Relativistic Quantum Mechanics and Field Theory...Relativistic Quantum Mechanics and Field Theory are among the most challenging and beautiful subjects in Physics. From their study

Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS College of William and Mary

Williamsburg, Virginia and

Continuous Electron Beam Accelerator Facility Newport News, Virginia

Wiley-VCH Verlag GmbH & Co. KGaA

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FRANZ GROSS College of William and Mary

Williamsburg, Virginia and

Continuous Electron Beam Accelerator Facility Newport News, Virginia

Wiley-VCH Verlag GmbH & Co. KGaA

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

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Printed in the Federal Republic ofGermany Printed on acid-free paper

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ISBN-13: 978-0-47 1-35386-7 ISBN-10: 0-471 -35386-8

To my parents, Genevieve and Llewellyn

and to the next generation, Glen, Sue, Kathy, Caitlin, and Christina


CONTENTS

Preface xiii

Part I QUANTUM THEORY OF RADIATION

1. Quantization of the Nonrelativistic String

1.1 The one-dimensional classical string

1.2 Normal modes of the string

1.3 Quantization of the string

1.4 Canonical commutation relations

1.5 The number operator and phonon states 1.6 The quanta as particles

1.7 The classical limit: Field-particle duality

1.8 Time translation

Problems

2. Quantization of the Electromagnetic Field

2.1 Lorentz transformations

2.2

2.3 2.4 Plane wave expansions

2.5 Massive vector fields 2.6 Field quantization

2.7 Spin of the photon

Relativistic form of Maxwell’s theory

Interactions between particles and fields

Problems

3. Interaction of Radiation with Matter

3.1

3.2

Time evolution and the S-matrix Decay rates and cross sections

1

3 3

7

10

12 13 15

18

22 25

28 28 31

39 42

44 46 50 56

57 57 65

vii

viii CONTENTS

3.3 Atomic decay 3.4 The Lamb shift

3.5 Deuteron photodisintegration Problems

Part II RELATlVlSTlC EQUATIONS

4. The Klein-Gordon Equation 4.1

4.2

4.3 4.4 4.5

4.6

4.7

4.8

The equation Conserved norm

Solutions for free particles Pair creation from a high Coulomb barrier

Two-component form Nonrelativistic limit

Coulomb scattering

Negative energy states

Problems

5. The Dirac Equation 5.1 The equation

5.2 Conserved norm 5.3 Solutions for free particles

5.4 Charge conjugation

5.5 Coulomb scattering

5.6 Negative energy states

5.7 Nonrelativistic limit

5.8 The Lorentz group 5.9 5.10 Bilinear covariants 5.11 Chirality and massless fermions

Covariance of the Dirac equation

Problems

6. Application of the Dirac Equation

6.1 Spherically symmetric potentials 6.2 Hadronic structure 6.3 Hydrogen-like atoms

Problems

69

73

81 85

89

91 91

94

95 97

102

106

108

110

116

119 119

122

123

126

129

131

134

140 146

152 157

169

163 163

171 177

183

CONTENTS iX

Part 111 ELEMENTS OF QUANTUM FIELD THEORY 185

7. Second Quantization

7.1 Schrodinger theory 7.2 Identical particles 7.3 Charged Klein-Gordon theory

7.4 Dirac theory 7.5 Interactions: An introduction

Problems

8. Symmetries I 8.1 Noether’s theorem

8.2 Translations

8.3

8.4 Parity 8.5 Charge conjugation

8.6 Time reversal 8.7 The PCT theorem

Transformations of states and operators

Problems

9. Interacting Field Theories

9.1 43 theory: An example

9.2 Relativistic decays

9.3 Relativistic scattering

9.4 9.5

9.6 Effective nonrelativistic potential 9.7 Identical particles

9.8 Pion-nucleon interactions and isospin 9.9 One-pion exchange 9.10 Electroweak decays

Introduction to the Feynman rules Calculation of the cross section

Problems

10. Quantum Electrodynamics 10.1 The Hamiltonian 10.2 Photon propagator: ep scattering 10.3 Antiparticles: e+e- -+ p+p-

187 188 191

194

198

20 1

204

206 206

212

2 14

216

219 223

23 1

236

238 238

240

243

247 252

257 25 8

264

268 273

279

282 282

286 296

x CONTENTS

10.4 e+e- annihilation 10.5 Fermion propagator: Compton scattering

Problems

11. Loops and Introduction to Renormalization

1 1.1 Wick’s theorem

11.2 QED to second order

11.3 Electron self-energy

11.4 Vacuum bubbles 11.5 Vacuum polarization

11.6

1 1.7 Dispersion relations

1 1.8 Vertex corrections

11.9 Charge renormalization

1 1.10 Bremsstrahlung and radiative corrections

Loop integrals and dimensional regularization

Problems

12. Bound States and Unitarity 12.1 The ladder diagrams

12.2 The role of crossed ladders

