to contain ideas based on experimental evidence and be described by mathematics. Theories are usually built in three stages, (i) the basic idea is formulated and interpreted and the physical foundation of the theory is built through experiments, (ii) a mathematical foundation is built which describes the experimental evidence in some form acceptable to peer review process. During this stage the physical meaning of the mathematical symbols come to represent the physical laws, (iii) a final stage in which the mathematical apparatus is set to work [Bohr63] to produce new results which can then be confirmed through experiments. This third stage leads to further understanding of the physical content of the theory and to further development the mathematical apparatus.

When Newton formulated the laws of mechanics, the corresponding mathematical tools also had to be formulated. These tools included differential and integral calculus. When quantum mechanics was formulated, the mathematical mechanisms were already in place in the form of the theory of linear operators.

This apparatus was summed up by Neils Bohr (1855–1962) as,

... in quantal formalism, the quantities by which the state of a physical system is ordinarily defined are replaced by symbolic operators subjected to a non–communtative algorism involving Planck’s constant. This procedure presents a fixation of such quantities to the extent which would be required for the deterministic description of classical physics, but allows us to determine their spectral distribution as revealed by evidence about atomic processes. In conformity with the non–pictorial character of the formalism, its physical interpretation finds expression in laws of an essentially statistical type... [Bohr63].

The mathematics of Quantum Mechanics and its relationship to the electromagnetic force — quantum electrodynamics is based on a few simple principals. Before these principals can be presented some background on the notation needs to be developed. It may seem unnecessary to define the simple mathematical term of a vector, but there is benefit, since the terminology has unique meaning in the field of quantum mechanics. [1]

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... all science as it grow toward perfection becomes mathematical in its ideas.

— Alfred North Whithead(1861–1947) (1911)

§12.1. VECTORS AND VECTOR SPACES

In quantum mechanics the theory abstract vector spaces provides the same tools as differential calculus does in classical mechanics. Although the concept of three dimensional vector spaces is familiar, the concept of an abstract vector space may not. There are two aspects to a vector space — geometric and algebraic. In the geometric aspect a vector can be described as a directed line segment which can represent a quantity with both magnitude and direction. A vector is distinguished from a scalar which is a quantity which has only magnitude. [2]

In algebraic aspect a vector can be described by a one–to–one correspondence between the unique set of vectors radiating from the origin of a coordinate system and the coordinates of the terminal points of each vector. This correspondence is represented by an ordered pair of real numbers ( )1 2 3, ,x x x in 3 dimensions or simply ix where the subscript

1,2 ,3i = . In higher dimensions the algebraic notation for a vector is given

by an n–tuple of real numbers ( )1, 2 , , nx x x… . Even though there is no

longer a physical analog of the n–dimensional vector, the language of these vectors can be maintained through the algebra of abstract vector spaces.

The collection of vector associated with all points in the coordinate space is called a vector space. The dimension of a given vector space is the dimension of the associated coordinate space of points. This dimension is the number of coordinates needed to define a geometric point in the vector space. A one dimensional vector space describes a line of points, while a two dimensional vector space described a plane of points and a three dimensional vector space describes the world in which we live (in non–relativistic terms). Vector space is higher than three dimensions become difficult to visualize, but the formal description of such vector spaces is not a problem.

In the algebraic representation a vector can be expressed as a linear combination of non–coplanar vectors ib or basis vectors. Using the basis of the vector space any vector can be built up from the basis elements of the vector space. It is the convention of abstract vector analysis though to

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choose the basis vectors perpendicular to each other. When this occurs the basis is called orthonormal otherwise it is an oblique basis.

The basis are then equivalent to a coordinate system for an n–dimensional point in the vector space. The vectors which have a length of 1 (defined in Eq. (12.4)) and which point along the perpendicular coordinate axes of an n–dimensional space constitute an orthonormal basis of the associated n–dimensional vector space. For any vector space larger than a single dimension, there will be an infinite number of orthonormal basis. Any vector in this space can be written in terms of any of those basis. In a 3 dimensional vector space the vector xx can be described as,

1 2 3b b b= α + β + γx , (12.1)

where , ,andα β γ are scalars and called the expansion coefficients of the vector xx in the vector containing the basis vectors 1 2 3, ,andb b b [Brau70].

§12.1.1. Abstract Vector Algebra

If vector analysis is to become the calculus of quantum mechanics, the algebra of vectors needs to be defined. Addition of vectors is defined as:

= +z x y . (12.2)

The sum of any two vectors is another vector. [3]

The multiplication of vectors is called the scalar or inner product. Given two vectors of 3 dimensions, ( )1 2 2, ,x x x=x and ( )1 2 3, ,y y y=y as:

3

1i i

i

x y=

= ⋅∑x yo . [4] (12.3)

The product of two vectors is a number. If the vector occupy a real vector space than the inner product results in a real number. If the vector occupy a complex vector space then the result is a complex number.

Using this notation, the length of a vector is denoted by x [5] and is given by,

3

1i i

i

x x=

= = ⋅∑x x xo . (12.4)

The angle between two vectors xx and yy is defined as,

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1cos− θ = ⋅

x yx y

o. (12.5)

A vector can also be multiplied by a number. This operation simply creates a new vector whose length is now n times the length of the original vector, such that,

3 ⋅ =x x x xo o . (12.6)

So long as the two vectors xx and yy occupy a real vector space of finite dimension then the inner product can be generalized to,

1

n

i ii

x y=

= ⋅∑x yo . (12.7)

If the vectors xx and yy are to be used to define the inner product in a complex vector space, there are difficulties with the definition of the length of a vector. If the distance function defined above in Eq. (12.4) is to be used, then the length of a complex vector will be,

2i i i= = −x x x xo (12.8)

which is an imaginary number if xx is real. This situation causes problems since the distance function should be a real value. Since quantum mechanics is based on the use of complex numbers, restricting the algebra of vectors to real numbers would eliminate its use in quantum theory. The definition of the inner product can be fixed with the following definition. Given two complex vectors xx and yy whose complex components are defined as ( )1 2, , , n= ξ ξ ξx … and ( )1 2, , , n= η η ηy … which refer to some basis, then the inner product of xx and yy is given as,

1

n

i ii

∗

=

= ξ ⋅ η∑x yo (12.9)

where ∗ξ is the complex conjugate of ξ . [6]

Using this definition the length of a vector in a complex vector space is not given as,

2

1 1

n n

i i ii i

∗

= =

≡ = ξ ⋅ξ = ξ∑ ∑x x xo (12.10)

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where iξ is the absolute value of iξ . Since 2 0iξ ≥ , x is always real. If

the vectors xx and yy are in a real vector space then i i∗ξ = ξ which reduces

the definition of the inner product to that in Eq. (12.7).

Unlike real vector spaces in complex vector spaces the inner product is not symmetrical, so that,

≠x y y xo o , (12.11)

rather,

( ) ( )∗=x y x yo o . (12.12)

This property will be the foundation of the commutator algebra used throughout quantum mechanics.

With the concept of length of a vector and the angle between two vectors defined, the theory of vector spaces can be applied to physical problems. One important concept in quantum mechanics is that of an orthogonal set of vectors (and later a set of functions represented by the vectors).

Two vectors xx and yy are orthogonal if and only 0=x yo . In two a three dimensional vector space two vector are orthogonal if the angle between them is 90o . This definition can be simply generalized to an n–dimensional vector space. If 0=x yo then ( ) ( ) 0∗= =x y x yo o . Although the inner product is not symmetric, the orthogonal condition is symmetric.

For a set of vectors 1 2, ,x x … the orthogonal condition is met when

,i j i j= δx xo for all i and j.

§12.2. LINEAR FUNCTIONALS

Corresponding to any linear vector space V there exists a dual space of linear functionals on V. The set of linear functionals on the vector space is itself an n–dimensional vector space. This principal can be applied to diverse applications of mathematics — from economics to combinatorics. A linear functional assigns a scalar function ( )F x for each vector xx such that,

( ) ( ) ( )F a b aF bF+ = +x y x y , (12.13)

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for any vector xx and yy and any scalar a and b. [7] The linear functionals form a vector space V. An important theorem in linear functionals is Riesz’s Theorem which states there is a one–to–one correspondence between linear functionals F in ′V and the vector xx in V such that all linear functionals have the form,

( )F f=x xo , (12.14)

where f is a fixed vector and xx is an arbitrary vector in the vector space V [Ries55], [Byro69]. [8] If this condition holds, the two vectors spaces V and

′V are said to be isomorphic.

§12.2.1. Linear Operators

An operator on a vector space maps one vector onto another vector or vectors. If L is an operator and xx is a vector, then =y xL is another vector. An operator is defined by specifying its actions on every vector in the vector space. A linear operator satisfies the relation,

( ) ( ) ( )1 1 2 2 1 1 2 2c c c c+ = +x x x xL L L (12.15)

Two operators are said to be equal, =L M if =x xL M for all vectors in the common domain of L and M. The sum of two operators can be defined as,

( )+ = +x x xL M L M (12.16)

and the product of two operators can be defined as,

( )=x xLM M M (12.17)

From this equation multiplication of operators is associative since ( ) ( )=L MN LM N , but as will become the foundation of quantum mechanics,

the multiplication of operators if not necessarily commutative, since ≠LM ML .

§12.3. DIRAC NOTATION AND LINEAR OPERATORS

There is a shorthand notation that used in quantum mechanics — the Dirac notation. [9] In this notation the vectors in the vector space V are called ket vectors and are denoted as x . The linear functionals in the dual space ′V are called bra vectors and are denoted as F . The numerical values of the linear functionals are denoted as,

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( )F F≡x x . (12.18)

According to Riesz’s Theorem, there is a one–to–one correspondence between bras and kets. This notation allows a simplification of the previous linear functional equations where f denoted a fixed vector and F the function applied to the arbitrary vector xx . In the bra–ket notation the bra F or the ket F can be used to determine which vector space ′V or V is

referred to, so that,

F F=x xo (12.19)

where F is the vector previously denoted as f.

The Reisz Theorem establishes an antilinear correspondence between bras and kets such that if F F↔ then,

1 2 1 2c F c F c F c F∗ ∗+ ↔ + , (12.20)

where the constant c∗ is the complex conjugate of constant c.

In a vector space of discrete vectors it is possible to represent the vectors as column or row vectors. In the previous notation the distinction between the row and column vector was not important. In the Dirac notation, this distinction becomes the method of defining the inner product of two vectors. If a column vector is given as,

1

2

3

xxx

≡

x

M

, (12.21)

and a row vector is given by,

( )1 2 3x x x≡x L , (12.22)

then the inner product can be defined as,

1

n

i ii

x x∗

=

≡ ⋅∑x x . (12.23)

The algebra of the Dirac notation can nor be summarized. The inner product of a ket is always real and given by,

0≥x x . (12.24)

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The distribution law holds for kets such that,

+ = +x y z x y x z . (12.25)

The inner product with one ket multiplied by a constant can be restated as,

α = αx y x y (12.26)

The inner product of x with y is the complex conjugate of the inner product of y with x , such that,

∗=x y y x . (12.27)

The result if this inner product x y is a complex number that is associated with each pair of vectors xx and yy .

The norm of a ket vector is the positive square root of their inner product, such that,

=x x x (12.28)

Using the Dirac notation and changing the notation for a vector to the symbols used in quantum mechanics, the description of a linear operator can be refined. [10]

A linear operator can then be represented as a matrix O which operates on the column vector ψ to produce the another column vector φ , such that,

ψ = φO , (12.29)

where the operator O in an n–dimensional vector space is given in the matrix representation as,

1,1 1,2 1,

2,1 2,2 2 ,,

,1 ,2 ,

n

ni j

n n n n

O O OO O O

O

O O O

= =

O

……

M M O ML

. (12.30)

Choosing an orthonormal basis , 1iu i n= … in which to expand the

vectors ψ and φ gives,

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1

n

j jj

a u=

ψ = ∑ and 1

n

k kk

b u=

φ = ∑ . (12.31)

Operating on Eq. (12.30) with the basis vector iu results in,

1 1

n n

i j j i ik kj k

u u a u u b b= =

= =∑ ∑O .

which has the form of a matrix equation given as,

,1

n

i j j ij

a b=

=∑O , (12.32)

with the expression,

,i j i ju u=O O , (12.33)

denoting the matrix element of the operator O. The expression in Eq. (12.33) says that the number ,i jO is the vector juO multiplied by the

vector iu .

Using this matrix representation for the operator and a column vector any linear operator can be uniquely specified, given a basis, by specifying the equation of Eq. (12.30) and Eq. (12.33). The effect of the operator O on a vector ψ can be determined by multiplying the vector by the operator, such that,

1,1 1,2 1, 1

2,1 2,2 2, 2

,1 ,2 ,

,

n

n

n n n n n

O

O O OO O O

O O O

ψ = φ

ψ ψ = × ψ

LL

M M O M ML

(12.34)

( ) ( )1,1 1 1,2 2 1, 1 ,1 1 ,2 2 , ,

.n n n n n n n nO O O u O O O u= ⋅ψ + ⋅ψ + + ⋅ψ + + ⋅ψ + ⋅ψ + + ⋅ ψ

= φ

L L L

This leads to a useful property for the analysis of quantum mechanical system. If for some particular operator O and a particular vector ψ , that the relation,

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ψ = λ ψO (12.35)

exists where λ is a complex number then ψ is said to be an eigenvector of O, with an eigenvalue of λ , which is the length of the new vector relative to the length of ψ . Since vector will be eigenvectors of certain operators and certain operators will have some vectors as eigenvectors and not others. The operator–eigenvector relationship will depend only on the vector and the operator, not on the basis in which the vectors are formed. [11]

In the mathematics of quantum mechanics the act of measuring is performed by operating on the eigenvector to produce a measurement represented by an eigenvalue. The possible set of results of measuring an observable is the set of eigenvalues of the corresponding operator [Polk85].

§12.3.1. Measurable Properties

The measurable properties of a quantum system are represented by a linear operators. A particular physical state of the system is represented by a vector is the vector space. If this vector is an eigenvalue of an operator associated with a measurable property of the system, then the state represented by the vector is an eigenstate of the measured property.

The act of measuring is performed by operating on the eigenvector to produce a measurement represented by the eigenvalue. The possible set of results of measuring an observable is the set of eigenvalues of the corresponding operator [Polk89].

If an operator O, represents a physical dynamical variable, then this operator must be Hermitian. Suppose an operator O acts on the vector ψ to produce another vector φ . When the operator acts on the ket vector to its right, it is denoted as,

ψ = φO , (12.36)

while the operator acts on the bra vector ψ to its left it is denoted as,

ψ = φO . (12.37)

Using the expression in Eq. (12.36) and Eq. (12.37) the products between ψ and ψO and ψ O and ψ are not in general equal such that, 1 2ψ ψO usually 2 1≠ ψ ψO , since one if the product of 1ψO with

2ψ and the other is the product of 2ψ O with 1ψ . If however the products are equal then the operator O is said to be Hermitian. Hermitian

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operators are important in quantum mechanics because their eigenvalue are always real.

Using the matrix representation a Hermitian operator can be defined as a complex valued matrix AA . Then the matrix A denotes a matrix obtained from AA by replacing each element of the complex valued matrix, z a ib= + with its conjugate z a ib= − . The matrix A is the conjugate of the matrix AA . The matrix AA is real =A A . The transpose of the conjugate of the matrix AA is denoted by ∗A , that is

( )T∗ =A A (12.38)

If

( ),i ja=A , (12.39)

then the transpose of AA is given by,

( ),T

j ia=A , (12.40)

and the conjugate of AA is given by,

( ),i ja=A , (12.41)

and the transpose of the conjugate is given by,

( ) ( ),

T Tj ia= =A A (12.42)

A matrix AA such that ∗=A A is a Hermitian matrix. There are several attributes of Hermitian matrices that are important in the mathematics of quantum mechanics: (I) if the eigenvalues of a matrix are distinct, then the associated eigenvectors are linearly independent; (ii) if AA is a Hermitian matrix, then the eigenvalues of AA are real, (iii) if AA is a real symmetric matrix, then the eigenvalues of AA are real; (iv) if AA is a Hermitian matrix, then the eigenvectors associated with the distinct eigenvalues are mutually orthogonal vectors [Pett78], [Akiv77].

