The book is intended to provide a general introduction to the physics of quantized fields and many-body physics. It is based on a two-semester sequence of courses taught at the University of Illinois at Urbana-Champaign at various times between 1985 and 1997. The students taking all or part of the sequence had interests ranging from particle and nuclear theory through quantum optics to condensed matter physics experiment.

Preface

1 Discrete Systems

 1.1 One-Dimensional Harmonic Crystal

1.1.1 Normal Modes

1.1.2 Harmonic Oscillator

1.1.3 Annihilation and Creation Operators for Normal Modes

 1.2 Continuum Limit

1.2.1 Sums and Integrals

1.2.2 Continuum Fields

2 Relativistic Scalar Fields

 2.1 Convcntions

 2.2 The Klein-Gordon Equation

2.2.1 Relativistic Normalization

2.2.2 An Inner Product

2.2.3 Complex Scalar Fields

 2.3 Symmetries and Noether's Theorem

2.3.1 Internal Symmetries

2.3.2 Space-Time Symmetries

3 Perturbation Theory

 3.1 Interactions

 3.2 Perturbation Theory

3.2.1 Interaction Picture

3.2.2 Propagators and Time-Ordered Products

 3.3 Wick's Theorem

3.3.1 Normal Products

3.3.2 Wick's Theorem

3.3.3 Applications

4 Feynman Rules

 4.1 Diagrams

4.1.1 Diagrams in Space-time

4.1.2 Diagrams in Momentum Space

 4.2 Scattering Theory

4.2.1 Cross-Sections

4.2.2 Decay of an Unstable Particle

5 Loops, Unitarity, and Analyticity

 5.1 Unitarity of the S Matrix

 5.2 The Analytic S Matrix

5.2.1 Origin of Analyticity

5.2.2 Unitarity and Branch Cuts

5.2.3 Resonances, Widths, and Lifetimes

 5.3 Some Loop Diagrams

5.3.1 Wick Rotation

5.3.2 Feynman Parameters

5.3.3 Dimensional Regularization

6 Formal Developments

 6.1 Gell-Mann Low Theorem

 6.2 Lehmann-Kaillen Spectral Representation

 6.3 LSZ Reduction Formulae

6.3.1 Amputation of External Legs

6.3.2 In and Out States and Fields

6.3.3 Borcher's Classes

7 Fermions

 7.1 Dirac Equation

 7.2 Spinors, Tensors, and Currents

7.2.1 Field Bilinears

7.2.2 Conservation Laws

 7.3 Holes and the Dirac Sea

7.3.1 Positive and Negative Energies

7.3.2 Holes

 7.4 Quantization

7.4.1 Normal and Time-Ordered Products

8 QED

 8.1 Quantizing Maxwell's Equations

8.1.1 1 Hamiltonian Formalism

8.1.2 Axial Gauge

8.1.3 Lorentz Gauge

 8.2 Feynman Rules for QED

8.2.1 Moiler Scattering

 8.3 Ward Identity and Gauge Invariance

8.3.1 The Ward Identity

8.3.2 Applications

9 Electrons in Solids

 9.1 Second Quantization

 9.2 Fermi Gas and Fermi Liquid

9.2.1 One-Particle Density Matrix

9.2.2 Linear Response

9.2.3 Diagram Approach

9.2.4 Applications

 9.3 Electrons and Phonons

10 Nonrelativistic Bosons

 10.1 The Boson Field

 10.2 Spontaneous Symmetry Breaking

 10.3 Dilute Bose Gas

10.3.1 Bogoliubov Transfomation

10.3.2 Field Equations

10.3.3 Quantization

10.3.4 Landau Criterion for Superfiuidity

10.3.5 Normal and Superfiuid Densities

 10.4 Charged Bosom

10.4.1 Gross-Pitaevskii Equation

10.4.2 Vortices

10.4.3 Connection with Fluid Mechanics

11 Finite Temperature

 11.1 Partition Functions

 11.2 Worldlines

 11.3 Matsubara Sums

12 Path Integrals

 12.1 Quantum Mechanics of a Particle

12.1.1 Real Time

12.1.2 Euclidean Time

 12.2 Gauge Invariance and Operator Ordering

 12.3 Correlation Functions

 12.4 Fields

 12.5 Gaussian Integrals and Free Fields

12.5.1 Real Fields

12.5.2 Complex Fields

 12.6 Perturbation Theory

13 Functional Methods

 13.1 Generating Functionals

13.1.1 Effective Action

 13.2 Ward Identities

13.2.1 Goldstone's Theorem

14 Path Integrals for Fermions

 14.1 Berezin Integrals

14.1.1 A Simple Supersymmetry

 14.2 Fermionic Coherent States

 14.3 Superconductors

14.3.1 Effective Action

15 Lattice Field Theory

 15.1 Boson Fields

 15.2 Random Walks

 15.3 Interactions and Bose Condensation

15.3.1 Rotational Invariance

 15.4 Lattice Fermions

15.4.1 No Chiral Lattice Fermions

16 The Renormailzation Group

 16.1 Transfer Matrices

16.1.1 Continuum Limit

16.1.2 Two-Dimensional Ising Model

 16.2 Block Spins and Renormalization Group

16.2.1 Correlation Functions

17 Fields and Renormalization

 17.1 The Free-Field Fixed Point

 17.2 The Gaussian Model

 17.3 General Method

 17.4 Nonlinear o Model

17.4.1 Renormalizing

17.4.2 Solution of the RGE

 17.5 Renormalizing

18 Large N Expansions

 18.1 O(N) Linear a-Model

 18.2 Large N Expansions

18.2.1 Linear vs. Nonlinear σ-Models

A Relativistic State Normalization

B The General Commutator

C Dimensional Regularization

 C.I Analytic Continuation and Integrals

 C.2 Propagators

D Spinors and the Principle of the Sextant

 D.1 Constructing the λ-Matrices

 D.2 Basic Theorem

 D.3 Chirality

 D.4 Spin(2N), Pin(2N), and SU(N) C SO(2N)

E Indefinite Metric

F Phonons and Momentum

G Determinants in Quantum Mechanics

Index