12.3 Relativistic two-body equations

12.4. Normalization of bound states

12.5 The Bethe-Salpeter equation

12.6 The spectator equation

12.7 Equivalence of two-body equations 12.8 Unitarity

12.9 The Blankenbecler-Sugar equation 12.10 Dispersion relations and anomalous thresholds

Problems

Part IV

13. Symmetries II

SYMMETRIES AND GAUGE THEORIES

13.1 Abelian gauge invariance

13.2 Non-Abelian gauge invariance 13.3 Yang-Mills theories 13.4 Chiral symmetry

13.5 The linear sigma model

304

307

316

319

320

326

330

336

338

342

349

353

358

363

37 1

373 374

382

388

39 1

393

395

399

400

404

405

41 1

413

415 416

419

424 421

430

CONTENTS xi

13.6 Spontaneous symmetry breaking

13.7 The non-linear sigma model

13.8 Chiral symmetry breaking and PCAC

Problems

14. Path Integrals 14.1

14.2 The S-Matrix

14.3 Time-ordered products 14.4 Path integrals for scalar field theories

14.5 Loop diagrams in 43 theory

14.6 Fermions

Problems

The wave function and the propagator

15. Quantum Chromodynamics and the Standard .-.Jdel 15.1 Quantization of gauge theories

15.2 Ghosts and the Feynman rules for QCD

15.3 Ghosts and unitarity

15.4 The standard electroweak model

15.5 Unitarity in the Standard Model

Problems

16. Renormalization 16.1 Power counting and regularization

16.2 d3 theory: An example

16.3 Proving renormalizability

16.4 The renormalization of QED

16.5 Fourth order vacuum polarization

16.6 The renormalization of QCD

Problems

17. The Renormalization Group and Asymptotic Freedom 17.1 The renormalization group equations

17.2 Scattering at large momenta 17.3 Behavior of the running coupling constant

17.4 Demonstration that QCD is asymptotically free

17.5 QCD corrections to the ratio R Problem

434

437

441

446

448 449

454

459

463

475

482

490

492 492

50 1

504 5 10

523

526

527 527

533

547

554

557

569

57 1

573 573

575

578

582

589

592

xii CONTENTS

Appendix A Relativistic Notation

A.1 Vectors and tensors

A.2 Dirac matrices A.3 Dirac spinors

Appendix B Feynman Rules

B.1 B.2 General rules B.3 Special rules

Decay rates and cross sections

Appendix C Evaluation of Loop Diagrams

Appendix D Quarks, Leptons, and All That

D.l Fundamental particles and forces D.2 Computation of color factors

593 593

594

596

598 598

600

602

606

609 609 61 1

References 615

Index 619

PREFACE

Relativistic Quantum Mechanics and Field Theory are among the most challenging and beautiful subjects in Physics. From their study we explain how states decay, can predict the existence of antimatter, learn about the origin of forces, and make the connection between spin and statistics. All of these are great developments which all physicists should know but it is a real challenge to learn them for the first time.

This book grew out of my struggle to understand these topics and to teach them to second year graduate students. It began with notes I prepared for my personal use and later shared with my students. About two years ago I decided to have these notes typed in w, little realizing that by so doing I had committed myself to eventually producing this book. My objectives in preparing this text 'Alect the original reasons I prepared my own notes: to write a book which (i) can be understood by students learning the subject for the first time, (ii) carries the development far enough so that a student is prepared to begin research, and (iii) gives meaning to the study through examples drawn from the fields of atomic, nuclear, and particle physics. In short, the goal was to produce a book which begins at the beginning, goes to the end, and is easy to read along the way.

The first two parts of this book (Part I: Quantum Theory of Radiation, and Part 11: Relativistic Equations) assume no previous experience with advanced quantum mechanics. The subjects included here are quantization of the electromagnetic field, relativistic one-body wave equations, and the theoretical explanation for atomic decay, all fundamental subjects which can be regarded as necessary to a well rounded education in physics (even for classical physicists). The presentation is modeled after the first third of a year-long course which I have taught at various times over the past 15 years and these topics are given in the beginning so that those students who must leave the course at the end of the first semester will have some knowledge of these important areas.