§12.3.2. Quantum Operators

In the linear vector space algebra, an operator F is defined which acts on a vector u to produce another vector v such that v u= F . The eigenvalue problem can then be stated as the search to find the vectors u

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and the numbers λ , such that, ,u u= λF that is, to find vectors which are transformed by the operator F into multiples of themselves. The vectors satisfying the expression are eigenvectors and the corresponding numbers λ , the eigenvalues. [12]

If an operator F is to represent a physical dynamical variable, then it will be Hermitian. Suppose that an operator F , acting on a vector u , produces a vector v . When the operators acts on the vector to its right, it is denoted as, .u v=F The corresponding operator which transforms u into v is denoted by, .u v=F

Given the quantities 2 1u uF and 1 2u uF they are not in general equal; the first is the product of 1uF with 2u and the second is the product of 1u with 2uF . An operator for which these quantities are

equal is called Hermitian. [13] The vector u is then the Hermitian conjugate of the vector v . Hermitian operators are important because their eigenvalues are real and therefore represent observable physical variables. If any dynamical variable (operator) F has eigenvectors nf

such that n n n=F f f f , then operators representing other dynamical

variables operate on the eigenstates of F . This relation can be stated in terms of the commutator,

[ ]1 2 1 2 2 1, = −F F F F F F , (12.43)

of the pair of operators 1F and 2F . [14] If 1F and 2F posses common eigenstates their commutator is zero (0), such that 1 2 2 1=F F F F . If the states n are common eigenstates of 1F and 2F , having the respective sets

of eigenvalues 1nf and 2nf , then 1 2 2 1=F F F F is satisfied for all operators.

It should be noted that the commutator defined in Eq. (12.43) is not a Hermitian operator. In order to illustrate the, the general matrix elements of the product 1 2k k is given as,

1 2 1 2n

n n n n n n′′

′ ′′ ′′ ′= ∑F F F F . (12.44)

A Hermitian operator is defined as any matrix whose elements are equal to their transpose conjugate. The transpose conjugate of Eq. (12.44) is,

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1 2 1 2n

n n n n n n∗ ∗ ∗

′′

′ ′ ′′ ′′= ∑F F F F (12.45)

which is, since 1F and 2F are Hermitian,

1 2 2 1n

n n n n n n′′

′′ ′′ ′ ′=∑ F F F F . (12.46)

Similarly, the transpose conjugate of 2 1n n′F F is 1 2n n′F F , therefore,

1 2 2 1 1 2 2 1n n n n ∗′ ′− = − −F F F F F F F F . (12.47)

This inequality, due to the minus sign may be corrected by multiplying the commutator by i, that is,

[ ] ( )1 2 1 2 2 1i i= −F F F F F F , (12.48)

which is Hermitian.

Restating the commutator algebra in terms of the Hamiltonian gives,

[ ], ,r r r r r

u v u vu v

q p p q ∂ ∂ ∂ ∂

= − ∂ ∂ ∂ ∂

∑ (12.49)

where rp and rq are any set of canonical variables [Dira25], with the

following algebra, [15]

[ ] [ ], ,u v v u= − , (12.50)

[ ], 0u c = , where c is a constant, (12.51)

[ ] [ ] [ ][ ] [ ] [ ]

1 2 1 2

1 2 1 2

, , , ., , , .

u u v u v u vu v v u v u v

+ = +

+ = + , (12.52)

[ ]

[ ] [ ][ ] [ ] [ ]

1 2 1 21 2 2 1 2 1

1 2 1 2

1 2 1 2 1 2

, ,

, , ,

, , , .

r r r r r r r

u u v u u vu u v u u u uq q p p p q

u v u u u v

u v v u v v v u v

∂ ∂ ∂ ∂ ∂ ∂ = + − + ∂ ∂ ∂ ∂ ∂ ∂

= + = +

∑

(12.53)

[ ] [ ] [ ], , , , , , 0u v w v w u w u v+ + = . (12.54)

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§12.3.3. Commutators and Poisson Brackets

During the development of the Lagrangian and Hamiltonian formalisms, the Poisson bracket notation was presented. The dynamical variables u and v in the Poisson bracket are functions on noncommuting p's and q's. This can be shown by defining a set of commuting variables

1, , kα α… and their associated partial derivatives as,

( ) ( )10

1lim , , , ,i ki

u u uε→

∂ = α α + ε α − α ∂α ε

… … (12.55)

With noncommuting variables a general derivative can be defined which involves an arbitrary function ( )c α , such that,

( ) ( ) ( )10

1lim , , , ,i ki

u c u c uε→

∂ α = α α + ε α − α ∂α ε

… … (12.56)

This derivative is called the Fréhet derivative [Park79]. If the function ( )c α commutes with all the 'sα then,

( ) ( )i i

u uc c∂ ∂α = α

∂α ∂α (12.57)

The Fréhet derivative follows the usual derivative rules, such that,

( ) ( ) ( ) ( )

i i i

u v u vc c c∂ + ∂ ∂

α = α + α∂α ∂α ∂α

, (12.58)

and,

( ) ( ) ( )

i i

au uc a c∂ ∂

α = α∂α ∂α

. (12.59)

A theorem provided by [Park79] will be used to equate the Poisson bracket and the commutator. The theorem is given as,

[ ] [ ], ,ii

u v u v∂α =

∂α. (12.60)

The proof of Park's theorem is given by induction by writing the general function ( )u α as a sum of the pieces, each of which is of the form

3 2 1 3 1 4a ⋅α ⋅α ⋅α ⋅α ⋅α ⋅α ⋅L and it is assumed that the commuting variables u meets this criteria. By adding another factor of jα to the left–and side of

Eq. (12.60) gives,

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( )[ ] ( ) ( ) [ ]

[ ]

0

1, lim , , ,

, , ,

,

ji j j j j i

i i

j j

j

u uv v u u v

v u u v

u v

ε→

′∂ α ′∂′ α = α + ε α − α − α + α α ∂α ε ∂α

′ ′ = α + α ′ = α

(12.61)

If Eq. (12.61) holds for u′ , then it holds for ju′α . By setting ku′ = α , the

theorem holds for the trivial case and has been proved by induction.

The relation between the Poisson bracket and the commutator can now be made by replacing i by j, v by iα and u by v to give,

[ ], ,j i ii

u v∂ α α = α ∂α, (12.62)

or

[ ], ,i j ii

u v∂ α α = α ∂α. (12.63)

The expression can now be substituted into Eq. (12.61) to give,

[ ], ,i ji j

u v u v∂ ∂ α α = ∂α ∂α

(12.64)

This identity yields the relationship between the Poisson bracket and the commutator by dividing the dynamical variables iα into two sets,

1, , fp p… and 1, , fq q… with,

[ ] ,,m n m nq q i= δh (12.65)

as iα and jα progress through the set of p's and q's,

( ) [ ],n n n n

u v u vi i u vq p p q∂ ∂ ∂ ∂

+ − =∂ ∂ ∂ ∂

h h (12.66)

or using Eq. (12.64),

[ ],n n n n

u v u vu v i

q p p q ∂ ∂ ∂ ∂

= + ∂ ∂ ∂ ∂ h (12.67)

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§12.3.4. Commutators and the Electromagnetic Field

A practical application of the commutator is useful at this point. The classical electromagnetic field contained in a bound cavity can be decomposed into eigenmodes using the commutator process. The thi complex amplitude of the electromagnetic field can be given as,

( ) ( ) ( )( )( )

122

i i i ii

i i

m q t ip ta t

m

ω +=

ωh, (12.68)

where im is the characteristic mass of the charged particle in the electromagnetic field and iq and ip are real canonical variables. In the description iω is the eigenfrequency for each eigenmode i. The real and imaginary components of the complex amplitude ( )ia t are related to the canonical variables ( )iq t and ( )ip t respectively.

The electromagnetic field can now be described by defining the complex amplitude as,

( ) ii ti ia t a e− ω= . (12.69)

This expression describes a complex variable ( )ia t moving with velocity −ω in a circle around the origin in the complex plane. The real and imaginary components correspond to the electric and magnetic components of the electromagnetic field and the circular motion corresponds to the oscillations of the energy contained in the field.

When the complex amplitude is quantized ( )ia t becomes the field operator ( )ia t which obeys the commutator rules,

( ) ( ) ,ˆ ˆ,i j i ja t a t∗ = δ . (12.70)

Each eigenmode i has a discrete set of eigenstates in where in is the

number of photons in mode i.

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I like relativity and quantum theories because I don't understand them and they make me feel as if space has shifted about like a swan that can’t settle, returning to sit still and be measured; and as if the atom were an impulsive thing always changing it mind.

— D. H. Lawrence in [Lede93]

1 The description of the mathematical techniques in this section is meant to provide a

very broad overview of the subject in preparation for the description of the interaction between the electromagnetic field and charged particles in conductors. The study of the quantum mechanics of fields — quantum field theory — is very complex and even the basics of the concepts are outside the scope of this monograph.

2 The notation used in the development of vector algebra follows that found in texts on vector spaces. This notation makes use of the following: (i) italic fonts (x) represent components of a vector; (ii) bold fonts (xx ) represent a vector, real or complex; (iii) bold Arial fonts (O) represent operators in the vector space; (iv) outline fonts (V) represent vector spaces. When the vector algebra is used in the description of quantum mechanics φ ψand represent vectors in the complex vector space.

Since the approach taken in this monograph is not based on the specific of the Heisenberg or Schrödinger formalisms, but rather the operator algebra of P.A.M. Dirac, the generic Dirac notation is presented without the background development found in [Dira58].

3 In quantum mechanics the sum of two vectors will be used to represent the superposition of two states of a system.

4 The notation used for the scalar and vector product varies dramatically. In this monograph the symbol aa bbo will be used to denote the product of the vectors aa and bb , while the symbol a b⋅ will be used to denote the product of the scalars a and b.

5In the analysis of abstract vector spaces the length or magnitude of a vector xx is given by xx so that the absolute value of a complex number can be denoted by xx .

6 If the number ξ is a complex variable such that ξ = +a ib then the complex

conjugate of ξ is given by ξ∗ = −a ib , where a and b are real scalars.

7 The set of linear functionals on an n–dimensional vector space is itself an n–dimensional vector space with respect to the common algebraic operations. The purpose foe developing the dual space is to describe a Hilbert space in which the coordinates of the vector space take the form of functions whose variables are themselves coordinates in the vector space. An infinite dimensional vector space, with inner product properties can then be used to describe the behavior of quantum mechanical objects [Dunf57], [Youn86].

8 The Reisz–Fischer Theorem states that the vector space of square integrable functions, that are functions with a finite norm, are complete. The can be proved by letting

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the functions ( ) ( )1 2, ,f x f x … be elements in a function space. If

( ) ( )2 2

, ,lim lim 0

b

n m n man m n mf f f x f x dx

→∞ →∞− = − =∫ then there exists a square (Lebesque)

integrable function ( )f x to which the sequence ( )nf x converges to a mean — that is

there exists an f such that ( ) ( ) 2

,lim 0

b

n man mf x f x dx

→∞− =∫ .

9 In his original publication [Dira58], Dirac assumed a one–to–one correspondence between bras and kets and it was stated this was a mathematical or physical assumption. The Reisz Theorem shows that this assumption is not needed [Byro70], [Youn86]. In quantum mechanics more general vector space are used, including the Rigged Hilbert space [Böhm78] which the one–to–one correspondence between bras and kets does not hold [Ball90].

10 At his point in the monograph the notation will change from that found in mathematics texts to a notation used in quantum mechanics texts.

11 Associated with each square matrix AA , ( )( ),i ja=A of order n is a function

( )f λ = − λA I called the characteristic function. The equation ( ) 0f λ = − λ =A I can be

expressed in the polynomial c c c cn n nn0 1

12

21 0λ λ λ λ+ + + + =− −

−L and is the characteristic equation of the matrix AA , where the λ's are the eigenvalues of the matrix AA . Any non–zero column vector XX i , such that ( ) 0i i− λ =A I X is an eigenvector of the matrix AA .

12Associated with each square matrix A, A a ij= (( )) of order n is a function

f A I( )λ λ= − called the characteristic function. The equation f A I( )λ λ= − = 0 can be

expressed in the polynomial c c c c cn n nn n0 1

12

21 0λ λ λ λ+ + + + + =− −

−L and is called the characteristic equation of matrix A are called the eigenvalues of A. Any non-zero column vector, X i , such that ( )A I Xi i− =λ 0 is called an eigenvector of matrix A.

13If A is a complex matrix, then A denotes the matrix obtained from A by replacing each element z a bi= + with its conjugate z a bi= − . The matrix A is the conjugate of matrix A. The matrix A is real iffA A= . The transpose of the conjugate of matrix A is denoted by A∗ that is A A T∗ = ( ) . If A a ij= (( )) then A aT

ji= (( )) , A a ji= (( ))

and( ) (( )) ( )A a ATji

T= = . The transpose of the conjugate of a matrix is equal to the

conjugate of the transpose of the matrix.

A matrix A such that A A= ∗ is a Hermitian matrix iff a aij ji= for all ij. Since

a aii ii= only if a ii is real. Some theorem important to this discussion are:

(1) If the eigenvalues of a matrix are distinct, then the associated eigenvectors are linearly independent.

(2) If A is a Hermitian matrix, then the eigenvalues of A are real.

(3) If A is a real symmetric matrix, then the eigenvalues of A are real.

(4) If A is a Hermitian matrix, then the eigenvectors associated with distinct eigenvalues are mutually orthogonal vectors. [Pett78], [Akiv77].

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14The use of Heisenberg's canonical commutation relation to describe the behavior of

quantum mechanical observable did not come about through direct development. In three papers on the matrix formulation of quantum mechanics [Heis25], [Born25] and [Born25a] together with papers of Heisenberg [Heis25a] and Born [Born26], the use of Hilbert space concepts formed the beginnings of the theory of quantum mechanics independent of the classical description of nature. This process was similar (in analogy only) to Fourier's use in an analytical model to describe the behavior of thermodynamics independent of the underlying physical process. It is the abstract modeling process that provides little or no natural understanding of quantum mechanics in terms of everyday experience.

15The method just used to derive the commutator algebra of the Hamiltonian, is based on transforming the classical Poisson Bracket formulation of classical mechanics. A more modern derivation based on obtaining operators for particular dynamical variables, does not depend on quantizing a classical theory. There are objections to the classical quantization method just presented. The first is an epistemological one. If the quantum mechanical equations can be obtained from classical mechanics, then the content of quantum theory must be logically contained in classical mechanics. The second objection is technical. The Poisson Bracket equations of classical mechanics are independent of the particular choice of generalized coordinates. There are several theorems proving the impossibility of such general coordinates quantization [Abra78], [Aren65], [Marg67].

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§13. POSTULATES OF QUANTUM MECHANICS

Now that some of the background has been established for formulating the electromagnetic field in terms of the quantum mechanical Hamiltonian, a small set of fundamental postulates will be given. These postulates form the basis of the quantum mechanical formalism developed over the past six decades. The complexities of quantum mechanics prohibits any detailed discussion of this subject. The intention of this section of the book is to given the read a quick overview of the vocabulary and lay the groundwork for the description of the quantum mechanical description of the electromagnetic field and its interaction with charges in a conductor.

At the quantum mechanical level, particles do not move along definitive paths imposed by a Euclidean coordinate system. Instead in each volume of space–time there exists a probability that at a given time, the particle may appear in an infinitesimal region 3d r with the probability

3 3d d∗ρ = ψ ψr r .

