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Quantum Field Theory Quantum Field Theory A Physicist’s Primer Gordon Walter Semenoff Copyright c 2019 Gordon Walter Semenoff PUBLISHED BY PUBLISHER BOOK-WEBSITE.COM Licensed under the Creative Commons Attribution-NonCommercial 3.0 Unported License (the “License”). You may not use this file except in compliance with the License. You may obtain a copy of the License at http://creativecommons.org/licenses/by-nc/3.0. Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an “AS IS” BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. First printing, May 2019 Contents I Many Particle Physics as a Quantum Field Theory 1 Many particle physics .........................................9 1.1 A preview of this chapter9 1.2 Introduction9 1.3 Non-relativistic particles 10 1.3.1 Identical particles............................................... 13 1.3.2 Spin.......................................................... 16 1.4 Second Quantization 17 1.5 The Heisenberg picture 20 1.6 Summary of this chapter 23 2 Degenerate Fermi and Bose Gases ........................... 25 2.1 A preview of this chapter 25 2.2 The limit of weak interactions 25 2.3 Degenerate Fermi gas and the Fermi surface 28 2.3.1 The ground state jO > ............................................ 28 2.3.2 Particle and holes............................................... 29 2.3.3 The grand canonical ensemble.................................... 31 2.4 Bosons 33 2.5 Summary of this chapter 38 3 Classical field theory and the action principle .................. 41 3.1 The Action Principle 41 3.1.1 The Action..................................................... 42 3.1.2 The action principle and the Euler-Lagrange equations.................. 43 3.1.3 Canonical momenta, Poisson brackets and Commutation relations........ 47 3.2 Noether’s theorem 48 3.2.1 Examples of symmetries........................................... 49 3.2.2 Proof of Noether’s Theorem........................................ 50 3.3 Phase symmetry and the conservation of particle number 51 3.4 Translation invariance 53 3.5 Galilean symmetry 54 3.6 Scale invariance 57 3.6.1 Improving the energy-momentum tensor............................. 58 3.6.2 The consequences of scale invariance............................... 59 3.7 Special Schrödinger symmetry 60 3.8 The Schrödinger algebra 61 3.9 Summary of this chapter 63 II Relativistic Symmetry and Quantum Field Theory 4 Space-time symmetry and relativistic field theory .............. 69 4.1 Quantum mechanics and special relativity 69 4.2 Coordinates 73 4.3 Scalars, vectors, tensors 75 4.4 The metric 76 4.5 Symmetry of space-time 77 4.6 The symmetries of Minkowski space 77 5 The Dirac Equation ............................................ 79 5.1 Solving the Dirac equation 81 5.2 Lorentz Invariance of the Dirac equation 84 5.3 Phase symmetry and the conservation of electric current 86 5.4 The Energy-Momentum Tensor of the Dirac Field 87 5.5 Summary of this chapter 90 6 Photons ...................................................... 93 6.1 Relativistic Classical Electrodynamics 93 6.2 Covariant quantization of the photon 94 6.2.1 Field equations and commutation relations........................... 94 6.2.2 Massive photon (Optional reading)................................ 102 6.3 Space-time symmetries of the photon 103 6.4 Quantum Electrodynamics 104 6.5 Summary of this chapter 106 III Functional methods and quantum electrodynamics 7 Functional Methods and Correlation Functions ................ 111 7.1 Functional derivative 111 7.2 Functional integral 113 7.3 Photon Correlation functions 114 7.3.1 Generating functional for correlation functions of free photons........... 117 7.3.2 Photon Generating functional as a functional integral.................. 119 7.4 Functional differentiation and integration for Fermions 121 7.5 Generating functionals for non-relativistic Fermions 125 7.5.1 Interacting non-relativistic Fermions................................ 127 7.6 The Dirac field 128 7.6.1 Two-point function for the Dirac field................................ 128 7.6.2 Generating functional for the Dirac field............................ 130 7.6.3 Functional integral for the Dirac field............................... 131 7.7 Summary of this chapter 131 8 Quantum Electrodynamics ................................... 135 8.1 Quantum Electrodynamics 135 8.2 The generating functional in perturbation theory 139 8.3 Wick’s Theorem 140 8.4 Feynman diagrams 141 8.5 Connected Correlations and Goldstone’s theorem 145 8.5.1 Connected correlation functions.................................. 145 8.5.2 Goldstone’s Theorem............................................ 