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2403F Relativistic Quantum Mechanics (Harvard)

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  • PHY2403F Lecture Notes

    Michael Luke

    (Dated: Fall, 2008)

    These notes are perpetually under construction. Please let me know of any typos or errors.

    I claim little originality; these notes are in large part an abridged and revised version of Sidney

    Colemans field theory lectures from Harvard.

  • 2Contents

    I. Introduction 5

    A. Relativistic Quantum Mechanics 5

    B. Conventions and Notation 9

    1. Units 9

    2. Relativistic Notation 10

    3. Fourier Transforms 14

    4. The Dirac Delta Function 15

    C. A Nave Relativistic Theory 15

    II. Constructing Quantum Field Theory 20

    A. Multi-particle Basis States 20

    1. Fock Space 20

    2. Review of the Simple Harmonic Oscillator 22

    3. An Operator Formalism for Fock Space 23

    4. Relativistically Normalized States 23

    B. Canonical Quantization 25

    1. Classical Particle Mechanics 26

    2. Quantum Particle Mechanics 27

    3. Classical Field Theory 29

    4. Quantum Field Theory 32

    C. Causality 37

    III. Symmetries and Conservation Laws 39

    A. Classical Mechanics 39

    B. Symmetries in Field Theory 41

    1. Space-Time Translations and the Energy-Momentum Tensor 42

    2. Lorentz Transformations 43

    C. Internal Symmetries 47

    1. U(1) Invariance and Antiparticles 48

    2. Non-Abelian Internal Symmetries 52

    D. Discrete Symmetries: C, P and T 54

    1. Charge Conjugation, C 54

    2. Parity, P 55

    3. Time Reversal, T 57

    IV. Example: Non-Relativistic Quantum Mechanics (Second Quantization) 59

    V. Interacting Fields 64

    A. Particle Creation by a Classical Source 64

    B. More on Green Functions 68

  • 3C. Mesons Coupled to a Dynamical Source 69

    D. The Interaction Picture 70

    E. Dysons Formula 72

    F. Wicks Theorem 76

    G. S matrix elements from Wicks Theorem 79

    H. Diagrammatic Perturbation Theory 82

    I. More Scattering Processes 87

    J. Potentials and Resonances 91

    VI. Example (continued): Perturbation Theory for nonrelativistic quantum mechanics 94

    VII. Decay Widths, Cross Sections and Phase Space 97

    A. Decays 100

    B. Cross Sections 101

    C. D for Two Body Final States 102

    VIII. More on Scattering Theory 105

    A. Feynman Diagrams with External Lines off the Mass Shell 105

    1. Answer One: Part of a Larger Diagram 106

    2. Answer Two: The Fourier Transform of a Green Function 107

    3. Answer Three: The VEV of a String of Heisenberg Fields 109

    B. Green Functions and Feynman Diagrams 110

    C. The LSZ Reduction Formula 114

    1. Proof of the LSZ Reduction Formula 116

    IX. Spin 1/2 Fields 124

    A. Transformation Properties 124

    B. The Weyl Lagrangian 130

    C. The Dirac Equation 134

    1. Plane Wave Solutions to the Dirac Equation 136

    D. Matrices 138

    1. Bilinear Forms 141

    2. Chirality and 5 143

    E. Summary of Results for the Dirac Equation 145

    1. Dirac Lagrangian, Dirac Equation, Dirac Matrices 145

    2. Space-Time Symmetries 145

    3. Dirac Adjoint, Matrices 146

    4. Bilinear Forms 147

    5. Plane Wave Solutions 148

    X. Quantizing the Dirac Lagrangian 150

    A. Canonical Commutation Relations 150

  • 4or, How Not to Quantize the Dirac Lagrangian 150

    B. Canonical Anticommutation Relations 152

    C. Fermi-Dirac Statistics 154

    D. Perturbation Theory for Spinors 156

    1. The Fermion Propagator 158

    2. Feynman Rules 160

    E. Spin Sums and Cross Sections 165

    XI. Vector Fields and Quantum Electrodynamics 167

    A. The Classical Theory 167

    B. The Quantum Theory 172

    C. The Massless Theory 175

    1. Minimal Coupling 178

    2. Gauge Transformations 181

    D. The Limit 0 1831. Decoupling of the Helicity 0 Mode 185

    E. QED 186

    F. Renormalizability of Gauge Theories 188

  • 5FIG. I.1 The results of a proton-antiproton collision at the Tevatron. The proton and antiproton beams

    travel perpendicular to the page, colliding at the origin of the tracks. Each of the curved tracks indicates a

    charged particle in the final state. The tracks are curved because the detector is placed in a magnetic field;

    the radius of the curvature of the path of a particle provides a means to determine its mass, and therefore

    identify it.