§13.1 BASIC THEORETICAL CONCEPTS

The theory of Quantum Mechanics is fraught with complexities, confusion and improper interpretation. There is a rich set of literature which addresses the interpretation of quantum mechanics. The primary feature of the quantum description of nature is the indetermism of individual atomic level events [Holl95]. Regularities and their associated predicitibalities emerge only when large ensembles of events are considered. The problem is formulating a clear understanding of the theory is stated in…

In prosecuting their quantization procedure, the Founding Fathers introduced the new notion of state not in addition to the classical state variables, but instead of them.

The result of the formulation is a wave function which characterizes the state of the system. Using this description of nature there is no mathematical mechanism to describe individual processes which results in indeterminism and unanalyzable behavior of atomic level processes.

The formalism of Quantum Mechanics developed in §12 — the Schrödinger equation and Hilbert vector space with the observables as operators in the vector space — was consolidated in the late 1920’s by Heisenberg, Bohr and Pauli in the Copenhagen School of physics. For the

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purposes of this section, it will be this theory of quantum mechanics that will be used.

The Copenhagen School states that the magnitude of the state function ψ determines the probability density ρ and the phase angle of the complex function ψ describes the particles nonrelativistic motion through space–time. The probability distribution of the particle together with its propagation properties produce all the observable quantities of the quantum mechanical system.

The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities that in principal are observable.

— W. Heisenberg [Heis25]

§13.2 THE FOUR POSTULATES OF QUANTUM MECHANICS (ACCORDING TO BOHR)

In quantum mechanics the definition of a system state is more subtle. The following postulates are considered fundamental in the quantum mechanical description of nature:

PostulatePostulate 11 — Each state of a physical system corresponds to an element, a state vector, in a Hilbert space H [1]. The length of the state vector u is unity, 1u u = . Elements that only differ by a phase factor ie φ represent the same state of the physical system.

PostulatePostulate 22 — Given that the state of the physical system corresponds to the state vector u , the probability that the system is

observed in the state vector v is 2v u .

1 A Hilbert space H is a complex vector space, in which an inner product ⋅ ⋅ is

defined as a mapping of H x H on a set of complex numbers. The inner product should satisfy the following conditions for elements of the vector space (u, v, and w):

(1) u u ≥ 0

(2) u v w u v u w+ = +

(3) u v u vα α=

(4) u v u v= ∗

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PostulatePostulate 33 — Every observable [2] corresponds to a linear, Hermitian operator on a Hilbert space. The possible results of a measurement are the eigenvalues of the corresponding operator.

PostulatePostulate 44 — The operators associated with a coordinate q and its canonically-conjugate momentum ip satisfy commutation rules

,i j ijq p i = δ h where ijδ equals 1 if i j= and zero otherwise.

Postu late 5Postu late 5 — The development in time of the state vector is given

by the first order differential equation ˆ( ) ( )di t tdt

ψ = ψh H , where H is a

linear Hermitian operator. The non–relativistic limit H is the operator corresponding to a classical Hamiltonian. [3]

§13.2.1 Postulate 1 and Postulate 2

The state vectors of Postulate 1 and their probability densities in Postulate 2 represent observable value of Schrödinger's wave equation or eigenvalue of Heisenberg's matrix mechanics. In either case the state of the quantum system must be single valued, finite and continuous — that is a well behaved function for all values of the generalized coordinates and time.

A system is in a stationary state when its observed properties do not change with time. Wave functions representing stationary states have time–dependence which can be factored into a configuration independent

2 The terminology used to describe a measurable physical quantity — dynamical

variables — are called observables. These entities are distinguished from their mathematical counterparts and the operators they are represented by. In a number of quantum mechanics texts the term observable is used to denote any Hermitian operator which posses a complete set of eigenfunctions.

3 The time dependent Schrödinger equation is the fundamental equation of motion in quantum theory, corresponding to Newton's equations in classical mechanics. In general

the form the time dependent Schrödinger equation is $Hψ∂ψ

∂= ih

t, where the Hamiltonian

of the system under consideration is derived from the classical Hamiltonian. In addition the wave equation ψ( , )rr t is not necessarily an eigen state of the Hamiltonian. Its physical

interpretation is that of a probability amplitude ψ( , )rr t dr2 of a measurement of the

position of a particle. The general solution to the Schrödinger equation is, ψ = −∑c eE E

iEt h

E

uu rr( ) / , where uu E denotes the spatial part of the wave equation and cE are

constants determined from the knowledge of the wave function at t = 0 .

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portion and a configuration dependent portion. The configuration dependent portion then fully described the system.

In the Schrödinger formulation of quantum mechanics the wave function was originally through to represent a physical matter wave traveling through space and time. The square of this matter wave function represents the measure of the intensity of the wave function. The modern view of Schrödinger's wave function is that of a probability density function for the instantaneous occurrence of a specific configuration at a specific time. This view of the wave equation requires that a quantum state be localized within some finite region of the configuration space. The norm and inner product of the vector space developed in the previous section can be used to describe behavior of the quantum states.

§13.2.2 Postulate 3

An operator is a mathematical prescription for transforming one function into another in the same manner a function is a prescription for transforming one number into another. The action of an operator on the wave function can be associated with the process of measuring an observable of the quantum system. The result of this measurement is the result of the transformation of the state of the system after the measurement has been performed. In general the state of the quantum system is different after the measurement since the process of measuring produces an irreducible perturbation of the system.

§13.2.3 Postulate 4

Although the other four postulates are important concepts in quantum mechanics, the commutator of two canonical variables and their relationship to Planck's constant needs further explanation. In Postulate 4, the commutation rule ,j i ijq p ic = δ represents the quantum mechanical

equivalent of the Poisson bracket notation of Eq. (8.27) and the subsequent commutator algebra. [4] The origin of Postulate 4 is based on

4 In this description of the uncertainty principle, the constant c is used in place of the

familiarh . In many texts the notation uses h directly, skipped the historical developments in which the Plank's constant was not yet measured. For a historically correct development see [Mand92] §3.3 For an electron, q = × −5 10 9 cm and p = × ⋅−2 10 19 g cm sec , which

gives h ≅ × ⋅−6 626 10 27. secg cm

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experimental observations rather than theoretical formulation [Dira58], [Heis30], [Dira25].

Heisenberg was searching for a theory that involved only measurable quantities of the quantum mechanical system [Mehr82]. In order to answer the question: what are the measurable quantities? What are the equations of motion for these quantities? Heisenberg made the distinction between the values that are measured and the mathematics used to describe the equations of the quantities. His model for this theory was Einstein's special theory of relativity, which focused on measurable quantities. In correspondence between Einstein and Heisenberg, Einstein insisted it makes no sense to assume that what is measurable can be specified without an underlying theory, that …

It is the theory which decides what we can observe [Heis71].

Considering the commutator rule in its expanded form,

i j j i ijq p p q ic− = δ , (13.1)

states that the position coordinate and the momentum coordinate when measured individually, produce real numbers with unlimited accuracy. It also states that the position and the momentum of a particle cannot be simultaneously measured with unlimited accuracy. If the momentum and the position values are represented by matrices P and Q then the Heisenberg uncertainty principle implies that,

( ) ( )2 2

2cQ Q P P− − ≥ . (13.2)

Postulate 4 can be derived from Heisenberg's uncertainty principle [5] which states that the product of two dynamical variables is greater than or

5 Heisenberg's Uncertainty Principle is one of the more popular misunderstandings of

modern science. Critics of this unpredictable description of nature have tried to interpret the uncertainty that appears in quantum mechanics as a consequence of the ignorance of the underlying physical process. However this ignorance may not be due to the inability of the observer to measure properly, but rather that the laws of nature — at the quantum mechanical level — set an absolute limit on the ability of the observer to predict the outcome of the measurement [Frit83]. Various descriptions of the principle can be found in the literature, including:

(1) It is not possible to measure both the position and the momentum of a particle accurately at the same time — This is somewhat misleading

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equal to one half the magnitude of the expectation value of the Hermitian commutator,

121 2k k c∆ ∆ ≥ . (13.3)

This expression can also be stated as: The product of the uncertainties for the position and momentum is never smaller than 2c [Jord86]. [6] In the context of the uncertainty principle, the uncertainty, ∆K of a variable

(2) The properties of a classical particle are no longer valid in the quantum mechanical description of motion — This is much too pessimistic a view.

(3) Attempts to localize a particle, inevitably give it a kick of such probable intensity that the momentum of the particle is changed — This is wrong.

A clear description of the Heisenberg Uncertainty Principle is given in [Penr89] pp. 248–50. The summary simply says that a particle's position and momentum are described by its Wave Packet having coordinates in both position and momentum spaces. Since the particle is moving, the wave packet evolves with time. The position description and the momentum description of the particle are related by the Fourier transform of each other [Bagg92], [Heis89]. When a precise measurement is taken in one space, a Dirac delta function is used as a sampling device. The Fourier transform of this delta function in one space is spread out in the other space. The result is that the more precisely the sample in one space, the more imprecisely the information in the other space.

Heisenberg himself summarized the confusion over his principle:

I remember discussions with Bohr which ... ended in almost despair; and when at the end of the discussion ... I repeated to myself again and again: Can nature possibly be as absurd as it seemed... ?

6 The traditional expression of Heisenberg’s uncertainty principal describes the relationship between two canonical variables, position q and momentum p, with ∆ ∆p q⋅ ≥ h. A second relationship is often found in the literature, but usually is given without a full explanation of its consequences. That relationship is between energy E and time t, ∆ ∆t E⋅ ≥ h. This uncertainty condition does not follow the canonical variable rules developed in this section. L. D. Landau was quoted as saying,

... there is obviously no such limitation ... I can measure the energy and look at my watch; then I know both the energy and time [Polk86].

Another way to view ∆ ∆t E⋅ ≥ h is to consider the uncertainty is terms of energy transfer. The limitation is on the measurement of the amount of energy transferred. As the system is measured for smaller amounts of time, the uncertainty in the amount of energy grows. This energy uncertainty relation is the basis of the electron tunneling observed in semiconductor devices. In normal electronics and electron can not penetrate a potential barrier that is of higher energy than the electron itself. An electron is allowed to borrow some energy to energy on the other side of the electric potential. Tunneling effects can be found in several areas of physics. The explanation of the alpha decays of heavy nuclei depends on tunneling. Such nuclei behave as if they have alpha particles confined inside them. Occasionally the alpha particle is ejected from the nucleus having penetrated as energy barrier which classically would have been insurmountable.

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K is defined as the root mean square deviation from the mean [Heis27]. In a state u , the mean value of K is,

u u=K K . (13.4)

The mean value of the operator ( )2−K K is then the mean square

deviation from the mean,

( ) ( )22 u u∆ = −K K K , (13.5)

and the uncertainty is then

( )2u u∆ = −K K K . (13.6)

If a new operator ′K is introduced such that,

′ = −K K K , (13.7)

the commutator may then be written as,

( ) ( )1 2 2 1 1 2 2 1i i ′ ′ ′ ′− = −K K K K K K K K . (13.8)

The uncertainty relation in Eq. (13.4) may be examined in more detail. If u is an arbitrary vector and λ is an arbitrary real constant, the magnitude of the vector ( )1 2i u′ ′+ λK K is greater than or equal to zero for all λ . Thus,

( ) ( )1 2 1 2 0u i i u′ ′ ′ ′− λ + λ ≥K K K K , (13.9)

with 1′K and 2′K being Hermitian operators. From this relation,

( ) ( )2 222 1 0cλ ∆ + λ + ∆ ≥K K , (13.10)

for all λ . This may be rewritten as,

( )( )

22

2 1 22 2

02 4

c c λ∆ + + ∆ − ≥ ∆ ∆

K KK K

. (13.11)

The inequality must hold in particular for the value of λ which makes the first term zero,

1 22cλ = − ∆ ∆K K . (13.12)

It follows that the second term is non–negative, so that,

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121 2 c∆ ∆ ≥K K . (13.13)

The preceding development states that the results of a measurement of a dynamical variable in a physical system prepared by the measurement of another dynamical variables depends on the commutator of the two variables. [7]

The essential hypothesis of quantum mechanics concerns the specification of these commutators.

In order to make this hypothesis explicit the difference between classical and non–classical dynamical variables must be made. The classical variables are defined in terms of the classical generalized coordinates of the system, iq and the conjugate momentum ip . The dynamical variable is then a function of these coordinates and momentum

( ),i iq pK . Examples of these variables are energy, position, momentum and angular momentum of a particle.

The commutation rules for these classical variables can be expressed in terms of the Poisson bracket from classical mechanics. For two dynamical variables ( )1 ,i iq pK and ( )2 ,i iq pK , the Poisson bracket is,

[ ] 1 2 2 11 2,

i i i i iq p q p ∂ ∂ ∂ ∂

= − ∂ ∂ ∂ ∂

∑ K K K KK K . (13.14)

7 The concept of a quantum mechanical system prepared by the observation —

intervention of an observer — has lead several authors to assert a connection between quantum mechanics and some forms of eastern mysticism [Zuka79], [Capr75]. The argument states that because position and momentum cannot be measured simultaneously the observer make a choice of one or the other through the act of measurement and that this choice is a conscious one that results in the mind of the observer becoming part of the observation process.

Eugene Wigner described the logical conclusion…

It was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to the consciousness of the observer... Remarkably, the very study of the external world lead to the conclusion that the content of the consciousness is the ultimate reality [Wign67], pp. 172.

The argument of the connection between mysticism and quantum mechanics is arrived at through reductive means by demonstrating we are all floating in a sea of mind [Zuka79] From the view point of a physicists, the idea of an observer

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The basic hypothesis a quantum mechanics is then that the commutator [ ]1 2,K K is ih times the dynamical variable ( )1 2,K K ,

( ) ( )1 2 1 2, ,i= hK K K K , [8] (13.15)

where the constant c has now been replaced with Planck's constant h , completing the development of Heisenberg's uncertainty principle. [9]

This expression is fundamental to the theory of quantum mechanics and is given here — stated as fact. However the development of this concept is tedious and requires an understanding of the mathematics of commutator matrices which is beyond the scope of this text. For those interested in the details of these development they can be found in [Dira58] §4 [Krag90] and [Heis30].

Heisenberg's emphasis on measurable quantities leads to some important conclusions about atomic physics. Through an unsuccessful attempt to describe the orbits of atoms, Heisenberg said the uncertainty principle ...

…helped to convince me of one thing: that one ought to ignore the problem of electron orbits inside the atom [Heis71].

8 It is important to understand that the operator on the right–hand side of this

expression must be Hermitian. If p and q are individually Hermitian, neither pq nor qp is

Hermitian, since ( ) ( )pq qp∗

= . Therefore the classical variables pq is represented by the

quantum operator ( )½ pq qp+ , which is Hermitian.

9 The use of the Poisson Bracket notation in quantum mechanics was introduced by P. A. M. Dirac in 1924. Dirac had made an unsuccessful attempt to introduce relativistic dynamics into Heisenberg’s quantum theory [Krag90]. The idea of using the Poisson Bracket came out of the blue after Dirac reread Whittaker’s Analytical Dynamics, [Whit37]. The notation in Whittaker’s text reads ( ),x y . The bracket notation x y, was

reserved for the Lagrangian symbol defined as k k k k

k

p q q p

x y x y

∂ ∂ ∂ ∂−

∂ ∂ ∂ ∂

∑ . Dirac replaced the

Poisson Bracket with the square brackets, which in turn pervaded quantum mechanics [Dira83].

This discovery can be traced to the fact that Hamiltonian dynamics can be formulated using a non–commuting Poisson Bracket algebra. By connecting the Poisson Bracket with Heisenberg’s products ( ) [ ],xy yx i x y− = h , the basis for Dirac’s paper “The Fundamental

Equations of Quantum Mechanics,” was laid. This work in turn resulted in the seminal text [Dira25].

