The Physics of Quantum Fields

This 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. The book does not cover as much ground as some texts. This is because I have tried to concentrate on the basic conceptual issues that many students find difficult. For a computation-method oriented course an instructor would probably wish to suplement this book with a more comprehensive and specialized text such as Peskin and Schroeder An Introduction to Quantum Field Theory, which is intended for particle theorists, or perhaps the venerable Quantum Theory of Many-Particle Systems by Fetter and Walecka. The most natural distribution of the material if the book is used for a two-semster course is as follows: 1 st Semester: Chapters 1-11. 2nd semester: Chapters 12-18.

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 Conventions.- 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-Källén 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 Hamiltonian Formalism.- 8.1.2 Axial Gauge.- 8.1.3 Lorentz Gauge.- 8.2 Feynman Rules for QED.- 8.2.1 Moller 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 Approacha.- 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 Superfluidity.- 10.3.5 Normal and Superfluid Densities.- 10.4 Charged Bosons.- 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 Renormalization 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 ? Model.- 17.4.1 Renormalizing.- 17.4.2 Solution of the RGE.- 17.5 Renormalizing ??4.- 18 Large N Expansions.- 18.1 O(N) Linear ?-Model.- 18.2 Large N Expansions.- 18.2.1 Linear vs. Nonlinear ?-Models.- A Relativistic State Normalization.- B The General Commutator.- C Dimensional Regularization.- C.1 Analytic Continuation and Integrals.- C.2 Propagators.- D Spinors and the Principle of the Sextant.- D.1 Constructing the ?-Matrices.- D.2 Basic Theorem.- D.3 Chirality.- E Indefinite Metric.- F Phonons and Momentum.- G Determinants in Quantum Mechanics.