To prepare a student for advanced work, the last two parts of this book include an introduction to many of the unique insights which relativistic field theory has contributed to modem physics, including gauge symmetry, functional methods (path integrals), spontaneous symmetry breaking, and an introduction to QCD.

xiii

xiv PREFACE

chiral symmetry, and the Standard Model. Part I11 also contains a chapter (Chap- ter 12) on relativistic bound state wave equations, an important topic frequently overlooked in studies at this level. I have tried to present even these more advanced topics from an elementary point of view and to discuss the subjects in sufficient detail so that the questions asked by beginning students are addressed. The entire book includes a little more material than can comfortably fit into a year long course, so that some selection must be made when used as a text.

To make the book easier to read, most proofs and demonstrations are worked out completely, with no important steps missing. Some topics, such as the quantization of fields, symmetries, and the study of the Lorentz group, are introduced briefly first, and returned to later as the reader gains more experience, and when a greater understanding is needed. This “spiral” structure (as it is sometimes referred to by the educators) is good for beginning students but may be frustrating for more advanced students who might prefer to find all the discussion of one topic in one place. I hope such readers will be satisfied by the table of contents and the index (which I have tried to make fairly complete). Considerable empha- sis is placed on applications and some effort is made to show the reader how to carry out practical calculations. Problems can be found at the end of each chapter and four appendices include important material in a convenient place for ready reference.

There are many good texts on this subject and some are listed in the Reference section. Most of these books are either classics, written before the advent of modern gauge theories, or new books which treat gauge theories but omit some of the detail and elementary material found in older books. I believe that most of this elementary material is still very helpful (maybe even necessary) for students, And have tried to cover both modern gauge theories and these elementary topics in a single book. As a result the book is somewhat longer than many, and omits some advanced topics I would very much like to have included. Among these omissions is a discussion of anomalies in field theories.

Many people have helped me in this effort. I am grateful to Michael Frank, Joe Milana, and Michael Musolf for important suggestions and help with indi- vidual chapters. I also thank my colleagues Carl Carlson, Nathan Isgur, Anatoly Radyushkin, and Marc Sher. S. Bethke and C. Wohl kindly gave permission to use figures 17.4 and 10.9 (respectively). Many students suffered through earlier drafts, found numerous mistakes, and made many helpful suggestions. Among these are: S. Ananyan, A. Colman, K. Doty, D. Gaetano, C. Hoff, R. Kahler, Z. Li, R. Martin, D. Meekins, C. Nichols, J. Oh, X. Ou, , M. Sasinowski, P. Spickler, Y. Surya, X. Tang, A. B. Wakley, and C. Wang. Roger Gilson did an excellent job transforming my original notes into T B . And no effort like this would be possible or meaningful without the support of my family. I am especially grateful to my wife, Chris, who assumed many of my responsibilities so I could complete the work on this book in a timely fashion. I could not have done it without her.

FRANZ GROSS

PART I

QUANTUM THEORY OF RADIATION


CHAPTER 1

QU ANTI ZATI 0 N OF THE NONRELATIVISTIC STRING

This book will discuss how nonrelativistic quantum mechanics can be extended to describe:

0 relativistic systems and

0 systems in which particles can be created and annihilated.

The key td both of these extensions is field theory, and we therefore begin with an introduction to this topic. In this chapter we will discuss the quantization of the nonrelativistic, one-dimensional string. This is a many-body system which is also simple and familiar. Quantization of this many-body system leads directly to the (new) concept of a quantum field, and many of the properties of quantum fields can be introduced and illustrated using the nonrelativistic one-dimensional string ?s an example. The goal of this chapter is to use this simple system to develop dn intuition and understanding of the meaning and properties of quantized fields. In subsequent chapters some of these ideas will be developed again in a more general, abstract way, and it is hoped that the intuition gained in this chapter will remove much of the mystery which might otherwise surround those more abstract discussions.

The discussion of relativistic systems begins in the next chapter, where the ideas developed here are immediately extended to the electromagnetic field.

1.1 THE ONE-DIMENSIONAL CLASSICAL STRING

We will approach the treatment of a continuous string by first considering a system of point masses connected together by “springs” and then letting the number of point masses go to infinity, and the distance between them go to zero, in such a way that a continuous system with a uniform density and tension emerges.