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The uncertainty relations confirm the knowledge of an orbit would imply knowledge of both position and momentum. For an electron in a hydrogen atom, the product of the position and momentum uncertainties is so large when compared to the orbit radius and momentum, that the orbit is only roughly defined [Jord86].

§13.2.4 Postulate 5 and Schrödinger's Equation

Although the focus of this monograph is on the interaction of charged matter with the electromagnetic field, a diversion into the fundamentals of quantum mechanics will provide some background of later sections. In March of 1926 Erwin Schrödinger wrote down a differential equation that described a nonrelativistic particle moving in a potential. Schrödinger was guided by Hamilton's description of classical mechanics and de Broglie's [10]

description of matter waves associated with the motion of atomic sized objects.

Using the formalism of special relativity de Broglie was able to equate space and time with momentum and energy. Planck’s formula states that energy is related to frequency by E h= ν . This relation also states that energy is related to the number of vibrations per unit of time. De Broglie proposed that momentum is similarly related to the number of vibrations per unit of space, which is simply the wavelength such that p h= λ . the de Broglie wavelength is then h pλ = .

de Broglie's equation states that the energy E of a matter wave [11] oscillates with frequency Eω = h with a wave function,

10 Prince Louis Victor de Broglie was a nobleman from an illustrious family of France.

As an amateur scientist unknown to the research community he wrote a doctoral dissertation — Rescherche sur la théorie des quanta, University of Paris, defended 25 November 1924 — that was sufficiently farfetched that the faculty of the Sorbonne was unable to evaluate it correctness [Serg80], [Crea86].

11 In de Broglie's [d’Brog24] thesis, the wavelength of light is given as λ ν= c , where ν is the frequency of the light wave. In quantum mechanics energy is quantized in units of E = hν giving ν = E h .

Substituting the quantum expression into the classical wavelength expression gives λ = c Eh . Since Einstein had already concluded that E mc= 2, de Broglie expression

becomes λ = c mch 2 or λ = h mc . Since mass velocity× ( )mc is equivalent to momentum,

particles like electrons and protons also have momentum equal to their mass times velocity. Since Einstein showed that matter cannot travel at the speed of light the actual

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i te − ωψ ∝ , (13.16)

which satisfies the differential equation,

i Et

∂ψ= ψ

∂h . (13.17)

If the matter wave has momentum pp , de Broglie's relation gives a wave number of =k p h which describes the wave function in terms of its position as,

ie ⋅ψ ∝ k r , (13.18)

which satisfies the relation,

i− ∇ψ = = ψk ph h .

Schrödinger guessed [12] the wave equation for a free particle with energy E would be,

2 2

2

2 2E

m mψ = ψ = − ∇ ψ

p h. (13.19)

An electron in a bounded potential well has potential energy ( )V r as

well as kinetic energy due to its motion. [13] Schrödinger generalized the wave equation using the potential and kinetic energies to give,

( )2

2

2E V

mψ = − ∇ ψ + rh

, (13.20)

velocity in de Broglie's expression can be represented byυ , which when substituted gives the wavelength of a matter wave as, λ = h mv .

12 It is impossible to derive — with any rigor — the quantum mechanical Schrödinger equation from classical mechanics. In most texts Schrödinger's equation is simply stated and then justified through its successful application. By following the reasoning given in Schrödinger's notebooks the classical equations of motion serve as the starting point for the derivation. When Schrödinger started to publish the results of his research, which are presented here, he elected to present a much more complex derivation, which did not refer to de Broglie waves or quantized energy [Crea86].

13 The use of a bounded potential well for the evaluation of quantum mechanical problem is a traditional approach to the solution of boundary value problems. Through this technique, the quantum mechanics of the hydrogen atom can be examined. Beyond this simple problem though, the potential well solution has little to contribute to unbounded problems such as quantizing the free electromagnetic field.

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which is Schrödinger's equation for a single particle with definite energy E. A more general time–dependent form can be obtained by eliminating E from the equation to give,

( )2

2

2i V

t m ∂ψ

= − ∇ + ψ ∂ rhh . (13.21)

It is customary to write this equation in the form,

i Ht

∂ψ= ψ

∂h , (13.22)

for the time–dependent case and,

H Eψ = ψ , (13.23)

when the energy is known. In both forms H is the derivative operator,

( )2

2

2H V

m= − ∇ + rh

. (13.24)

§13.2.4.1 The Expectation Value of an Operator

In order to proceed with Schrödinger’s description of the interaction between matter and fields, some further diversions are needed. Since the objects in quantum mechanics and the operators that work on them are statistical in nature the definition of the expectation value of an observable property is needed.

The expectation value of a property corresponding to an operator A is given by,

3A d∗= ψ ψ∫ r . (13.25)

Differentiating Eq. (13.25) with respect to time gives,

3d AA A A ddt t t t

∗∗ ∗ ∂ψ ∂ψ ∂

= ψ + ψ + ψ ψ ∂ ∂ ∂ ∫ r . (13.26)

Substituting Eq. (13.22) and its complex conjugate, i Ht

∗∗∂ψ

= ψ∂

h into

Eq. (13.26) gives,

( ) ( ) 3d i AA H A AH d

dt t∗ ∗ ∂ = ψ ψ − ψ ψ + ∂∫ rh . (13.27)

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Using the Hermitian conditions of H,

( ) ( ) 3 3H A d HA d∗ ∗ψ ψ = ψ ψ∫ ∫r r . (13.28)

Using Eq. (13.28), Eq. (13.27) reduces to,

( ) 3d i AA HA AH ddt t

∗ ∂ = ψ − ψ + ∂∫ rh , (13.29)

where ( )HA AH− is the commutator [ ],H A so,

[ ],d AA H Adt t

∂= +

∂. (13.30)

§13.2.5 Lorentz Force Law from Schrödinger's Equation

In classical mechanics, Newton's equations of motion assume that at each instant of time a particle is located at definite position with a definite velocity, momentum and rate or change of momentum or force. In quantum mechanics, only the eigenvalues or expectation values of the equations of motion have precise values.

The expectation value can be used to generalize the description of motion and provide an expression for the Lorentz force law. The expectation value for a linear momentum equals the mass of the particle times the rate of change of the expectation value of its position. The expectation value for a force is given by the time rate of change of the expectation value of the corresponding momentum [Duff83].

Given a particle of mass m moving is a scalar potential V the time derivative of the expectation value for its position on the x–axis involves

,x H . Using the explicit form of the operator ,x x x = − H H H gives

the commutator as,

2 2 2 2 2 2 2 2

2 2 2 2 2 2

2

, ,2 2

2 ,2

x x Vx x xVm x y z m x y z

m x

xim i x im

∂ ∂ ∂ ∂ ∂ ∂ = − + + + + + + − ∂ ∂ ∂ ∂ ∂ ∂

∂ = ∂ ∂

= =∂

p

h h

h

h h h

H

(13.31)

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Since x is constant in the partial derivatives with respect to t, 0x t∂ ∂ = , gives the expectation value of the momentum as,

0x xd im x m p pdt im

= + =

hh . (13.32)

The time rate of change of the expectation value of the x–component of the momentum xp is given by , xp H which using the explicit form gives,

2 2 2 2

2 2 2

2 2 2 2

2 2 2

,2

,2

.

xp Vm x y z i x i x

Vi x m x y z i xV

i x

∂ ∂ ∂ ∂ ∂ = − + + + + ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂+ + + − ∂ ∂ ∂ ∂ ∂

∂= −

∂

h h h L

h h hL

h

H

(13.33)

Since 0xp t∂ ∂ = ,

0x xd i V Vp Fdt i x x

∂ ∂= − + = − =

∂ ∂h

h . (13.34)

Similar equations can be developed for the y and z components. In vector form the time derivative of the momentum is given as,

d Vdt

= − ∇ =p F , (13.35)

which says that the rate of change of the expectation value of the momentum of the particle's wave function equals the expectation value of the force acting on the particle. Although this equation does not directly equate to Newtonian mechanics, since the particle's wave packet does not propagate with unchanging certainty, it does approximate Newtonian mechanics. Eq. (13.35) was derived in 1927 by Paul Ehrenfest (1886–1933) and is called Ehrenfest's theorem. [14] These equations are not the classical

14 Ehrenfest's theorem can be stated in a general manner using the momenta and

generalized coordinates that follow the expression ( ) [ ]1 ,i id q dt i q= h H and

( ) [ ]1 ,i id p dt i p= h H . A system of particles placed in a conservative potential field

can be represented by the Hamiltonian ( ) ( )22k k k

k

p m V= +∑ rH where the momentum of

the kth particle is given as ( )1, ,k k NF V≡−∇ r r… . Using the commutator relations developed

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limit of the quantum mechanical equations of motion, since Eq. (13.35) holds for any state of a particle and since they are the equations for the expectation values about which the uncertainty occurs, the standard deviations x∆ and ∆p are no longer negligible. The classical limit only applies for states in which this uncertainty is negligible, so that the particles travel along well–defined trajectories [Gott66].

§13.2.5.1 Lorentz Force Derivation

The Schrödinger formulation can now be extended to a field which carries both energy and momentum — such as the electromagnetic field [Duff84]. This field may contribute its energy and momentum to a particle which interacts with it in the same way the classical electromagnetic fields interact with charged particles. The effect of this energy interaction can be represented by an interaction coefficient and a scalar potential. The momentum effect can be represented by a coefficient and a vector field.

Given a particle of mass m and velocity v moving in a field, the potential energy resulting from the interaction between the particle and the field is V and the total energy of the interaction is,

2E mv V= + . (13.36)

If the particle were traveling in the absence of the field — it was a free particle — its linear momentum would be mv, but the interaction between the particle and the field alters the momentum by a factor dependent on the interaction coefficient q such that the momentum is now given by,

mv q= +p A , (13.37)

which gives,

( ) ( )

( ) ( )2 2 ,

.j j j jj

m v mv mv q q

p qA p qA

= ⋅ = − ⋅ −

= − −∑p A p A

(13.38)

previously [ ] ( ),k ki m p=r hH and p i Fk k,H = h . Substituting these expressions into the

original time derivative gives, d dt p mk krr = and d p dt Fk k= which is Ehrenfest's

theorem.

By combining the two previous expressions the quantum analog of Newton's Second law of motion can be given as, F md dtk k= 2 2rr which states that the expectation value

of the position and momentum coordinates are analogous to the classical laws of motion [Biln74].

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Substituting Eq. (13.38) into Eq. (13.36) gives the Hamiltonian of the particle as,

( ) ( )12 j j j j

j

E p qA p qA Vm

= − − + ≡∑ H , (13.39)

which can be rewritten as,

( )( ) ( )2j j j jj

p qA p qA m E V− − = −∑ . (13.40)

The operator for the thj component of the momentum is given by,

jj

pi x

∂=

∂h

(13.41)

Since the vector field AA is a function of position and possibly time it is generally independent of these components, which gives the corresponding operator for the vector field as,

j j jj

qA p qAi x

∂− = −

∂h

. (13.42)

Combining each side of Eq. (458) and summing over the j components and applying the result to the particles wave function ψ gives,

( ) ( )

( )

,

2 .

j j j jj jj j

qA qA p qA p qAi x i x

m E V

∂ ∂− − ψ = − − ψ ∂ ∂

= − ψ

∑ ∑h h (13.43)

which is the Schrödinger equation for the motion of a particle is a field described by AA and V.

The potential energy V of the field can be expressed as the interaction coefficient q times a scalar function φ such that,

V q= φ . (13.44)

The Hamiltonian then becomes,

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( ) ( )

( )( )

( ) 2

1 ,21 ,21 .2

j j j jj

p qA p qA qm

q q qm

q q qm

= − − + φ

= − − + φ

= ⋅ − ⋅ + ⋅ + ⋅ + φ

∑

p A p A

p p p A A p A A

H

(13.45)

The operator describing the motion of a particle with mass m and an interaction coefficient q traveling in a field described by AA and φ is the

Hamiltonian H [Duff84].

By restating the momentum as,

j jj

pi x i

∂= = ∇

∂h h

, (13.46)

allows the momentum to be represented as a vector,

i

= ∇p h. [15] (13.47)

Since the momentum pp is not a function of time,

0t

∂=

∂p

. (13.48)

Applying the momentum operator to the scalar wave equation gives,

[ ] ( )

( )

, , ,

,.

i iq q q q

iq q qq

φ ψ = φ ∇ ψ = φ∇ψ−∇ φψ = φ∇ψ− φ∇ψ− ψ∇φ

= − ∇φ ψ

ph

h h (13.49)

and applying the momentum operator to the vector field gives,

15 This operator notation of a vector can also be used for the kinetic energy and total

system energy where T m= −∇ 2 2 and E i t= ∂ ∂ . These operators do not necessarily commute. From the definition p i xx = − ∂ ∂ it follows x p p x x p ix x− = =, where x is the

position operator.

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( ) ( )1 ,2

1 , ,2

1 .2

12

j j j j

j j j j j

j j j j j j j j

j j j j j j

i p qA p qAmi qA qA

m i i i

qA qA qA qAm i i i i

qA qA qAm i i i

− − ψ = = ∇ − ∇ − ∇ ψ = ∇ − ∇ − ∇ − ∇ ∇ − ∇ − ψ

= ∇ − ∇ − ∇ψ − ∇ −

p Lhh h hL h

h h h h

h h h

( )

( ) ( ) .2

j j

j j j j j j j

j j j j j j

qAi

qA qA q A qAi i i

q p qA A A p qAm

∇ − ∇ψ− − ∇ − ∇ − ψ − − ∇ ∇ − ψ

= − ∇ + ∇ − ψ

h L

h h hL

(13.50)

Using Eq. (13.50) the time derivative of the expectation value of the momentum operator is,

( ) ( ) ( ) ( )

( )

,21 .2

j j j j j j

j j j j

d q p qA A A p qA qdt m

q v A A v q

= − ∇ + ∇ − − ∇φ

= ∇ + ∇ − ∇φ

∑

∑

p (13.51)

where, j j jp qA mv− = has been introduced, which gives,

j jd q v A qdt

= ∇ − ∇φ∑p . (13.52)

When the wave packet is small enough to make each jA∇ and ∇φ

constant within the integrals then v = v and v = v which gives,

( )

( )

,

.

j j

j j

d q v Adt

q v A

= ∇ − φ

= ∇ − φ

∑p (13.53)

By the chain rule of differentiation d vdt t

∂= + ⋅∇

∂ while differentiating

mv q= +p A gives,

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( ) ,

.

d d mv qdt dt

dvm q v

dt t

= +

∂ = + + ⋅ ∂

p A

AA

(13.54)

which converts Eq. (13.52) to,

( )

( )

,

.

dvm q q v vdt t

q qvt

∂ = −∇φ− + ∇ ⋅ − ⋅∇ ∂ ∂ = −∇φ− + × ∇× ∂

A A A

A A (13.55)

Using the usual vector identity ( ) ( ) ( )A B C B A C A B C× × = ⋅ − ⋅ and defining the familiar Maxwell field potentials, t=−∇φ−∂ ∂E A and

= ∇ ×B A , gives the Lorentz force equation derived from Schrödinger's equation as,

( )dvm q vdt

= + ×E B . (13.56)

§13.2.5.2 Current Associated with Propagation

Before proceeding with quantizing the electromagnetic field, another example of a derivation from Schrödinger's equation will be useful. The propagation of energy and momentum through the classical electromagnetic field has a quantum mechanical counterpart. Starting with Schrödinger's equation for the motion of a quantum particle it can be shown the a current is associated with the propagation of the particle through space [Eise69].

