Elementary Particle Theory. Volume 2, Quantum Electrodynamics

Description

In a successful theory of elementary particles, at least three important conditions must be fulfilled: (1) relativistic invariance in the instant form of dynamics; (2) cluster separability of the interaction; (3) description of processes involving creation and destruction of particles. In the first volume of our book we discussed interacting quantum theories in Hilbert spaces with a fixed set of particles. We showed how it is possible to satisfy the first two requirements (relativistic invariance and cluster separability).1 However, these theories were fundamentally incomplete, due to their inability to describe physical processes that change the types and/or number of particles in the system. Thus, condition 3 from our list was not fulfilled. Familiar examples of the creation and annihilation processes are emission and absorption of light (photons), decays, neutrino oscillations, etc. Particles are produced especially intensively at high energies. This is due to the famous Einstein formula E = mc2, which says, in particular, that if the system has sufficient energy E of relative motion, then this energy can be transformed into the mass m of newly created particles. Even in the simplest two-particle case, the energy of the relative motion of these reactants is unlimited. Therefore, there is no limit to the number of new particles that can be created in a collision. To advance in the study of such processes, the first thing to do is to build a Hilbert space of states H , which is capable of describing particle creation and annihilation. Such a space must include states with arbitrary numbers (from zero to infinity) of particles of all types. It is called the Fock space. This construction is rather simple. However, the next step – the definition of realistic interaction operators in the Fock space – is highly nontrivial. A big part of our third volume will be devoted to the solution of this problem. Here we will prepare ourselves to this task by starting with a more traditional approach, which is known as the renormalized relativistic quantum field theory (QFT). Our discussions in this book are limited to electromagnetic phenomena, so we will be interested in the simplest and most successful type of QFT – quantum electrodynamics (QED). In Chapter 1, Fock space, we will describe the basic mathematical machinery of Fock spaces, including creation and annihilation operators, normal ordering and classification of interaction potentials. A simple toy model with variable number of particles will be presented in Chapter 2, Scattering in Fock space. In this example, we will discuss such important ingredients of QFT as the S-matrix formalism, renormalization, diagram technique and cluster separability. Our first two chapters have a mostly technical character. They define our terminology and notation and prepare us for a more in-depth study of QED in the two following chapters. In Chapter 3, Quantum electrodynamics, we introduce the important concept of the quantum field. This idea will be applied to systems of charged particles and photons in the formalism of QED. Here we will obtain an interacting theory, which satisfies the principles of relativistic invariance and cluster separability, where the number of particles is not fixed. However, the “naïve” version of QED presented here is unsatisfactory, since it cannot calculate scattering amplitudes beyond the lowest orders of perturbation theory. Chapter 4, Renormalization, completes the second volume of the book. We will discuss the plague of ultraviolet divergences in the “naïve” QED and explain how they can be eliminated by adding counterterms to the Hamiltonian. As a result, we will get the traditional “renormalized” QED, which has proven itself in precision calculations of scattering cross sections and energy levels in systems of charged particles. However, this theory failed to provide a well-defined interacting Hamiltonian and the interacting time evolution (= dynamics). We will address these issues in the third volume of our book. As in the first volume, here we refrain from criticism and unconventional interpretations, trying to keep in line with generally accepted approaches. The main purpose of this volume is to explain the basic concepts and terminology of QFT. For the most part, we will adhere to the logic of QFT formulated by Weinberg in the series of articles and in the excellent textbook. A critical discussion of the traditional approaches and a new look at the theory of relativity will be presented in Volume 3. References to Volume 1 of this book will be prefixed