Intro
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