Using the now familiar Hamiltonian T V= −H and the energy operator E an expression relating the two can be formed as,

Eψ = ψH , (13.57)

so that,

( )21 ,2

i V tt m

∂ψ = ψ = − ∇ + ∂ rH . (13.58)

If V is independent of t then putting ( ) ( ) ( ),t u g tψ =r r allows the time and space components of the expression to be separated and rewritten as,

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( )

( )( ) ( ) ( )21 1

2g ti V u E

g t t u m∂ = − ∇ + = ∂

r rr

, (13.59)

where E is the energy eigenvalue. Solving this expression for the time component ( )g t gives,

( ) iEtu e −ψ = r . (13.60)

The complex conjugate of the Schrödinger equation is given by,

( )21 ,2

i V tt m

∗∗∂ψ = − ∇ + ψ ∂

r , (13.61)

The rate of change of the probability density ∗ρ = ψ ψ is given by,

( ) ( )2 2

2i

t t t m

∗∗ ∗ ∗ ∂ρ ∂ψ ∂ψ = ψ + ψ = − ∇ ψ ψ − ψ ∇ ψ ∂ ∂ ∂

. (13.62)

The right–hand side of this expression can be written in terms of a vector current or probability current,

( ) ( ) ( ),2itm

∗ ∗ = ∇ψ ψ − ψ ∇ψ j r , (13.63)

while the integral of the left–hand side gives the current density as,

( ) ( ),t i ∗ ∗ρ = ψ ∇ψ−ψ∇ψr . (13.64)

The divergence of the vector current is given as,

( ) ( )2 2

2im t

∗ ∗ ∂ρ ∇ ⋅ = ∇ ψ ψ − ψ ∇ ψ = − ∂j . (13.65)

Using the following definition for the divergence: the divergence of a vector function is the negative of the time rate of change in the mass density of the fluid for which the function is the momentum density [Duff84]. This allows the use Gauss's theorem for converting a surface integral into a volume integral gives,

3 3

V S

d d dSt

∂ρ = − ∇ ⋅ = −

∂ ∫ ∫ ∫r j r jÑ . (13.66)

by using the familiar Maxwell relation for current and charge, the conservation law of charges and currents can be restated as,

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0t

∂ρ+ ∇ ⋅ =

∂j . (13.67)

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I...never accept anything as true if I had not evident knowledge of its being so; that is, to accept only what presented itself to my mind so clearly and distinctly that I had no occasion to doubt it.

— Rene Descartes in Discourse on Method [Elli66]

Copyright 2000, 2001 14–1

§14. FOUNDATIONS OF QUANTUM FIELD THEORY

Quantum Field Theory (QFT) is currently the most successful framework for describing in interactions between forces and matter at the subatomic level. It provides both theoretical and computational techniques. It’s prediction of the interaction between the quantized electromagnetic field and electrons is accurate to within on part in 810 . Three of the four forces of nature can be explained with QFT, using the Standard Model. Only Gravity remains outside the description of QFT. This chapter will not present the origin or complete details of QFT. These can be found in numerous texts, including: [Bjor64], [ Bjor65], [Bobo59], [Chan90], [Itzy80], [Jauc55], [Mand84], [Ramo89], [Ryde85], [Saku67], [Schw61], [Schw58], [Schw94].

QFT is a generalization of the ordinary quantum mechanics of point particles and in principle applies the laws of quantum theory to non–mechanical systems with infinite degrees of freedom. A sub–theory of QFT is Quantum Electrodynamics (QED). QED is the quantum theory of the electron’s interaction with the electromagnetic field. There are aspects of QED that will be needed to describe the interaction of the radiated electromagnetic field with the electrons in the conductive metal of the radio antenna. This chapter will lay the foundations for the description of this interaction.

In classical physics such systems are called fields, with the electromagnetic and gravitational fields being examples [Roma69]. A field theory presupposes a continuum of space–time points. This continuum is formally called a differentiable manifold, which can be described by coordinates and to which an afine connection and a metric can be assigned. The specification of the values of all the relevant quantities to each space–time point specifies the configuration of the field. The differential equations constraining the values of the quantities at different space–time points forms the field theory.

§14.1 PROBLEMS WITH QFT

During the early stages in the development of QFT it became clear that there were problems in the description of the electron itself. Even Lorentz’s classical theory of the electron produced infinities when the radiation produced by the accelerating electron was included in the forces acting on the electron [Lore52].

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These self energies were first addressed by Dirac in 1927 [Dira27]. Dirac wrote…

…hardly anything has been done up to the present on QED. The questions of the correct treatment of a system in which the forces are propagated with the velocity of light instead on instantaneously, of the production of an electromagnetic field by a moving electron and of the radiation of this field on the electron have not yet been touched.

Dirac combines special relativity and quantum mechanics to create QED. This theory correctly described spin and the magnetic moment of the electron and provided the relativistic corrections to the spectra of hydrogen atoms. This theory also predicted anti–matter which was discovered in 1932 by Anderson [Ande33].

However other problems began to occur. When higher order corrections to the theory were applied the infinities reappeared [Schw94]. These divergence’s in the theory lay deep within the structure of QED. QED contained integrals which diverged as → 0x on in the momentum representation → ∞k . These divergence’s were equivalent to Lorentz’s self energy problem with the electron.

Over the next several decades the technique of renormalization was used to hide the problem. Although useful work was done there was always a nagging question that these corrections were masking the true problem — that the theory did not correctly describe the workings of nature. In 1949 Tomonaga, Schwinger and Feynman formulated the solution to these divergence’s for which the shared the Nobel Prize. In the early 1960’s attempts to unify the weak and strong forces lead to new divergence’s [Pais86], [Crea86].

The quantum field theory developed in the 1930's [Dira27] relied on the quantization of the classical field. Although this approach worked for the electromagnetic field, after the addition of perturbation techniques, it failed to work for the strong force field of interacting nucleons. A new approach was taken in the 1950's which generalized field theory with the abandonment of perturbations and the underlying Lagrangian and instead introduced a set of axioms for the quantum field.

§14.2 SIMPLE APPROACH TO QFT

It is the former description of the quantum field that is presented here, one in which the electromagnetic field has been quantized from the

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Copyright 2000, 2001 14–3

classical description, using the Fourier expansion of the field's vector potential.

In the previous sections quantum mechanics was used to describe a system in which the potential energy is known. However such a theory has limitations that prevent it from being a complete description of nature. Two shortcomings are its incompatibility with special relativity and its inability to describe systems where the number of particles is not a constant. A relativistic quantum field theory has been developed to remedy these problems. This monograph will not make use of this theory but rather present a non–relativistic description based on Hamiltonian mechanics.

Like the classical Hamiltonian formalism in classical mechanics, the quantum mechanical Hamiltonian is not so much a specific equation, but a framework for quantum mechanics in general.

The quantum mechanical form of the Hamiltonian can be construed to describe the interaction between a radiation field and a charged particle in much the same manner as the interaction between a charged particle and an assemblage of particles moving with the speed of light. [1]

§14.3 MECHANICAL ANALOGY

One method for developing the mathematics of quantum field theory is by quantizing a set of independent harmonic oscillators. This technique can be simplified by considering a one–dimensional continuous string composed of N point masses connected by springs. [2]. The string can be

1By deriving the quantum mechanical representation of the electromagnetic field from the Hamiltonian least action principal, it is sufficient to represent the field through the electromagnetic potential AA . This potential representation allows for gauge invariance to be utilized during the quantization process. When the field interaction is quantized, the matter which interacts with the field is represented by a probability amplitude, dependent on space and time coordinates. The probability of finding an electron at a point is ψ 2

. Only

this modulus can be measured, but its phase is indeterminate. In Quantum Electrodynamics now acquires a new property of symmetry and invariance. This invariance is equivalent to the conservation of charge. This property produces an important behavior. This phase invariance of the matter field can be attributed to a local invariance dependent on the point in space–time where it occurs — if and only if the quantum field of electrons is coupled to a quantum field of interaction obeying the same gauge invariance as the classical electromagnetic potential [Tann93]

2This method of developing the quantum field description using a mechanical model has a long history [Bril46]. The first effort of analyzing a one–dimensional vibrating lattice

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envisioned as lumpy with a total length of L containing N points each with equal mass m coupled with a spring constant k. If each point in this lumpy string is capable of being displaced around its equilibrium in a periodic manner by a distance l, than the string can be constrained to oscillate with a total length of =L Nl .

was done by Newton in his attempt to derive a formula for the velocity of sound [Cajo62]. Newton assumed sound was propagated in air in the same manner an elastic wave was propagated along a group of point masses. He assumed the simplest set of masses connected by a restoring force shown in the same manner as in Figure 5.0. Neighboring masses were assumed to attract each other with a elastic force e. Using m as the mass of each of the particles and d to be the distance between each particle when it was in a state of equilibrium, the velocity of propagation V of an elastic wave was given by Newton as, V d e m ed= = ρ , where ρ is the density of the air surrounding the particles. Using ed

as the isothermal modulus of air, the figure Newton computed was less than the experimental value. In 1822 Laplace pointed out that the expansions and contractions of air took place adiabatically and the adiabatic elastic constant should be used instead of the isothermal. The calculation using the proper constant produced an excellent agreement with the experimental data.

Using this approach to describe the vibrations in elastic media continued with John and Daniel Bernoulli. Using Bernoulli’s formulation of the supposition principal, which states that the general motion of a vibrating system is given by a superposition of it proper vibrations, the beginnings of theoretical physics became distinct from mechanics. Lagrange and Euler continued to develop — with much controversy — the work of the Bernoilli’s through the end of the eighteenth century. At that point the basis for theoretical physics and modern mathematical analysis was laid. The theory of proper functions, Fourier expansions, partial differential equations, wave propagation and the atomic theory of solids and crystal structures were first associated with the vibrations in strings.

In 1830 Cauchy used Newton’ model in an attempt to describe the dispersion of optical waves. Cauchy assumed light waves were just high frequency elastic waves. The length of the waves was estimated to be comparable with the distance between the point masses of a crystal lattice and thus their velocity was assumed to be independent of the wavelength.

Using theory of a vibrating string, quantum mechanics provides a set of rules which can be applied in a general way to mechanical systems. The application of these rules to classical fields results in a quantum field theory. However elementary (meaning non–relativistic) quantum mechanics is not consistent when it is combined with classical field theory unless the classical electromagnetic field is also quantized [Heis30], [Hene62], [Schw58]. Following the development technique of quantum mechanics, quantum field theory is linked to classical field theory through the correspondence principle. Using this principle, the excitations of the quantum field behave as if they were particles. In the case of the electromagnetic field the particle is a photon. Buy using a mechanical model containing point particles, connected by a linear restoring force, a visualization of the field can be constructed. The extension of this mechanical model to a quantum field theory has several limitations, but this will not be important for the description of the quantized electromagnetic field.

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L

kl

z

z n

0 N

Figure 5.0 — An ideal string of length L with N point masses. Each mass is connected by a spring with a spring constant k. Each point mass is constrained to oscillate around its equilibrium point by a distance l. This displacement is given by the function ( )φ ,i z t

If the point masses are constrained to vibrate in only one direction, z, then the displacement of the nth particle can be denoted by nz . The kinetic energy of all the point particle can be given by:

+

=

= ∑ &1

212

0

N

nn

T mz . (14.1)

It is assumed that there is potential energy associated with the distortions of the string and that this, to the lowest order in the nz direction is given by:

( ) ( )+

−=

= −∑1

2112

0

N

n nn

V kN L z z , (14.2)

with the constraint += =0 1 0Nz z .

The classical action for this mechanical system is given by:

( ) ( )+

−=

= − − ∑∫ &1

221 112 2

1

b

a

t N

n n nnt

S mz kN L z z dt . (14.3)

If the discrete system is extended to a continuum the individual displacement an be described by a function of both position and time. If the function φn is the displacement of the nth oscillator from its equilibrium point then the boundary constraints are φ = φ0 N and φ = φ0 Nd dt d dt . The kinetic and potential energies of the string are then given by,

−

=

φ =

∑21

0

12

Nn

n

dT m

dt, (14.4)

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and

( )−

+=

= φ − φ∑1

21

0

12

N

n nn

V k . (14.5)

By expanding the number of points in the discrete string to a large number a continuous system can be developed. If → ∞N and → 0l and the length of the string remains fixed, such that =L Nl and the mass per unit length is m l and the string tension iskl , then the displacement of the string in the z direction and energy of the string can be given as a continuous field ( )φ ,z t , where,

( ) ( ) ( )φ = φ → φ, ,n nt z t z t , (14.6)

giving the kinetic and potential energy as,

( )−

=

∂φφ = → ∂ ∑ ∫

221

0 0

,1 12 2

LNn

n

z td mT m l dt

dt l t, (14.7)

( )−

+

=

∂φφ − φ = → ∂ ∑ ∫

2211

0 0

,1 12 2

LNn n

n

z tV kl l kl dt

l t. (14.8)

The Lagrangian and Hamiltonian of the continuous string can now be constructed using these expressions.

( ) ∂φ ∂φ = − = − = ∂ ∂

∫ ∫L L2 2

0 0

1 1,

2 2

L LmT V kl dz z t dz

l t t, (14.9)

( ) ∂φ ∂φ = + = + = ∂ ∂

∫ ∫H H2 2

0 0

1 1,

2 2

L LmT V kl dz z t dz

l t t. (14.10)

where L and H are the Lagrangian densities.

The previous expressions can be placed in the standard form of the Euler–Lagrange equations by absorbing the spring constant kl through the transformation φ → φkl and the introduction of the wave velocity

=2 2v kl m , giving the Lagrangian and Hamiltonian as,

∂φ ∂φ ∂φ ∂φ = − = ∂ ∂ ∂ ∂

∫L L2 2

20

1 1,

2

L

dzv t t t z

, (14.11)

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∂φ ∂φ = + ∂ ∂

∫H2 2

20

1 12

L

dzv t t

. (14.12)

The equations of motion for the string are now given using the principal of least action [Gold51]. Using the Lagrangian principles developed previously the least action principle can be stated as,

∂φ δ = δ φ = ∂ ∫L , , 0i

iS tt

, (14.13)

giving the Euler–Lagrange equations of motion as,

∂ ∂

− =φ ∂φ ∂

L L0

i i

dddtdt

. (14.14)

In order to proceed with the quantization of the field equations, the discrete oscillators describing the string's motion — and eventually describing the electromagnetic field — will be increased to infinity by allowing → ∞N and → 0l such that the Euler–Lagrange equations can be written as,

∂ ∂ ∂ ∂ ∂ ∂

− → + φ ∂φ ∂φ∂φ ∂ ∂ ∂ ∂∂ ∂ ∂

L Lsmall l

i i

d L Ll kl

ddt t zt zdt

, (14.15)

with the term l kl being discarded.

The result of this mechanical example is the basis for quantized the electromagnetic field. The concept of a continuous field can be developed to describe a system containing an infinite number of particles. In the mechanical model the field equations ( )φ ,z t describes the displacement of

each particle from its normal position ± 12z l . Although a mechanical

example was used these concepts can be generalized to describe any field which is displacement of some dynamical system.