Start, then, with a “lumpy” string of overall length L made up of N points, each with mass m, coupled together by springs with a spring constant k. Assume that the oscillators move about their equilibrium positions in a periodic pattern, which is best realized by thinking of the string as closed on itself in a circle, as

3

4 QUANTIZATION OF THE NONRELATIVISTIC STRING

shown in Fig. 1.1. The oscillators are constrained to vibrate along the circum- ference of the ring (which has a radius very much greater than the equilibrium separation t? so that the system will be treated as a linear system with periodic boundary conditions). The 0th and Nth oscillators are identical, so that if is the displacement of the ith oscillator from equilibrium, then

&j0 d 4 N 1 periodic boundary conditions.

d t d t -= -

The kinetic energy (KE) and potential energy (PE) are

N-1 2

K E = i m x ( % )

PE = Z k C (&+I - &)' .

i = O 2

N-1

t = O

Now, take the continuum limit by letting l' -+ 0, N -+ 00, such that the length L = N I , mass per unit length p = m/L, and string tension T = ke are fixed. Then the displacement and energy of the string can be defined in terms of a continuous field &(zl t ) , where

&(t) = &zi, t ) --t 6(.tl t )

The Lagrangian and Hamiltonian are

L = KE - PE = l L d z { f p (2)' - iT (z)'] = l L d z C ( z l t )

H = K E + P E = lL d z ( i p (g)2 + iT (g)'] = lL d z 7 i ( z l t )

where C and H are the Lagrangian and Hamiltonian densities. In this example, the field function d ( z , t ) is the displacement of an infinitesimal mass from its equilibrium position at z. In three dimensions, 6 would be a vector field.

1.1 THE ONE-DIMENSIONAL CLASSICAL STRING 5

Fig. 1.1 enlarged. The equilibrium position of the oscillators are the solid lines separated a distance e.

Drawing of the circular string with the location of the oscillators in the interval [ i , i + I]

Anticipating later applications, we redefine 4 by absorbing fi:

&J?;4=q5. Then introducing the wave velocity

gives

(1.2)

The equations of motion for the string can be derived from the Lagrangian using the principle of least action [for a review, see Goldstein (1977)J. This principle states that the “path” followed by a classical system is the one along which its action A is an extremum. For the “lumpy” string, made up of discrete coupled oscillators, this condition is


where& = d & / d t . Working out the variation gives the Euler-Lagrange equations for the motion of each oscillator:

= o ( i = O t o N - 1 ) . d aL aL d t a& a&

However, in the continuum limit as the number of oscillators N + 00,

1 (&+I - $2) - ($2 - &-I)

e e = ke [ I3 ac

(1.3) where z+ = z + i.k is the midpoint of the interval z2+1 - zi, and z- = z - the midpoint of the interval zi - zi-1. These are appropriate arguments for the two derivatives which arise in the next to the last step of Eq. (1.3). Also:

d a L - a2&, t ) dt a;2 = m4i = - -

a t 2

Hence the Euler-Lagrange equations can be expressed directly in terms of C, the Lagrangian density

a ac

where the eJ?' factor can be discarded. More generally, C can also be a function of 4 as well as a d / & and adlaz,

and for a scalar field in three dimensions, where T , = (z, y, z ) are the three spatial components, we obtain directly

where summation over repeated indices is implied. Assuming that 6 (V24) = V, (64) and integrating by parts (assuming boundary terms are zero because the

1.2 NORMAL MODES OF THE STRING 7

boundary conditions are periodic and that the variations 64 at the initial and final times are zero) give

Using the notation

x p = ( t , 2, Y, 2) P = = o , 1 , 2 , 3 ,

which can be readily generalized to relativistic systems (it will later be the con- travariant four-vector), gives the famous Euler-Lagrange equations for a continuous field

where summation over repeated indices is assumed. For the one-dimensional string treated in this chapter, the Euler-Lagrange equations give the familiar wave equation

Nhere the wave velocity was defined in Eq. (1.1). In summary, we have shown in this section how a quantity referred to as a

continuous field emerges as the natural way to describe a system with infinitely many particles. In this example, the field is 4 ( z , t ) , and it gives the displacement of each particle from its equilibrium position at z. Since we absorbed f i into the field, its units (for a one-dimensional system) are L d w . In the narural system of units, where ti = c = 1, it is dimensionless (for a discussion of the natural system of units, see Sec. 1.3 and Prob. 1.1). In three dimensions, the dimensions of such a field are L-’, which can be deduced directly from the observation that J d 3 r L: has units of energy. Regardless of its dimensions, it is useful to remember that a field is always the “displacement” (in a generalized sense) of some dynamical system, and that therefore ab/at is a generalized velocity.