    A. Relativistic Quantum Mechanics

    Usually, additional symmetries simplify physical problems. For example, in non-relativistic

    quantum mechanics (NRQM) rotational invariance greatly simplifies scattering problems. Why

    does the addition of Lorentz invariance complicate quantum mechanics? The answer is very simple:

    in relativistic systems, the number of particles is not conserved. In a quantum system, this has

    profound implications.

    Consider, for example, scattering a particle in potential. At low energies, E mc2 whererelativity is unimportant, NRQM provides a perfectly adequate description. The incident particle

    is in some initial state, and one can fairly simply calculate the amplitude for it to scatter into any

    final state. There is only one particle, before and after the scattering process. At higher energies

    where relativity is important things gets more complicated, because if E mc2 there is enoughenergy to pop additional particles out of the vacuum (we will discuss how this works at length in

    the course). For example, in p-p (proton-proton) scattering with a centre of mass energy E > mpic2

    (where mpi 140 MeV is the mass of the neutral pion) the process

    p+ p p+ p+ pi0

  • 6is possible. At higher energies, E > 2mpc2, one can produce an additional proton-antiproton pair:

    p+ p p+ p+ p+ p

    and so on. Therefore, what started out as a simple two-body scattering process has turned into

    a many-body problem, and it is necessary to calculate the amplitude to produce a variety of

    many-body final states. The most energetic accelerator today is the Tevatron at Fermilab, out-

    side Chicago, which collides protons and antiprotons with energies greater than 1 TeV, or about

    103mpc2, so typical collisions produce a huge slew of particles (see Fig. I.1).

    Clearly we will have to construct a many-particle quantum theory to describe such a process.

    However, the problems with NRQM run much deeper, as a brief contemplation of the uncertainty

    principle indicates. Consider the familiar problem of a particle in a box. In the nonrelativistic

    description, we can localize the particle in an arbitrarily small region, as long as we accept an

    arbitrarily large uncertainty in its momentum. But relativity tells us that this description must

    break down if the box gets too small. Consider a particle of mass trapped in a container with

    reflecting walls of side L. The uncertainty in the particles momentum is therefore of order h/L.

    In the relativistic regime, this translates to an uncertainty of order hc/L in the particles energy.

    For L small enough, L

  • 7FIG. I.2 A particle of mass cannot be localized in a region smaller than its Compton wavelength, c = h/.

    At smaller scales, the uncertainty in the energy of the system allows particle production to occur; the number

    of particles in the box is therefore indeterminate.

    body problem! In principle, one is always dealing with the infinite body problem. Thus, except in

    very simple toy models (typically in one spatial dimension), it is impossible to solve any relativistic

    quantum system exactly. Even the nature of the vacuum state in the real world, a horribly complex

    sea of quark-antiquark pairs, gluons, electron-positron pairs as well as more exotic beasts like Higgs

    condensates and gravitons, is totally intractable analytically. Nevertheless, as we shall see in this

    course, even incomplete (usually perturbative) solutions will give us a great deal of understanding

    and predictive power.

    As a general conclusion, you cannot have a consistent, relativistic, single particle quantum

    theory. So we will have to set up a formalism to handle many-particle systems. Furthermore, it

    should be clear from this discussion that our old friend the position operator ~X from NRQM does

    not make sense in a relativistic theory: the {| ~x} basis of NRQM simply does not exist, sinceparticles cannot be localized to arbitrarily small regions. The first casualty of relativistic QM is

    the position operator, and it will not arise in the formalism which we will develop.

    There is a second, intimately related problem which arises in a relativistic quantum theory,

    which is that of causality. In both relativistic and nonrelativistic quantum mechanics observables

  • 8correspond to Hermitian operators. In NRQM, however, observables are not attached to space-

    time points - one simply talks about the position operator, the momentum operator, and so on.

    However, in a relativistic theory we have to be more careful, because making a measurement forces

    the system into an eigenstate of the corresponding operator. Unless we are careful about only

    defining observables locally (i.e. having different observables at each space-time point) we will run

    into trouble with causality, because observables separated by spacelike separations will be able to

    interfere with one another.

    Consider applying the NRQM approach to observables to a situation with two observers, One

    and Two, at space-time points x1 and x2, which are separated by a spacelike interval. Observer

    FIG. I.3 Observers O1 and O2 are separated by a spacelike interval. A Lorentz boost will move the observer

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