The next step in quantizing the mechanical model is to develop the solutions to the wave equation which satisfies the boundary conditions,

( )± −ωφ ≈ n ni k z te , (14.16)

where the oscillators are periodically spaced with,

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π

=2

n

nk

L, (14.17)

which gives the wave equation of,

∂ φ ∂ φ

− =∂ ∂

2 2

2 2 2

10

v t z, (14.18)

where,

ω

=2

22n

n

vk

. (14.19)

Since both positive and negative frequencies are allowed in the wave equation, the following notation will be used to distinguish one from the other. For positive frequencies, +ωn the solution to the wave equation can be written as,

( ) ( )−ωφ =1

, n ni k z tn z t e

L, (14.20)

and for negative frequencies, −ωn the solution can be written as,

( ) ( )+ωφ =1

, n ni k z tn z t e

L. (14.21)

The expression for the negative frequencies represents the complex conjugate of the positive frequencies and will be written as ( )∗φ ,n z t . The

normalization of these n frequency states is given by,

( ) ( )∗φ φ = δ∫ ,0

, ,L

n m n mz t z t dz , (14.22)

which can be rewritten as,

( ) ( ) − ω−φ φ = δ∫ 2

,0

, , n

Li t

n m n mz t z t dz e . (14.23)

The field can now be expanded in a manner similar to Eq. (10.13),

( ) ( ) ( ) ( ) ( ) ∞

∗ ∗

=−∞

φ = φ + φ∑, 0 , 0 ,n n n n nn

z t c a z t a z t , (14.24)

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where ( )0na are the coefficients of each normal mode in the expansion.

and nc is a real normalization factor. The time dependence of each normal mode can be incorporated by,

( ) ( ) − ω= 0 ni tn na t a e , (14.25)

resulting in,

( ) ( ) ( ) ∞

−∗

=−∞

φ = +∑, n ni k z i k zn n n

n

z t c a t e a t e . (14.26)

The final step is to write the equation for the harmonic oscillator for each point in the mechanical system as,

( ) ( )+ ω =&& 2 0n n na t a t . (14.27)

By quantizing the harmonic oscillator the transition from classical field theory to quantum field theory will be made. Since the Lagrangian approach to the dynamics of the field has been previously used, the energy of the mechanical systems dynamical variables ( )na t can be evaluated. By

rewriting the orthonormal relations in Eq. (14.22)as,

( ) ( ) ( ) ( ) ( ) ( )∗ ∗φ φ = δ = δ∫2 2

, ,0

0 , 0 , 0L

n n n m n m n n m na z t a z t dz a a t , (14.28)

and

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )− ω− − − −φ φ = δ = δ∫ 2

, ,0

0 , 0 , 0 0 n

Li t

n n n m n m n n n m n na z t a z t dz a a e a t a t ,

(14.29)

allows the kinetic energy to be written as,

( ) ( ) ( ) ( ) ( )∗ ∗− − − −

∂φ∂φ=

∂ ∂

= + +

∫

∑ & & & & &

20

222

1,

2

12 .

2

L

n n n n n n n n n n

T dzv t t

c a t c c a t a t c c a t a tv

(14.30)

and the potential energy to be written as,

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( ) ( ) ( ) ( ) ( )∞

∗ ∗− − − −

=−∞

∂φ∂φ=

∂ ∂

= + +

∫

∑

20

22 2

1,

2

12

2

L

n n n n n n n n n n nn

V dzv z z

k c a t c c a t a t c c a t a t

(14.31)

By equating ( ) ( )= − ω&n n na t i a t the Hamiltonian of the dynamical system

can be written as,

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

∞∗

=−∞

∗ ∗− − −

∞∗

=−∞

= +

ω= + +

ω + − + + ω

=

∑

∑

L

L

H2

2 22

22

2

22

2

,

1,

2

2.

nn n n n

n

nn n n n n n n

nn n n

n

T V

c k a t a tv

c c k a t a t a t a tv

c a t a tv

(14.32)

By using natural units in which the frequency ω has units of energy, the

normalization factor nc can be chosen such that, = ω2 2n nc v which

simplifies the Hamiltonian to,

( ) ( )∞

∗

=−∞

= ω∑H n n nn

a t a t . (14.33)

§14.3.1 Canonical Coordinates of the String

By selecting an alternative set of coordinates the Fourier coefficients of the Hamiltonian can be expressed as generalized position and momentum. The Lagrangian of the dynamical system allows the generalized position to be given as,

( ) ( ) ( )∗ = + ω12

n n n

n

q t a t a t , (14.34)

and the generalized momentum to be given as,

( ) ( ) ( ) ( )∗ = − = ω12

nn n n

n

dq tp t a t a t

dt. (14.35)

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which now allows the Fourier coefficients to be expressed in terms of a coordinate system,

+ ω

=ω2

n n nn

n

ip qa , (14.36)

and,

∗ − + ω=

ω2n n n

n

n

ip qa . (14.37)

Using the Fourier coefficients the Hamiltonian can be rewritten as,

∞

=−∞

= + ω ∑H 2 2 212 n n n

n

p q . (14.38)

This expression of the Hamiltonian is simply the sum of the Hamiltonians of each independent harmonic oscillator in the string. Hamilton's equations of motion can now be stated from the previous development as,

∂

= =∂

& Hn n

n

q pp

, (14.39)

and,

∂

= − =−ω∂

& H 2n n n

n

p qq

. (14.40)

which is the equations of motion for an uncouple oscillator.

§14.3.2 Quantizing the Mechanical System

Using the canonical variables developed above, a transformation from their Poisson bracket form into a commutator form can be performed using the techniques previously developed [Schi68]. For generalized coordinates and momenta [3], these canonical variables are related by,

3The process of treating the generalized coordinates pn and qn as quantized variables is called first quantization.

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[ ] = δh ,,n n n mq p i , (14.41)

and

[ ] [ ]= =, , 0n n n nq q p p . (14.42)

Using natural units = =h 1c these variables can be restated as,

∗ = δ ,,n n n ma a , (14.43)

and,

[ ] ∗ ∗ = = , , 0n m n ma a a a . (14.44)

The field equations can now be restated as,

( ) ( ) ( ) ∞

−ω − −ω∗

=−∞

φ = +ω

∑ 1,

2n n n ni k z t i k z t

n nn n

z t v a e a eL

. (14.45)

The Hamiltonian of the string can be written as,

∞∗ ∗

=−∞

∞∗

=−∞

ω = +

= ω +

∑

∑

H

12

,2

.

nn n n n

n

n n nn

a a a a

a a

(14.46)

§14.4 CANONICAL MOMENTUM OF THE STRING

The commutation rules between a and ∗a imply a relationship between the fields φ . If the φ field is considered a canonical coordinate, then using the Lagrangian defined in Eq. (14.11) can be used to define the canonical momentum as,

( ) ∂ ∂φΠ = =

∂φ ∂ ∂ ∂

L2

1,z t

v tt

. (14.47)

By generalizing the commutation rules given in Eq. (14.41) and Eq. (14.42) to the continuous fields coordinates φ and Π ,

( ) ( ) ( )′ ′ φ Π = δ − , , ,z t z t i z z , (14.48)

( ) ( )′ Π Π = , , , 0z t z t , (14.49)

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( ) ( )′ φ φ = , , , 0z t z t . (14.50)

It is useful at this point to demonstrate these commutation relationships explicitly. Expanding the canonical momentum of Eq. (14.47) in terms of the harmonic oscillator’s dynamical variable gives,

( ) ( ) ( ) ∞

−ω − −ω∗

=−∞

Π = − ω + ωω

∑1 1,

2n n n ni k z t i k z t

n n n nn n

z t i a e i a ev L

(14.51)

Expanding Eq. (14.49) using Eq. (14.51) gives,

( ) ( )

[ ] ( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

+ − ω +ω

′− + + ω −ω∗ ∗

′− − ω −ω∗

′− + + ω −ω∗

′−

− +ω ′ Π φ = − ω −

′= − ≡ − Ι

∑

∑

LL

L.

,

,, , , ,

2 ,

,

1, .

n m n m

n m n m

n m n m

n m n m

n

i k z k z i tn m

i k z k z i tn mn

i k z k z i tn m m n m

i k z k z i tn m

ik z z

n

a a e

a a eiz t z t

L a a e

a a e

i e i z zL

(14.52)

Since the exponential term inside the summation of Eq. (14.52) is complete and orthonormal, it can be expressed in terms of the exponential such that,

( ) ′′ ′Ι =∫0

, n n

Li k z i k zz z e dz e . (14.53)

Eq. (14.53) is the definition of the Dirac delta function,

( ) ( ) ( )′−′ ′Ι = = δ −∑1, nik z z

n

z z e z zL

, (14.54)

which results in Eq. (14.48).

The commutation relationships in Eq. (14.49) and Eq. (14.50) can be shown as,

( ) ( )

( ) ( )

( ) ( )

( ) [ ]

′− − ω −ω∗

′− − + ω −ω∗

′−

+ ′ φ φ = ω ω

= − ≡ω

∑

∑

L2

.

2

,1, , , ,

2 ,

11 1 0.

2

n m n m

n m n m

n

i k z k z i tn m

i k z k z i tn m n m n m

ik z z

n n

a a evz t z t

L a a e

ve

L

(14.56)

Foundations of Quantum Field Theory

14–14 Copyright 2000, 2001

They assert that all originates from the sky and its fires, and that first the fire changes into the breezes of the air and from the water is produced, and from water comes the earth and is reverse order everything reproduces from the earth, first water, next the air and then the heat — these are four elements never cease to interchange with each other, passing from the sky to the earth, and from the earth to the stars of the firmament.

— Lucrezio, I§A.C

Copyright 2000, 2001 15–1

§15. QUANTIZING THE CLASSICAL RADIATION FIELD

Up to this point the description of the electromagnetic field using the vector potential AA has simply been a convenience. Although both the vector and scalar potentials were utilized to expand the classical radiation function, their existence has always been mathematical. The macroscopic description of the electromagnetic field requires two Euclidean vectors — the electric field EE and the magnetic field BB . From these two vectors, the field potential AA can be inferred.

The transition from classical electrodynamics to quantum electrodynamics can be performed through the following processes. Suppose a volume V is divided into a large number N of smaller volumes, with the nth cell specified by the coordinate rrn . The distribution of the electromagnetic filed within the nth volume at time t can be approximated by the potential vectors, ( ) ( )1 , , , ,nt tA r A r… .

In order to transform the classical description of the electromagnetic field into a quantum mechanical form, the potential vectors ( ),n tA r will be restated as operators

acting on the vectors of a Hilbert space in such a manner that all the observables in the system are Hermitian operators which have real eigenvalues. The second step in the quantization process is to allow the number of small volumes to tend to infinity, allowing the system to have an infinite number of degree’s of freedom. [1]

The classical description of the fields given by the Hamiltonian formulation provides the transition to the quantum mechanical theory of the electromagnetic field equations. Using the canonical variables which are the generalized coordinates X kk,α and the

generalized momenta Pkk ,α , the scalar potential AA and the EE and the BB fields become

quantum field operators.

There is no place in this new kind of physics for the field and matter, for the field is the only reality.

— Albert Einstein [Cape61]

The quantum operators ( ),tA r form one of the elements of the mathematical

description of the quantum field. They will be referred to as field variables. These operators are defined over the infinite dimension Hilbert space, whose algebra is fixed by the commutation rules and by assuming the existence of a particular state vector describing the vacuum state of the system.

As a result of the quantization, the electromagnetic field’s potential is,

( ) ( ) ( ) 22

, , , ,1

2 ˆ ˆ, i ik i k i k i k i

i k

ct e a t e e a t e

V⋅ ∗ ∗ − ⋅

=

π= +

ω∑∑ k r k r

k

A rh

(15.1)

Quantizing the Classical Radiation Field

15–2

By replacing the classical vector potential ( ),tA r by the linear operator ( )ˆ ,tA r ,

which is a function of the Euclidean coordinates, every physical quantity of the system then becomes a Hermitian operator — also called an observable — whose real eigenvalues represent a precise measurement of this quantity.

For the radiation field using the transverse gauge, ∇ ⋅ =AA 0 and with the absence of external sources, indicated by φ = 0 , the electric and magnetic fields are given by the no familiar expressions, EE AA= − ∂ ∂t and BB AA= ∇ × , where AA satisfies the wave equation,

∇ − =22

210AA

AAc t

∂∂

, (15.2)

resembles the dynamic behavior of a collection of harmonic oscillators. In the quantum formulation of the radiation field, the oscillation system described by Eq. (15.1) has a variety of energy states, each associated with a quantum state. These states can be interpreted in terms of specific quanta.

§15.1 QUANTIZING THE SCHRÖDINGER EQUATION

The classical Lagrangian density which results in the Schrödinger equation is given by,

( ) ( ) ( )2

2 ,2

di V t

dt m∗ ∗ ∗ψ

= ψ − ∇ ∇ψ ⋅ ∇ψ − ψ ψrhhL , (15.3)

where L is not Hermitian. The Euler–Lagrange equation for ψ gives,

( )2

2 ,2

di V t

dt m

∗∗ ∗ψ

− = − ∇ ψ + ψrhh (15.4)

and for ψ∗ ,

( )2

2 ,2

di V t

dt m

ψ= − ∇ ψ + ψr

hh (15.5)

where these equations are the Schrödinger equation and its complex conjugate. The momentum conjugate for ψ is given by,

πδ

δ∂ψ∂

∂

∂ψ

ψ= = = ∗L

tddt

iL h (15.6)

Since ddtψ∗

does not occur in Eq. (15.3), ψ∗ has no conjugate momentum.

The Hamiltonian density is now given by,

( ) ( )2

d i iV

dt m

ψ= π − = ∇π ⋅ ∇ψ − πψ

hhH L (15.7)

Quantizing the Classical Radiation Field

Copyright 2000, 2001 15–3

The canonical equations of motion are now,

2

,

.2

i i

i

d d

dt x

x

i iV

m

ψ ∂ ∂ ∂= = −

δπ ∂π ∂ ∂π∂ ∂

= − ψ + ∇ ψ

∑

hh

H H H

(15.8)

2

,

.2

i i

i

d d

dt x

x

i iV

m

π ∂ ∂ ∂= = −

δπ ∂π ∂ ∂π∂ ∂

= − π + ∇ π

∑

hh

H H H

(15.9)

§15.2 QUANTIZING THE RADIATION FIELD

§15.2.1 FIELD COMMUTATION MODES

§15.2.2 ZERO POINT ENERGY

1 This limit expression method provides a simplified approach to quantizing the electromagnetic field,

but it does have serious drawbacks. The major problem is that the limit method only considers local fields which are described by function of space–time coordinates. It has been shown that this approach is the only description of the quantized field.

Copyright 2000, 2001 16–1

§16. GAUGE THEORY AND THE CREATION OF PHOTONS

In the previous chapter the existence of a gauge theory allowed the electromagnetic field to be described in an invariant manner. Although the existence of this invariance had been known for some time, its full significance was not recognized until the late 1960's [Kane93]. In quantum theory gauge invariance takes on a new meaning which leads to a cleared understanding of the photon as the carrier of the electromagnetic force.

Using Postulate 3 to restate that quantum observables depend on ψ2,

the structure of the invariance can now be given as,

− α′ψ → ψ = ψie , (16.1)

where α is a constant. This transformation is global since ( )ψ ,x t

transforms the same everywhere in space–time. A local transformation could be constructed in which the phase of ψ at each point in space–time can be chosen arbitrarily such that,

( ) ( ) ( ) ( )− χ′ψ = ψ = ψ,, , ,i x tx t x t e x t (16.2)

An exception to this local gauge transformation occurs when interference effects are measured and the interference intensities are sensitive to phase differences [Saku85].

It turns out that the local gauge transformation in Eq. (16.2) does not exist in that form. If the Schrödinger wave function for a particle satisfies the time dependent equation,

( ) ∂ − − ∇ + ψ = + φ ψ ∂ A 21

2i e i e

m t, (16.3)

where e is the charge magnitude of the electron.

§16.1 ANNIHILATION AND CREATION OPERATORS

In Dirac's theory, the interaction between the electromagnetic field and atomic sized matter became an interaction between a large number of photons and the matter. Any change in the energy of the atom due to the electromagnetic field could be described as the creation and annihilation of photons in the electromagnetic field.

