This textbook is intended to be used in an introductory course in quantum field theory. It assumes the standard undergraduate education of a physics major and it is designed to appeal to a wide array of physics graduate students, from those studying theoretical and experimental high energy physics to those interested in condensed matter, optical, atomic, nuclear and astrophysicists. It includes a thorough development of the field theoretic approach to nonrelativistic many-body physics as a step in developing a broad-based working knowledge of some of the basic aspects of quantum field theory. It presents a logical, step by step systematic development of relativistic field theory and of functional techniques and their applications to perturbation theory with Feynman diagrams, renormalization, and basic computations in quantum electrodynamics.
Author(s): Gordon Walter Semenoff
Series: Graduate Texts in Physics
Edition: 1
Publisher: Springer Nature Singapore
Year: 2023
Language: English
Pages: 403
Tags: Many particle physics, Second quantization, Fermi and Bose gases, Feynman diagrams, Renormalization group, Non-relativistic space-time, Relativistic symmetry, Photon Hamiltonian, Dirac field, Quantum Electrodynamics, Goldstone’s Theorem, Elastic scattering
Contents
1 Prologue
2 Many Particle Physics as a Quantum Field Theory
2.1 Introduction
2.2 Non-relativistic Particles
2.2.1 Identical and Indistinguishable Particles
2.2.2 The Example of Weakly Interacting Particles
2.2.3 Hamiltonian and Stationary States
2.2.4 Particles with Spin
2.3 Second Quantization in the Schrödinger Picture
2.4 Second Quantization in the Heisenberg Picture
3 Degenerate Fermi and Bose Gases
3.1 The Limit of Weakly Interacting Particles
3.2 Degenerate Fermi Gas
3.2.1 The Ground State |mathcalO>
3.2.2 Particles and Holes
3.2.3 The Grand Canonical Free Energy
3.3 Degenerate Bose Gas
3.3.1 Landau's Criterion for Superfluidity
3.3.2 Vacuum Expectation Value
3.4 Spontaneous Symmetry Breaking
4 The Action Principle and Noether's Theorem
4.1 The Action
4.1.1 The Euler–Lagrange Equations
4.2 Canonical Momenta, Poisson Brackets and Commutation Relations
4.3 Noether's Theorem
4.3.1 Conservation Laws and Continuity Equations
4.3.2 Definition of Symmetry
4.3.3 Examples of Symmetries
4.3.4 Proof of Noether's Theorem
4.4 Phase Symmetry and the Conservation of Particle Number
5 Non-relativistic Space–Time Symmetries
5.1 Translation Invariance and the Stress Tensor
5.2 Galilean Symmetry
5.3 Scale Invariance
5.3.1 Improving the Stress Tensor
5.3.2 The Consequences of Scale Invariance
5.4 Special Schrödinger Symmetry
5.5 Summary
6 Space–Time Symmetry and Relativistic Field Theory
6.1 Quantum Mechanics and Special Relativity
6.2 Coordinates
6.3 Scalars, Vectors, Tensors
6.4 The Metric
6.5 Symmetry of Space–Time
6.6 The Symmetries of Minkowski Space
6.7 Natural Units
6.8 Relativistic Fields
7 The Real Scalar Quantum Field Theory
7.1 Constructing a Relativistic Lagrangian Density
7.2 Field Equation and Commutation Relations
7.3 Noether's Theorem and Poincare Symmetry
7.4 Correlation Functions of the Real Scalar Field
7.5 The Free Scalar Field
7.6 Consequences of Spacetime Symmetry
7.7 Spectral Theorem
7.8 Normalization of the Spectral Function
7.9 Analyticity
7.9.1 The Reeh–Schlieder Theorem