1.2 NORMAL MODES OF T H E STRING

As a preparation to quantizing the string, we find its normal modes. The solutions of the wave equation which satisfy the periodic boundary conditions are


where periodicity requires

and the wave equation gives w, = V k, .

Note that there are both positive and negative frequency solutions. We will adopt the convention that w, is always positive, and use -w, for negative frequency solutions. The states with positive frequency are written

2 2 2

The negative frequency states have a time factor e i w n t , and since k, is both positive and negative, it is convenient to denote the negative frequency states by qj;(z, t). The normalization condition which these states satisfy is

However, by direct evaluation it is also true that

#.L

co the states are not orthogonal in the usual sense. The most general real field can be expanded in normal modes as follows:

00

4(zr t> = C cn {an(o)4n(zr t ) + a; (0)4; (~1 t ) } n=-m

where a,(O) are the coefficients of each normal mode in the expansion (1.10) and the real normalization factor c, will be chosen later. It will sometimes be convenient to incorporate the time dependence of each normal mode into a generalized an(t>v

a,(t) = a n ( ~ ) e - a w n t (1.11)

as was done in the second line of (1.10). The condition that 4(z1 t) is real means that the coefficient of 4; must be the complex conjugate of the coefficient of 4,.

Equation (1.11) shows that each normal mode behaves as an independent simple harmonic oscillator (SHO) satisfying the equation

iin(t) + w i a n ( t ) = o

1.2 NORMAL MODES OF THE STRING 9

To quantize the field, it is only necessary to quantize these oscillators. Before doing this, however, we evaluate the energy in terms of the dynamical

variables an (t). Using the “orthogonality” relations (1.8) and (1.9), which can be written

dta:,(O)4:,(z,t)arn(O)4rn(z,t) = bn,m Ian(o>l2 = bn,m lan(t>12

1” dz an(0)4n(zj t)arn(O)4rn(z, t ) = b-n , rnan(o )a -n(o ) e-2iwnt

= b-n , rnan( t )a -n ( t ) 3

we obtain

Using iLn(t) = - i w n a n ( t ) gives

In natural units where h = 1. E = tw = w, and the frequency w has units of energy. It is convenient to choose cn to make an dimensionless. If we choose

1/2

c n = ( & ) 7

the Hamiltonian assumes a simple form

m

n=--m


An alternative choice of coordinates will enable us to quantize these oscillators. For this we need generalized positions and momenta, which must be real. Choose

1 qn( t ) = - [an( t ) + 4 ( t ) ] Jzw,

The a’s can then be expressed in terms of the real p’s and q’s

and the Hamiltonian becomes

(1.12)

which is a sum of independent oscillator Hamiltonians. This is confirmed by substituting (1.12) into Hamilton’s equations of motion

d H q - - = p n a p ,

n -

wnich gives back the familiar equations of motion for uncoupled oscillators.

1.3 QUANTIZATION OF T H E STRING

We now quantize the string by the canonical procedure: the canonical variables are made into operators which are defined by transforming the Poisson bracket relations into commutation relations [for a review of this procedure see, for example, Schiff (1968), Sec. 241. For the generalized coordinates and momenta this leads to the following commutation relations:

(1.13)

In what follows we will always set h = c = 1. This defines the so-called natural system of units, which is very convenient. It is important to realize that the correct factors of h and c can always be uniquely restored at the end of any calculation, if desired. These units are discussed in Prob. 1.1 at the end of this chapter.

1.3 QUANTIZATION

From the commutation relations (1.13) we obtain

OF THE STRING 11

(1.14)

where the complex conjugate of a complex number (sometimes called a c-number) must be generalized to the Hermitian conjugate of an operator (sometimes called a q-number), and the operators an are independent of time. The time dependence is in the fields, which are also operators*:

= +(+)(z, t ) + t ) , (1.15)

where the positive frequency part, $(+I, contains the sum over an (later to be identified as annihilation operators) and the negative frequency part, $(-), is the sum over a; (the creation operators). In this case 4 is Hermitian because it is associated with a physical observable (the displacement), but in general a field need not be a Hermitian operator. We will study such fields in Part III of this book.