Using the Fourier decomposition techniques developed in the classical expansion of the vector potential in the previous section, the linear combinations of αk ,Q and αk,P are now given by,

Gauge Theory and the Creation of Photons

16–2 Copyright 2000, 2001

( )−α α α= ω +

ωk k kh, , ,

1ˆ

2c Q iP , (16.4)

( )+α α α= ω −

ωk k kh, , ,

1ˆ

2c Q iP . (16.5)

The commutation relations are given as,

− +

α α α α α α

′ ′α α

= − +

= δ δ

k k k k k k

k k

h h, , , , , ,

, ,

ˆ ˆ, , ,2 2

,

i ic c Q P P Q

(16.6)

− − + +′ ′α α α α = = k k k k, , , ,ˆ ˆ ˆ ˆ, , 0c c c c (16.7)

These commutation rules are evaluated at equal times, giving,

− + − +′ ′α α α α ≡ k k k k, , , ,ˆ ˆ ˆ ˆ, ( ), ( )c c c t c t . (16.8)

A new operator is created to aid in the development of the radiation field expressions,

+ −α α α≡k k k, , ,ˆ ˆN c c . (16.9)

The new operator gives,

− − + − + − −′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′α α α α α α α α

− + − + − −′ ′ ′ ′ ′ ′ ′ ′ ′ ′α α α α α α

−′ ′α α α

= − = −

= δ δ

k k k k k k k k

k k k k k k

k k k

, , , , , , , ,

, , , , , ,

, , ,

ˆ ˆ ˆ ˆ ˆ ˆ ˆ, ,

ˆ ˆ ˆ ˆ ˆ ˆ, , ,

ˆ

c N c c c c c c

c c c c c c

c

(16.10)

Similarly,

+ +′ ′ ′ ′α α α α α = − δ δ k k k k k, , , , ,ˆ ˆ, , .c N c (16.11)

The quantized vector potential can now be written as,

( )− α ⋅ + α − ⋅α α

α =

= ε + ε∑ ∑ k r k rk k

k

A r ( ) ( ), ,

1,2

ˆ ˆ( , ) ( ) ( )i it c t e c t e (16.12)

Although this expression for the vector potential is nearly identical to Eq. (5.43) its meaning is very different. The vector potential in Eq. (16.12) is an operator that acts on the state vectors in the occupation number space of the electromagnetic field. This operator has parameters of space and time, but he operator is not a function of the space and time

Gauge Theory and Creation of Photons

Copyright 2000, 2001 16–3

coordinates of the photons representing the propagating electromagnetic field [Saki67].

The determination of the quantum mechanical Hamiltonian requires the calculation of the total energy of the radiation field,

( )= +∫ E BH 2 2rad

1 ˆ ˆ2

dv , (16.13)

in which E and B are expressed in terms of canonical variables αk ,Q

and αk,P .

The resulting expression for the quantum mechanical Hamiltonian is now,

( )α αα=

= + ω∑ ∑ k kk

H 2 2rad , ,

1,2

1ˆ ˆˆ2

P Q , (16.14)

which is exactly the same form as the classical Hamiltonian.

§16.2 PHOTONS STATES

The algebra of the operators described in the previous section will now be applied the physical situation where the number of photons with a given momentum and polarization is increased or decreased using the operators, +

αk,c and −αk,c . The wave vector k will be defined as the photon

momentum divided by h and α again represents the polarization state (spin state) of the photon.

The operator αk,N now represents the number of photons in state

αk, . In order represent a system in which there are many photons in

many states, the product of the various states is given by,

α α α α α α=k k k k k k… … L L1 1 2 2 1 1 2 2, , , , , ,, , , ,

i i i iN N N N N N (16.15)

This state vector corresponds to the physical situation in which there are αk1 1,N photons in state αk1 1, . The number of photons, αk,N , is called

the occupation number for state αk, .

The state in which no photons are present is 0 . This state will serve

as the building block in which a general number of photons can be place by,

Gauge Theory and the Creation of Photons

16–4 Copyright 2000, 2001

( ) α+

α

α αα α

= ∏k

kk k

k k

…,

1 1 2 2

,

, ,, ,

ˆ, , 0

i i

i i

i i i i

nc

N NN

, (16.16)

where the creation operator +αk,c is applied to the general state vector,

+α α α α α α α α= + +k k k k k k k k… … …

1 1 2 2 1 1 2 2, , , , , , , ,ˆ , , , , 1 , , , , 1i i i i i i i i

c N N N N N N N .

(16.17)

where +αk ,ˆ

i ic creates a photon in state αk ,i i , leaving the occupation

number of states other than αk ,i i unchanged and the operator αk,c is

applied to the general state vector,

−α α α α α α α α= −k k k k k k k k… … …

1 1 2 2 1 1 2 2, , , , , , , ,ˆ , , , , , , , , 1i i i i i i i i

c N N N N N N N ,(16.18)

where −αk,c annihilates a photon in state αk ,i i leaving the occupation

number of states other than αk ,i i unchanged. The operator αk ,i iN does

not change the occupation number, since it is diagonal and simply gives the eigenvalue of the number of photons in state αk ,i i .

Using Bose–Einstein statistics in which the number of photons in a particular state is unrestricted, a many particle system can be constructed. [1]

§16.3 PHOTONS AS RADIATED FIELD EXCITATIONS

In this manner the radiation field oscillator analogy is equivalent to the wave equation field operators which annihilate and create a particle at a given point in the field.

The determination of the eigenvalues of this Hamiltonian is given by energy levels of linear oscillators, which is simply the radiation Hamiltonian,

1Bose–Einstein statistics describes the mean number of bosons (photons) in mode n as

N n nmn

= ∑ ρ where ( )/ 2 1nm e e−β ω −β ωρ = −h h. The probability P n( ) of having n photons in

the mode as a function of n and N (Bose distribution law) is given as,

( ) ( ) 2( ) 1 1 1P n N N N= + + . Contrary to the Poisson distribution law for classical

particle states, the Bose distribution law decreasing with increasing n [Cohe89].

Gauge Theory and Creation of Photons

Copyright 2000, 2001 16–5

( )

( )

12

+ − − +′ ′ ′ ′α α α α

α=

αα=

+

= + ω

∑ ∑

∑ ∑

k k k kk

kk

h

H rad , , , ,1,2

1, 2

1,2

ˆ ˆ ˆ ˆ= ,

.

c c c c

N (16.19)

where again − +′ ′α α α=k k k, , ,ˆ ˆN c c is the operator that measures the occupation

number of a state of the system and has αk,N as its eigenfunctions.

The annihilation −α,ˆkc and creation +

α,ˆi ikc operators can now be used to

construct the plane wave solution of the potential equation.

The plane wave,

α

ω − ⋅α

ε= π

ωk r

kA( )

( ), 4 ,

2i te (16.20)

which appears in the radiation field vector potential operator,

− + ∗α α α α

α=

= +∑ ∑ k k k kk

A A A, , , ,1,2

ˆ ˆ ˆ( ),c c (16.21)

as coefficients of the photon annihilation operators may be treated as the wave function of photons having momenta kk and polarization αε( ) .

§16.3.1 Total Hamiltonian

The total Hamiltonian of charged matter's interaction with the quantized radiation field is,

= + +H H H Hobj int rad , (16.22)

where H obj is the Hamiltonian of the electrons,

= +

∑ rH 2obj

1( )

2 i ii

p Vm

, (16.23)

where the time dependent interaction terms is,

( ) = − + + φ

∑ A r A r

22

2( ) , ( , ) ( , ) ( , )2 2i i i

i

e eV t p t r t e t

mc mc, (16.24)

which is summed over all electrons in the matter, H rad is the radiation

field Hamiltonian and H int is the Hamiltonian of the interaction between the radiated field and the matter, given by,

Gauge Theory and the Creation of Photons

16–6 Copyright 2000, 2001

= − ⋅ + ρ

∫ j r A r r A rH2

2int 2( ) ( ) ( ) ( )

2e e

dvc mc

. (16.25)

The relation between these three Hamiltonian's is shown in Figure 5.0.

V rrb g

e−

e−

objH

H obj

H int

H rad

Figure 5.0 — The Total Hamiltonian for a charged particle interacting with the electromagnetic field. The Hamiltonians for the radiation field and charged particle are combined with the interaction Hamiltonian to form the Total Hamiltonian. This description of the interaction of the electromagnetic field with charged matter is based on representing the radiation field's vector potential as a gauge particle field, whose interaction is governed through the dynamics of Hamiltonian mechanics.

§16.3.2 Photon Polarization

Like the polarization of the classical electromagnetic field, the quantified field contains a polarization element. In this case the quanta of the field, the photon, possesses a spin polarization. The photon like any other particle can possess angular momentum. In traditional descriptions of spin there are two components, the spin angular momentum and the orbital angular momentum. In the case of the photon, the momentum representation of the wave function ( )A k satisfies the transversallity

condition ( ) ⋅ = 0A k k . As a result the wavefunction cannot be arbitrarily

specified for every component of the vector ( )A k at the same time. The

result is that the photon's angular momentum and the photon's spin cannot be independently distinguished. The definition of the spin as the angular momentum of a particle at rest is inappropriate for the photon since is has no rest frame. Therefore only the total angular momentum of the photon has any meaning [Land82].

The photon state is characterized by its polarization vector αε( ) . Like the polarization of the classical electromagnetic field any polarization ε

Gauge Theory and Creation of Photons

Copyright 2000, 2001 16–7

can be constructed as a superposition of two vector ( )ε 1 and ( )ε 2 , which are orthogonal with a third vector such that ( ) ( )∗ε ⋅ ε =1 2 0 , where the polarization is given by,

( ) ( )∗ε = ε ⋅ ε1 21 2e e (16.26)

where 2

1e and 2

2e are the probabilities that the photon has a polarization ( )ε 1 and ( )ε 2 respectively.

Since αε( ) transforms like a vector, the general theory of angular momentum allows the polarization vector to be associated with a unit of angular momentum. In the vocabulary of quantum mechanics, the photon has one unit of spin angular momentum. To find the spin components the circular polarization vectors are given by,

±ε = ε ± ε∓( ) (1) (2)(1/ 2)( ).i (16.27)

The transformation properties of the polarization vector can be seen by rotating the right hand polarization vector εR about an angle θ ,

′ε → ε = ε θ + ε θ′ε → ε = − ε θ + ε θ

1 1 1 2

2 2 1 2

cos sin ,

sin cos . (16.28)

Therefore,

( )

( ) − θ

− θ

′ ε → ε = − ε θ + ε θ + −ε θ + ε θ

= − ε + ε

= ε

1 2 1 2

1 2

1cos sin sin cos ,

21

,2

.

R R

i

iR

i

i e

e

(16.29)

Under an infinitesimal rotation around the propagation vector kk , by an amount δφ , the circular polarization vectors are changed by,

±

±

δε = δφ ε ε

= δφε

∓ ∓∓

( ) (2) (1)

( )

( / 2)( ),

.

i

i (16.30)

so that the circular polarization vectors ±ε( ) are associated with the spin components = ±1m , where the quantization axis has been chosen in the propagation direction kk . The photon spin is either parallel or antiparallel to the propagation direction.

Gauge Theory and the Creation of Photons

16–8 Copyright 2000, 2001

The description of the polarization state with ±ε( ) is called the circular polarization representation in contrast to the linear polarization representation. The orthogonal representations for ±ε( ) are given by,

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

± ± ∗ ±

± ∗ ± ±

±

ε ⋅ ε = − ε ⋅ ε =

ε ⋅ ε = − ε ⋅ ε =

⋅ ε =k

∓

∓

1,

0,

0.

(16.31)

The quantum mechanical expansion of the vector potential AA 's canonical variables leads to the idea that the excitations of the radiation field can be regarded as particles of zero mass and spin 1 — the photon.

The field variables are simply the Fourier sum of the individual wavefunctions, where the coefficients multiplying each of the individual wavefunctions represent the probability of the creation and destruction of a quantum of that particular wavelength at any given point in the field and is referred to as the second quantization of the field. In this model the quantum field is equivalent mathematically to an infinite collection of harmonic oscillators. [2] Since photons can act together in one energy state — since photons obey Bose-Einstein statistics — macroscopic effects can be observed from large assemblies of coherent photons. [3]

What really interests me is whether God could have made the world in a different way; that is, whether the necessity of the logical simplicity leaves any freedom at all.

— A. Einstein [Holt75]

2Although the excitations of the electromagnetic field behave in many ways like particles, carrying energy and momentum, this process has limitations whose description is beyond this monograph. A similar situation occurs in the theory of harmonic vibrations of crystalline solids [Zima69]. The coupled motions of the crystal's atoms can be isolated into independent normal modes. each of these modes can be treated as a quantized harmonic oscillator. The excitations of the normal modes carry energy and momentum and are called phonons. The term was introduced by Frenkel [Fren32] with the following remark

It is not in the least intended to convey the impression that such phonons have a real existence. On the contrary, the possibility of their introduction rather servers to discredit the belief in the real existence of photons.

3Bose-Einstein statistics describe particles which occupy the same state as described by Pauli's exclusion principle. Photons obey Bose-Einstein statistics, where electrons obey Fermi-Dirac statistics and are excluded from occupying the same state while orbiting the nucleus of an atom.

Copyright 2000, 2001 17–1

§17. VACUUM STATE FLUCTUATIONS

Before proceeding with the development of the radiation density of the electromagnetic field, the state containing no photons needs to be addressed further. In order for the creation and annihilation operators to function they must operate on the state of the radiation field, including the vacuum state.

Because to commutator rules given in Eq. (16.10) neither the individual occupation number ,i i

N αk or the total photon number operator

, , ,1,2 1,2

ˆ ˆN c c+ −α α α

α = α=

= =∑ ∑ ∑ ∑k k kk k

N commutes with the EE , BB , and AA fields. [1]

Like the fundamental commutation relation for a harmonic oscillator, [ ],q p i= h , which prevents the simultaneous vanishing of the potential

energy and the kinetic energy, in quantum mechanics noncommuting operators cannot be simultaneously determined to arbitrary accuracy's. In the situation where the number of photons is fixed than there are uncertainties in the field strength of the radiation field. This uncertainty follows the root mean square measurements developed in the previous section.

As a result the ground state of the system has an absolute energy which is non–zero and with variances in the kinetic and potential energies of 2q∆ and 2p∆ . This odd behavior can be shown by taking the electric field operator t= − ∂ ∂E A and evaluating the expectation value of the field strength. Although the expectation value,

0 0 0=E (17.1)

because,

1The commutators of the field variables EE , BB , and AA can be derived through the a technique described in [Cohe89]. Let mV and nW be two field components. These can be

expressed as linear combinations of the annihilation and creation operators ,c+αk and ,c αk

as, , , , ,1,2

ˆ ˆm m mV v c v c∗ +α α

α =

= +∑ ∑ k k k kk

and , , , ,1,2

ˆ ˆn n nW w c w c∗ +α α

α =

= +∑ ∑ k k k kk

. The commutator is

given as,

[ ] , , , , , , , , , , , , , , , ,ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, , , , ,m n m m m m m m m mV W v w c c v w c c v w c c v w c c∗ + ∗ + ∗ ∗ +α α α α α α α α = + + + ∑∑ k l k l k l k l k l k l k l k l

k l

, which reduces to, [ ] ( ), , , ,,m n m n m nV W v w v w∗= −∑ k k k kk

.

The Quantum Vacuum

17–2 Copyright 2000, 2001

,ˆ 0 0c α =k , (17.2)

the mean square fluctuation of the electric field is given by,

2

0 0 0 0 0 0⋅ − = ⋅ = ∞E E E E E . (17.3)

The variance of the electric field in the vacuum is then proportional to

h and diverges as maximum

2 32

02

k

k dk∆ =π ∫Eh

. In a state with definite photon

occupation numbers, the electric and magnetic fields are indefinite and fluctuating. The probability distributions for the EE and BB fields are analogous to the position and momentum of an oscillator in and energy eigenstate. The result of Eq. (17.3) is that if the occupation number is fixed than the field strength is completely uncertain. [2]

This divergence problem can often be circumvented by recognizing that a practical electromagnetic field is coupled to a material object over a finite bandwidth. The mean square fields associated with the thk photon state averaged over the normalization volume dv and all states up to the thk state gives,

( )2 21

, , , 2

1 1 4dv dv N

dv dv vα α α

π= = ω +∫ ∫k k kE B h . (17.4)

In the practical world the field strength is measured by an instrument of finite dimension and the strength of the field is averaged over a volume

defined by 1

Vdv

V ∆=

∆ ∫E E where V∆ is a small volume containing the

measuring device.