with “1-”. For example, (1-7.14) is formula (7.14) from Volume 1. Contents......Page 6 List of figures......Page 10 List of tables......Page 12 Postulates, statements, theorems......Page 14 Conventional notation......Page 16 Preface......Page 18 1.1.1 Sectors with fixed numbers of particles......Page 20 1.1.3 Noninteracting representation of Poincaré group......Page 22 1.1.4 Creation and annihilation operators for fermions......Page 23 1.1.5 Anticommutators of particle operators......Page 25 1.1.7 Particle number operators......Page 26 1.1.8 Continuous spectrum of momentum......Page 27 1.1.9 Normal ordering......Page 28 1.1.10 Noninteracting energy and momentum......Page 30 1.1.11 Noninteracting angular momentum and boost......Page 31 1.1.12 Poincaré transformations of particle operators......Page 32 1.2.1 Conservation laws......Page 34 1.2.2 General form of interaction operators......Page 36 1.2.3 Five types of regular potentials......Page 39 1.2.4 Products and commutators of regular potentials......Page 42 1.2.5 More about t-integrals......Page 44 1.2.6 Solution of one commutator equation......Page 46 1.2.7 Two-particle potentials......Page 47 1.2.8 Momentum-dependent potentials......Page 50 2.1.1 Fock space and Hamiltonian......Page 52 2.1.2 S-operator in second order......Page 54 2.1.3 Drawing diagrams in toy model......Page 55 2.1.4 Reading diagrams in toy model......Page 58 2.1.5 Scattering in second order......Page 59 2.2.1 Renormalization of electron self-scattering in second order......Page 60 2.2.2 Renormalization of electron self-scattering in fourth order......Page 62 2.3.1 Products of diagrams......Page 65 2.3.2 Connected and disconnected diagrams......Page 66 2.3.3 Divergence of loop integrals......Page 69 2.4.1 Cluster separability of interaction......Page 71 2.4.2 Cluster separability of S-operator......Page 73 3.1 Interaction in QED......Page 76 3.1.2 Simple quantum field theories......Page 77 3.1.3 Interaction operators in QED......Page 79 3.2.1 S-operator in second order......Page 81 3.2.2 Covariant form of S-operator......Page 85 3.2.3 Feynman gauge......Page 87 3.2.4 Feynman diagrams......Page 89 3.2.5 Compton scattering......Page 91 3.2.6 Virtual particles?......Page 92 4.1.1 No self-scattering condition......Page 94 4.1.3 Renormalization by counterterms......Page 97 4.1.4 Diagrams of electron–proton scattering......Page 98 4.1.5 Regularization......Page 99 4.2.1 Electron’s self-scattering......Page 100 4.2.2 Electron self-scattering counterterm......Page 102 4.2.3 Fitting coefficient (δm)2......Page 103 4.2.4 Fitting coefficient (Z2 − 1)2......Page 104 4.2.5 Photon’s self-scattering......Page 105 4.2.6 Photon self-energy counterterm......Page 106 4.2.7 Applying charge renormalization condition......Page 108 4.2.8 Vertex renormalization......Page 109 4.3.2 Vertex diagram......Page 112 4.3.3 Ladder diagram......Page 114 4.3.4 Cross ladder diagram......Page 117 4.3.5 Renormalizability......Page 120 A Useful integrals......Page 122 B.1 Pauli matrices......Page 126 B.2 Dirac gamma matrices......Page 127 B.3 Dirac representation of Lorentz group......Page 128 B.4 Construction of Dirac field......Page 131 B.5 Properties of functions u and v......Page 133 B.6 Explicit formulas for u and v......Page 134 B.7 Useful notation......Page 137 B.8 Poincaré transformations of fields......Page 138 B.9 Approximation (v/c)2......Page 139 B.10 Anticommutation relations......Page 141 B.11 Dirac equation......Page 142 B.12 Fermion propagator......Page 144 C.1 Construction of photon quantum field......Page 148 C.2 Properties of function eμ(p, τ)......Page 149 C.3 Useful commutator......Page 150 C.5 Photon propagator......Page 152 C.6 Poincaré transformations of photon field......Page 154 D.1 Current density......Page 158 D.3 Second-order interaction in QED......Page 161 E.1 Relativistic invariance of simple QFT......Page 174 E.2 Relativistic invariance of QED......Page 175 F.1 Schwinger–Feynman integration trick......Page 182 F.2 Some basic four-dimensional integrals......Page 183 F.3 Electron self-energy integral......Page 186 F.4 Vertex integral......Page 189 F.4.1 Calculation of M......Page 191 F.4.2 Calculation of Mσ......Page 192 F.4.3 Calculation of Mστ......Page 193 F.4.4 Complete integral......Page 194 F.5 Integral for ladder diagram......Page 197 F.5.1 Calculation of LI......Page 198 F.5.2 Calculation of LII......Page 200 F.5.3 Calculation of LIII......Page 201 F.5.4 Complete integral......Page 203 G.1 Second perturbation order......Page 204 G.2 Vertex contribution in fourth order......Page 206 H Checks of physical dimensions......Page 210 Bibliography......Page 212 Index......Page 214