The Hamiltonian is also an operator, and its precise form depends on the order of at and a, which was unimportant when these were c-numbers. Perhaps the most "natural" form for H is

I

n = - w

However, the sum over $Wn gives an infinite contribution to the energy (the zero-point energy), which can be removed simply by redefining the energy. This redefinition will lead to the idea of a n o m l ordered product, which will be defined and discussed in Sec. 1.6 below. For now we will simply adopt the following form for H:

w

H = C wnaJ,an (1.16) n = - w

Note that H is the sum of the dimensionless operators aAan, each multiplied by the energy wn of the normal mode which it describes.

'To avoid singularities, we will exclude the state n = 0 from this sum. Later, when we take the limit L 4 00 (the continuum limit), the sum will include states of arbitrarily small energy.


1.4 CANONICAL COMMUTATION RELATIONS

The commutation relations between a and at also imply relations between the fields 4. Suppose we regard 4 as a canonical coordinate. Then, the canonical momentum is [using L defined in Eq. (1.2)]

Then, generalizing the commutation relations (1.13) to a continuous field, we expect to find relations of the form

(1.17)

where the 6(z - z’) function is the generalization of the Kronecker brim which ap- pears in (1.13). These important commutation relations are known as the canonical commutation relations, sometimes referred to as the CCR’s.

To prove the relations (1.17), we use the explicit form for K:

Then

However, the functions h e i k n z are complete (i.e., any periodic function can be expanded in terms of them) and orthonormal, and hence

1.5 THE NUMBER OPERATOR AND PHONON STATES 13

which is the property of the &function, and hence

(1.18)

This proves the first of the relations (1.17). To prove the others, note that

[ l - 1 ] = 0 a

A field theory may be quantized with either the CCR’s (1.17) or the commutation relations (1.14) between the operators a and at. As we have seen, these two methods are equivalent. Should either be regarded as more fundamental than the other? Many prefer to start from the CCR’s because of their close connection with the fundamental relations (1.13), but in this book the relations (1.14) between the a’s will be chosen as the starting point for quantizing new field theories. The reason for this choice is that the relations (1.14) are directly related to the oscillators which describe the independent dynamical degrees of freedom associated with the field, and therefore always have the same form, while the fields them- selves sometimes include degrees of freedom which are not independent (such as the vector degrees of freedom of the electromagnetic field) and in these cases the form of the CCR’s must be modified so that these dependent degrees of freedom are removed from the commutation relations. This will be apparent in the next chapter where the quantization of the electromagnetic field is discussed.

1.5 THE NUMBER OPERATOR AND PHONON STATES

Next, we find the eigenstates of the Hamiltonian (1.16). The first step is to find the eigenstates of the operator

Nn = aLan

known as the nwnber operator. These are easy to find from the commutation relation for the a’s.

Since N = ata is Hermitian (from now on we suppress n), it has a complete set of orthonormal eigenstates. Denote these by Im). Then

Nlm) = mlm)

(m’lm) = 6m’,rn


At this point we know only that m is real. Now consider the state atlm). From the commutation relations (1.14) we

have

Hence

Nat lm) = { [N, at] + a”} Im)

= (at +at,) Im) = (m + 1)atIm)

aqm) = C+(m + 1) , where C+ is a number to be determined. A similar argument gives

alm) = C-lm - 1)

The numbers C+ and C- can be determined from the norms

Similarly, Ic-I2 = (m la+al m) = m .

The axiomatic development of quantum mechanics requires that all quantum mechanical states lie in a Hilbert space with a positive definite norm. Hence we require that m > 0, or if m = 0,

al0) = 0 . Furthermore, since m can be lowered by integers, all positive m must be integers; otherwise, we could generate negative values for m from positive values by lowering m repeatedly by one unit.

Hence, it is possible to choose phases (signs) so that (m 2 0)

alm) = f i l m - 1) atalm) = mlm) .

(1.19)

This means that all the states can be generated from a “ground state” 10) (sometimes called the “vacuum”) by successive operations of at :

For a mechanical system like the string, these states Im) are referred to as phonon states, and if a = a,, we will show that m can be interpreted as the number of phonons of energy w,, where the quantum of energy carried by the phonon is

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Relativistic Quantum Mechanics and Field Theory...Relativistic Quantum Mechanics and Field Theory are among the most challenging and beautiful subjects in Physics. From their study