The mean square fluctuation can now be given as,

( )40 0 c l⋅ ≈ ∆E E h , (17.5)

where l∆ is the linear dimension of the volume.

2The infinities of quantum field theory can be dealt with renormalization of the

underlying theory, allowing the infinities to be ignored. P. A. M. Dirac [Dira58] commented on the renormalization by saying ... the rules ... do not fit in with the logical foundations of quantum mechanics. They showed therefore not be considered as a satisfactory solution to the difficulties. In the final sentence Dirac states, The difficulties, being of a profound character, can be removed only by some drastic change in the foundations of the theory, probably a change as drastic as the passage from Bohr's orbit theory to the present quantum mechanics.

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Copyright 2000, 2001 17–3

§17.1 RADIATION DENSITY OF THE QUANTIZED FIELD

All the peculiarities of the quantized radiation field must somehow be brought back to the classical description. The properties of a quantum electromagnetic field are similar to the classical electromagnetic field properties when the quantum numbers, ,N αk , defining the stationary

states of the field oscillators, are large.

In nature's infinite book of secrecy

A little I can read

— W. Shakespeare

The total field energy, E , per unit volume is then proportional to the

number of photons is state kk , such that,

2 3 4/ .N c≈ ωk E h (17.6)

With this number is large, the field energy, E , is,

2( ) ,c c t∆E hÓ (17.7)

which allows the field averaged over time intervals t∆ to be treated as classical.

The field strength for a classical electromagnetic wave of wavelength 2πλ is comparable to the quantum field operator E , such that,

2 4ˆ ,c≈ λE h (17.8)

where the average is taken over the volume 3λ . The time average of 2E can be equated to the energy density of the electromagnetic wave, so that,

2ˆ ,c

n=λ

E h (17.9)

where n is the number of photons per unit volume.

In order to observe the classical behavior of the radiation field, the quantum mechanical effects must be negligible for large numbers of photons, such that,

3

1,n

λÓ (17.10)

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17–4 Copyright 2000, 2001

resulting in the description of physical phenomena based on classical electrodynamics when the number of photons per unit volume is much greater than one.

For a radiation field whose source is generating a 100 MHz ( 48cmλ ≈ ), with a power of 140,000 watts, the number of photons per unit volume at a distance of 5.0 miles from the transmitter is on the order of

171.0 10× . In order to calculate the number of photons at a distance from the radiation source, the ratio the volume containing the photons to the overall volume of the radiation field will be used to adjust the total energy density. FigureFigure 5.05.0 describes the volume element of interest.

r rλ

I e t0

−ω

Figure 5.0 — Volume element of the radiation field density containing a calculated number of photons radiating from the radio antenna distance r away. In order to maintain the correspondence principle, the number of photons in this unit volume must be very large compared to the quantum mechanical description of the electromagnetic field.

The volume of a sphere [Owen61], in spherical coordinates is given by,

2

2 3

0 0 0

4sin

3

R

V r dr d d Rπ π

= θ θ φ = π∫ ∫ ∫ . (17.11)

For a unit sphere of radius 1R = , the volume is 32.819 cmV ≅ . For a sphere of radius 5 miles (160,934 cm) , the volume is 16 31.7416 10 cmV ≅ × . The ratio of the total volume of the radiation sphere to the small volume area of interest is,

5 3

1216 3

1.106 106.35 10

1.7416 10cm

ratiocm

−×= = ×

×. (17.12)

This ratio will be used to adjust the total number of photons to the number contained in the volume of interest.

The Quantum Vacuum

Copyright 2000, 2001 17–5

Solving for the total number of photons n radiated by the transmitter gives,

( ) ( ) ( )( )( )

( )( )

5 72

-25 10

13

28

-15

1 10 W 1 10 erg/sec 48cm

1.05 10 erg 1 10 cm/sec

4.8 10 erg cm/sec4.57 10

1.05 10 erg cm/sec

En

c

× ×λ= =

× ×

×= = ×

×

h (17.13)

Adjusting the total number of photons in the spherical volume for the number of photons in the volume of interest gives,

28 12 174.57 10 6.35 10 1.0 10n −= × × × ≅ × . (17.14)

§17.2 RADIATION DAMPING AND SELF FIELDS

Now that the quantum field description of the electromagnetic field has be shown to be extendible to the classical radiation field, the effect on the electrons in the receiving antenna will be examined from the point of view of quantum field theory.

In the description of the classical equations of motion of electrons in a conducting material the problem of electrodynamics could be divided into two categories. The first in which the external electromagnetic field is applied to charges — the receiving antenna — and their resulting motions calculated and the second where the motion of charges resulted in an electromagnetic field — the transmitting antenna. The description produced in both situations only approximate the actual behavior of nature.

In the case where an external field is incident on a charge, the motion of the charges involve the emission of radiation. This radiation carries off energy, momentum and angular momentum and therefore influences the subsequent motion of the charged particles. In the macro–world these effects cause negligible errors in the calculations of measurable quantities. Although these solutions are workable the basic problem remains unsolved for the micro–world description of the electrons motion.

If an external field causes a particle of charge e to have an acceleration a for a period t then the radiated energy is approximately,

2 2

rad 2

23e a t

Ec

∼ , (17.15)

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17–6 Copyright 2000, 2001

which is Larmor's formula. [3] If this radiated energy is very small compared to the incident energy 0E then the radiate effects will be unimportant.

If however the radiated energy is approximately equal to the incident energy, that is rad 0E EÁ , the effects of the radiative reaction on the motion of the charged particle will be appreciable. This relationship between the incident energy 0E and the radiated energy can be defined in one of two ways. If a charged particle is initially at rest and a force is applied to it for a limited time t, than the particle will be accelerated continuously for that time period. The relevant energy can then be given as,

( )2

0E m at∼ . (17.16)

The determination of whether the radiative effects are important becomes,

2 2

22

23e a t

ma tc

? , (17.17)

or,

2

2

23

et

mcτ ≡ @ , (17.18)

which is called the characteristic time of the interaction. The t is long compared to τ then the radiation corrections are unimportant. For electrons the characteristic time 246.26 10 sec−τ = × . This can be restated as a distance, since light will travel 1310 cm− in this time. The radiative corrections are unimportant if the interaction is less than this distance.

3Larmor's formula describes the total radiated power produced by an accelerated

charge [JACK75]. This expression can be derived starting from Poynting's vector, 2

4S c= × = π ⋅E B E n . The power radiated per unit of solid angle is given as,

( ) ( ) 22 4dP d e cΩ = π × × βn n & where ( )v t cβ = is the distance traveled by the charged

particle. If Θ is the angle between the acceleration vector v& and nn then the radiated

power is, ( ) 22 24 sindP d e c vΩ = π Θ& . The total instantaneous power radiated by the

accelerating particle can be found by integrating over all solid angles to give,

( ) 22 32 3P e c v= & .

The Quantum Vacuum

Copyright 2000, 2001 17–7

For a charged particle undergoing a periodic motion at a characteristic frequency of 0ω the mechanical energy associated with the particles motion can be identified with 0E and is given by,

2 20 0E m dω∼ , (17.19)

where the accelerations are typically 20a dω∼ and the time intervals are

01t ω∼ . The determination of the radiative effects are now given as,

2 2 2

2002

0

23e d

m dcω

ωω

? , (17.20)

or

0 1ω τ? . (17.21)

where τ is again the characteristic time. Since 01 ω is a time associated with the mechanical motion, if this inverse frequency or time is long compared the characteristic time than the radiative effects or unimportant.

For classical electrodynamics times of τ or distances of cτ are sufficiently small so that the radiative effects can be ignored.

The Quantum Vacuum

17–8 Copyright 2000, 2001

§17.3 OPEN QUESTIONS ABOUT THE QFT

In the previous sections the number of photons in a volume was given by a simple expression which was derived from the average energy density of the quantized radiation field. This number was introduced after a lengthy background development which included many diversions. The result of the arduous journey is a surprisingly simple expression,

2ˆ cn=

λE h . As usually is the case in a subject like this, the final result is a

bit disappointing.

The path to this conclusion started with Maxwell's classical fields equations, which originally depended on a propagation media — the ether. These fields then became a material object themselves through the development of action-by-contact. Quantizing the field produced a coupled oscillator description of the radiation. Through the mathematics of the oscillator, particles carrying the force were introduced — making the transition to quantum field theory and the beginnings of quantum electrodynamics.

All that really happened — in the end — is that the quantum mechanical description of the electromagnetic radiation field has been verified to produce the proper result when compared to the classical field equation description — a restatement of the obvious.

But there is more than meets the eye here:

n The transition to a particle based description of the radiation field has been made, laying the groundwork for Quantum Electrodynamics.

n The details of the potential field Hamiltonian and gauge invariance have been explored.

n The background has been developed to ask the original question in light of a new understanding.

WHAT CAUSES THE ELECTRONS IN THE RECEIVING ANTENNA TO MOVE?

The answer to this question seems no more plausible than the classical answer — a force causes the electrons to move/ The actual cause is just as mysterious as before:

n Are there photons flying across empty space and colliding with the electrons in the antenna?

n Are there actual photons be emitted by the accelerated electrons in the transmitting antenna?

The Quantum Vacuum

Copyright 2000, 2001 17–9

The quantum explanation uses the photon as the gauge particle because the mathematics allows it to do so — not because they are physically there.

The Quantum Vacuum

17–10 Copyright 2000, 2001

And in the end, when man has fully partaken of the fruit of the tree of knowledge, there will be this difference between the first Eden and the last, that man will not become as a god, but remain forever humble.

— P. W. Bridgeman [Brid29]

Bibliography–1

§18. BIBLIOGRAPHY

There are two basic classes of material in this bibliography, (i) text books on specific subjects which are focused on material that is used in the description and analysis of electromagnetic phenomenon, journal articles that support the text book materials; (ii) text books and journal articles on the historiography of physics, which are used to provide the historical background so often missing from the texts and course instruction of modern physics education. [† ]

None of the material presented in this monograph is original in the sense that it did not exist prior to this work. The following sources have been used to produce a compendium of information. Direct references to material in the monograph are noted in the text. In those cases, some or all the material given is derived from the reference source referenced in brackets. In other portions of the monograph, source material is used in an indirect manner. In those case no direct reference is made, but the source is listed in this bibliography.

During the reading of the materials found in the bibliography, it was clear there were distinguished works in the field while other materials were expansions of these works. I have marked in italics, [Arfk85], the material I feel is the foundation of later works. Many modern electrodynamics texts restate material found in earlier works. Because of the wealth of sources in the field, it is felt that older texts are closer to the source and therefore more faithful to the original concepts. In many instances, this may not be true, since the subject matter was still undergoing discovery. There are earlier texts however that provided clearer discussions of the material than that found in recent publications. The reader is encouraged to delve further into the subject using this bibliography as a starting point – remembering it is only the tip of the iceberg.

Since the subject matter presented in this monograph would not be considered state of the art, most of the references are texts rather than journal articles. This should not diminish the readers enthusiasm for the material since there are still many unanswered questions given as problems in the better texts.

† The primary reference source for texts in the mathematics and physical sciences is

Guide to the Literature of Math and Physics, Including Related Works on Engineering, Nathaniel G. Parke III, McGraw Hill, 1947, reprinted Dover, 1954. This book is currently out of print, but contains a wealth of reference material, worth searching the used books stores for.

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[Aber73] “Gauge Theories”, E. S. Abers and B. W. Lee, Physics Reports, 9 C9 C (1), North Holland, 1973.

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[Abri63] Methods of Quantum Field Theory in Statistical Physics, A. A. Abriosov, L. P. Gorkov and I. E. Dzyaloshinski, Prentice Hall, 1963; Reprinted, Dover, 1975.

[Adai87] The Great Design, R. K. Adair, Oxford University Press, 1987.

[Adle60] Foundations of Science and Mathematics, M. J. Alder and P. Wolff, Encyclopedia Britannica, 1960.

[Aepi5] De Integratione, et Separatione Variabilium, in Aequationibus Differentialibus, Duas Variabiles continentibus, Commentatio. F. U. T. Aepinus, Rostock: J. J. Adler, 1755.

[Aepi59] Testamen theoriae electricitatus et magnetismi (Aepinus’s essay on the Theory of Electricity and Magnetism), F. U. T. Aepinus, St. Petersburg, 1759, translated by P. J. Conner, Princeton University Press, 1979.

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[Agas71] Faraday as a Natural Philosopher, J. Agassi, University of Chicago Press, 1971.

[Ahar59] “Significance of Electromagnetic Potentials in the Quantum Theory,” Y. Aharonov and D. Bohm, Physics Review, 115115 , pp. 485–491.

[Ahar75] The Special Theory of Relativity, J. Aharoni, Oxford University Press, 1975.

[Aitc82] An Informal Introduction to Gauge Theories, I. J. R. Aitchison, Cambridge University Press, 1982.

[Aitc82a] Gauge Theories in Particle Physics, I. J. R. Aitchison and A. J. G. Hey, Hilges, 1982.

[Akiv77] An Introduction to Linear Algebra and Tensors, M. A. Alivis and V. V. Goldberg, Prentice–Hall, 1972; Reprinted, Dover, 1977.

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[Ande67] Principals of Relativity Physics, J. L. Anderson, Academic Press, 1967.

[Ande71] Modern Physics and Quantum Mechanics, E. E. Anderson, W. B. Saunders, 1971.

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[Aren65] “Algebraic Difficulties of Preserving Dynamical Relations when Forming Quantum–Mechanical Operators,” R. Arens and D. Babbitt, in Journal of Mathematical Physics, Volume 6, pp. 1071–5, 1965.

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[Armi57] Copernicus, the Founder of Modern Astronomy, A. Armitage, Thomas Yoseloff, 1957.

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[Bell70] Introduction to Matrix Analysis, R. Bellman, McGraw–Hill, 1970.

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[Born71] The Born–Einstein Letters, translated by Irene Born, Walker, 1971.

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[Bowl90] Femtophysics: A Short Course on Particle Physics, M. G. Bowler, Pergamon Press, 1990.

[Boye71] “Energy and Momentum in Electromagnetic Field for Charged Particles Moving with Constant Velocities,” T. H. Boyer, American Journal of Physics, 3939 (3), pp. 257–270, 1971

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[Brac95] The Key to Newton’s Dynamics: The Kepler problem and the Principia, J. B. Brackenridge, University of California Press, 1995.

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[Brow92] Quantum Field Theory, L. S. Brown, Cambridge University Press, 1992.

[Byro69] Mathematics of Classical and Quantum Mechanics, (Volume 1) F. W. Byron and R. W Fuller, Addison–Wesley, 1969; Reprinted, Dover 1992.

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[Buch85] From Maxwell to Microphysics, Aspects of Electromagnetic Theory in the Last Quarter of the Nineteenth Century, J. Z. Buchwald, University of Chicago Press, 1985.

[Buch88] “The Michelson Experiments in the Light of Electromagnetic Theory Before 1900,” J. Z. Buchwald, in The Michelson Era of American Science: 1870–1930, American Institute of Physics, 1988.

[Buch94] The Creation of Scientific Effects: Heinrich Hertz and Electric Waves, J. Z. Buchwald, University of Chicago Press, 1994.

[Bury29] Timaeus, Plato, translated by R. G. Bury, Harvard University Press, 1929.

[Butk68] Mathematical Physics, E. Butkov, Addison–Wesley, 1968.

[Cabr83] “Magnetic Monopoles: Evidence Since the Dirac Conjecture,” B. Cabrera and W. P. Trower, Foundation of Physics, 1313 , pp. 196–216, 1983.

[Cant81] Conceptions of Ether: Studies in the History of Ether Theories, edited by, G. N. Cantor and M. J. S. Hodge, Cambridge University Press, 1981.

[Cahn89] The Experimental Foundations of Particle Physics, R. N. Cahn and G. Goldhaber, Cambridge University Press, 1989.

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And further, by these, my son be admonished; of making many books there is no end...

— Ecclesiastes