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PHY2403F Lecture Notes Michael Luke (Dated: Fall, 2011) 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 Coleman's field theory lectures from Harvard. 2 Contents 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 Na¨ıve 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 40 A. Classical Mechanics 40 B. Symmetries in Field Theory 42 1. Space-Time Translations and the Energy-Momentum Tensor 43 2. Lorentz Transformations 44 C. Internal Symmetries 48 1. U(1) Invariance and Antiparticles 49 2. Non-Abelian Internal Symmetries 53 D. Discrete Symmetries: C, P and T 55 1. Charge Conjugation, C 55 2. Parity, P 56 3. Time Reversal, T 58 IV. Example: Non-Relativistic Quantum Mechanics (\Second Quantization") 60 V. Interacting Fields 65 A. Particle Creation by a Classical Source 65 B. More on Green Functions 69 3 C. Mesons Coupled to a Dynamical Source 70 D. The Interaction Picture 71 E. Dyson's Formula 73 F. Wick's Theorem 77 G. S matrix elements from Wick's Theorem 80 H. Diagrammatic Perturbation Theory 83 I. More Scattering Processes 88 J. Potentials and Resonances 92 VI. Example (continued): Perturbation Theory for nonrelativistic quantum mechanics 95 VII. Decay Widths, Cross Sections and Phase Space 98 A. Decays 101 B. Cross Sections 102 C. D for Two Body Final States 103 VIII. More on Scattering Theory 106 A. Feynman Diagrams with External Lines off the Mass Shell 106 1. Answer One: Part of a Larger Diagram 107 2. Answer Two: The Fourier Transform of a Green Function 108 3. Answer Three: The VEV of a String of Heisenberg Fields 110 B. Green Functions and Feynman Diagrams 111 C. The LSZ Reduction Formula 115 1. Proof of the LSZ Reduction Formula 117 IX. Spin 1/2 Fields 125 A. Transformation Properties 125 B. The Weyl Lagrangian 131 C. The Dirac Equation 135 1. Plane Wave Solutions to the Dirac Equation 137 D. γ Matrices 139 1. Bilinear Forms 142 2. Chirality and γ5 144 E. Summary of Results for the Dirac Equation 146 1. Dirac Lagrangian, Dirac Equation, Dirac Matrices 146 2. Space-Time Symmetries 146 3. Dirac Adjoint, γ Matrices 147 4. Bilinear Forms 148 5. Plane Wave Solutions 149 X. Quantizing the Dirac Lagrangian 151 A. Canonical Commutation Relations 151 4 or, How Not to Quantize the Dirac Lagrangian 151 B. Canonical Anticommutation Relations 153 C. Fermi-Dirac Statistics 155 D. Perturbation Theory for Spinors 157 1. The Fermion Propagator 159 2. Feynman Rules 161 E. Spin Sums and Cross Sections 166 XI. Vector Fields and Quantum Electrodynamics 168 A. The Classical Theory 168 B. The Quantum Theory 173 C. The Massless Theory 176 1. Minimal Coupling 179 2. Gauge Transformations 182 D. The Limit µ ! 0 184 1. Decoupling of the Helicity 0 Mode 186 E. QED 187 F. Renormalizability of Gauge Theories 189 5 jet #2 jet #3 jet #1 jet #4 e+ FIG. I.1 The results of a proton-antiproton collision at the Tevatron at Fermilab. 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. I. INTRODUCTION 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 where relativity 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 enough energy to pop additional particles out of the vacuum (we will discuss how this works at length in 2 the course). For example, in p-p (proton-proton) scattering with a centre of mass energy E > mπc (where mπ ∼ 140 MeV is the mass of the neutral pion) the process p + p ! p + p + π0 6 2 is possible. At higher energies, E > 2mpc , 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 Large Hadron Collider at CERN, outside Geneva, which collides protons and antiprotons with energies of several TeV, or several 2 thousands times mpc , 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 µ trapped in a container with reflecting walls of side L. The uncertainty in the particle's momentum is therefore of order ¯h=L. In the relativistic regime, this translates to an uncertainty of order ¯hc=L in the particle's energy. For L small enough, L <∼ ¯h/µc (where ¯h/µc ≡ λc, the Compton wavelength of the particle), the uncertainty in the energy of the system is large enough for particle creation to occur - particle anti-particle pairs can pop out of the vacuum, making the number of particles in the container uncertain! The physical state of the system is a quantum-mechanical superposition of states with different particle number. Even the vacuum state - which in an interacting quantum theory is not the zero-particle state, but rather the state of lowest energy - is complicated. The smaller the distance scale you look at it, the more complex its structure. There is therefore no sense in which it is possible to localize a particle in a region smaller than its Compton wavelength. In atomic physics, where NRQM works very well, this does not introduce any problems. The Compton wavelength of an electron (mass µ = 0:511 MeV=c2), is 1=0:511 MeV × 197 MeV fm ∼ 4 × 10−11 cm, or about 10−3 Bohr radii. So there is no problem localizing an electron on atomic scales, and the relativistic corrections due to multi-particle states are small. On the other hand, the up and down quarks which make up the proton have masses of order 10 MeV (λc ' 20 fm) and are confined to a region the size of a proton, or about 1 fm. Clearly the internal structure of the proton is much more complex than a simple three quark system, and relativistic effects will be huge. Thus, there is no such thing in relativistic quantum mechanics as the two, one, or even zero 7 L L (a) L>> h/µc (b) L <<h/µc FIG. I.2 A particle of mass µ cannot be localized in a region smaller than its Compton wavelength, λc = ¯h/µ. 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 fj ~xig basis of NRQM simply does not exist, since particles cannot be localized to arbitrarily small regions.
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