147 8.6 Fourier transform 147 8.7 Furry’s theorem 150 8.8 One-particle irreducible correlation functions 151 8.9 Some calculations 152 8.9.1 The photon two-point function.................................... 152 8.9.2 The Dirac field two-point function.................................. 155 8.9.3 Traces of gamma matrices....................................... 158 8.9.4 Feynman Parameter Formula..................................... 158 8.9.5 Dimensional regularization integral................................. 159 8.10 Quantum corrections of the Coulomb potential 160 8.11 Renormalization 162 8.11.1 The Ward-Takahashi identities..................................... 163 8.12 Summary of this Chapter 164 9 Formal developments ........................................ 167 9.1 In-fields, the Haag expansion and the S-matrix 167 9.2 Spectral Representation 168 9.2.1 Gauge invariant scalar operators.................................. 169 9.2.2 The Dirac field................................................. 170 9.3 S-matrix and Reduction formula 173 9.3.1 Some intuition about asymptotic behaviour.......................... 173 9.3.2 In and out-fields in QED.......................................... 174 9.3.3 Proof of the LSZ identities........................................ 178 9.3.4 An Example: Electron-electron scattering............................ 180 9.4 More generating functionals 182 9.4.1 Connected correlation functions.................................. 182 9.4.2 One-particle irreducible correlation functions........................ 184 9.4.3 Charge conjugation symmetry ....................................... 186 Many Particle Physics as a I Quantum Field Theory 1 Many particle physics ................9 1.1 A preview of this chapter 1.2 Introduction 1.3 Non-relativistic particles 1.4 Second Quantization 1.5 The Heisenberg picture 1.6 Summary of this chapter 2 Degenerate Fermi and Bose Gases .. 25 2.1 A preview of this chapter 2.2 The limit of weak interactions 2.3 Degenerate Fermi gas and the Fermi surface 2.4 Bosons 2.5 Summary of this chapter 3 Classical field theory and the action prin- ciple ................................ 41 3.1 The Action Principle 3.2 Noether’s theorem 3.3 Phase symmetry and the conservation of particle number 3.4 Translation invariance 3.5 Galilean symmetry 3.6 Scale invariance 3.7 Special Schrödinger symmetry 3.8 The Schrödinger algebra 3.9 Summary of this chapter 1. Many particle physics 1.1 A preview of this chapter In this chapter, we will formulate our first example of a quantum field theory. We begin with the study of a quantum mechanical system with non-relativistic, identical, interacting particles. We will define the problem at hand as that of needing to find a solution of the Schrödinger equation which describes that system, subject to the appropriate boundary conditions. We will discuss the two cases of particle exchange statistics, Fermions and Bosons. We shall then find a way to rewrite the quantum many-particle problem as a quantum field theory. For this, we introduce field operators and the problem is posed as a field equation and commutation relations that the field operators of the quantum field theory should satisfy. 1.2 Introduction In this chapter we will attempt to develop intuition for the answer to the question “what is a quantum field theory”. We will do this by studying a system with many particles. For now, we will assume that the particles are non-relativistic. The generalization to relativistic particles and relativistic quantum field theory will be discussed in later chapters. We will assume that the problem in front of us is quantum mechanical, that is, that we want to find a solution of the Schrödinger equation for the system as a whole and then use that solution, the wave function, to answer questions about the physical state of the system. In order to describe the quantum mechanical problem for a large number of particles in an elegant way, we will develop a procedure which is called “second quantization”. In non-relativistic quantum mechanics, when the total number of particles is finite, second quantization gives an alternative, but at the same time completely equivalent formulation of the problem of solving the Schrödinger equation. This formulation is convenient for some applications, such as perturbation theory which is widely used to study many-particle systems and it can be relevant to many interesting physical scenarios. Metals, superconductors, superfluids, trapped cold atoms and nuclear matter are important examples. The formalism is particularly useful in that it allows us to take the “thermodynamic limit” which is an idealization of such a system that considers 10 Chapter 1. Many particle physics the limit as both the volume of the system
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