7.10 Conformal Symmetry
8 Emergent Relativistic Symmetry
8.1 Phonons
8.2 The Debye Theory of Solids
8.3 Relativistic Fermions in Graphene
9 The Dirac Field Theory
9.1 The Dirac Equation
9.2 Solving the Dirac Equation
9.3 Lorentz Invariance of the Dirac Equation
9.4 Spin of the Dirac Field
9.5 Phase Symmetry and the Conservation of Charge
9.5.1 Conserved Number Current
9.5.2 Relativistic Noether's Theorem for the Dirac Equation
9.5.3 Alternative Proof of Noether's Theorem
9.6 Spacetime Symmetry
9.6.1 Translation Invariance and the Stress Tensor
9.6.2 Lorentz Transformations
9.6.3 Stress Tensor and Killing Vectors
10 Photons
10.1 Relativistic Classical Electrodynamics
10.2 Quantization
10.2.1 Negative Normed States
10.2.2 Physical State Condition
10.2.3 Null States and the Equivalence Relation
10.3 Space–Time Symmetries of the Photon
10.4 Massive Photon
10.5 Quantum Electrodynamics
10.5.1 C, P and T
11 Functional Methods
11.1 Functional Derivative
11.2 Functional Integral
11.3 Generating Functional for Free Scalar Fields
11.3.1 Wick's Theorem for Scalar Fields
11.3.2 Generating Functional as a Functional Integral
11.4 The Interacting Real Scalar Field
12 More Functional Integrals
12.1 Functional Integrals for the Photon Field
12.2 Functional Methods for Fermions
12.3 Generating Functionals for Non-relativistic Fermions
12.3.1 Interacting Non-relativistic Fermions
12.4 The Dirac Field
12.4.1 2 Point Function for the Dirac Field
12.4.2 Generating Functional for the Dirac Field
12.4.3 Functional Integral for the Dirac Field
12.5 Functional Quantum Electrodynamics
13 The Weakly Coupled Real Scalar Field
13.1 Counterterms
13.2 Computation of the 2 Point Function
13.3 Feynman Diagrams
13.4 Simplifications of Feynman Diagrams
13.5 Computation of a One-Loop Feynman Integral
13.5.1 Dimensional Regularization
13.5.2 Wick Rotation
13.5.3 Feynman Parameters
13.5.4 Integration in 2ω-Dimensions
13.5.5 Asymptotic Expansion at 2ωsim4
13.5.6 Inverse Wick Rotation
13.5.7 The Mass Tadpole
13.5.8 Euclidean Quantum Field Theory
13.5.9 The 2 Point and 4 Point Functions
13.6 Subtraction Schemes
13.7 Renormalization Group
13.8 Appendix: Integration Formulae
13.8.1 Euler's Gamma Function
13.8.2 Feynman Parameter Formula
13.8.3 Dimensional Regularization Integral
14 More Theory of the Real Scalar Field
14.1 The S Matrix
14.1.1 The T Matrix
14.2 The LSZ Formula
14.3 Elastic Two-Particle Scattering
14.4 Connected and Irreducible Generating Functionals
14.4.1 Connected Correlation Functions and the Linked Cluster Theorem
14.4.2 Connected Correlation Functions
14.4.3 Cancelation of Vacuum Diagrams
14.4.4 Irreducible Correlation Function
14.5 Derivation of the LSZ Formula
15 Perturbative Quantum Electrodynamics
15.1 Counterterms
15.2 The Generating Functional in Perturbation Theory
15.2.1 Wick's Theorem for Photons and Electrons
15.3 Feynman Diagrams
15.4 Feynman Rules
15.5 The Electron 2 Point Function
15.6 Feynman Rules in Momentum Space
15.7 The Photon 2 Point Function
15.8 Quantum Corrections of the Coulomb Potential
15.9 The Electron 2 Point Function
15.10 Radiative Correction of the Vertex
15.10.1 Electromagnetic form Factors
15.10.2 Anomalous Magnetic Moment
15.11 Photon Production, the Soft Photon Theorem
15.12 Furry's Theorem
15.13 The Ward–Takahashi Identities
16 Epilogue
Index
