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.
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1 Discrete Systems.- 1.1 One-Dimensional Harmonic Crystal.- 1.2
Continuum Limit.- 2 Relativistic Scalar Fields.- 2.1 Conventions.- 2.2 The
Klein-Gordon Equation.- 2.3 Symmetries and Noethers Theorem.- 3 Perturbation
Theory.- 3.1 Interactions.- 3.2 Perturbation Theory.- 3.3 Wicks Theorem.- 4
Feynman Rules.- 4.1 Diagrams.- 4.2 Scattering Theory.- 5 Loops, Unitarity,
and Analyticity.- 5.1 Unitarity of the S Matrix.- 5.2 The Analytic S Matrix.-
5.3 Some Loop Diagrams.- 6 Formal Developments.- 6.1 Gell-Mann Low Theorem.-
6.2 Lehmann-Källén Spectral Representation.- 6.3 LSZ Reduction Formulae.- 7
Fermions.- 7.1 Dirac Equation.- 7.2 Spinors, Tensors, and Currents.- 7.3
Holes and the Dirac Sea.- 7.4 Quantization.- 8 QED.- 8.1 Quantizing Maxwells
Equations.- 8.2 Feynman Rules for QED.- 8.3 Ward Identity and Gauge
Invariance.- 9 Electrons in Solids.- 9.1 Second Quantization.- 9.2 Fermi Gas
and Fermi Liquid.- 9.3 Electrons and Phonons.- 10 Nonrelativistic Bosons.-
10.1 The Boson Field.- 10.2 Spontaneous Symmetry Breaking.- 10.3 Dilute Bose
Gas.- 10.4 Charged Bosons.- 11 Finite Temperature.- 11.1 Partition
Functions.- 11.2 Worldlines.- 11.3 Matsubara Sums.- 12 Path Integrals.- 12.1
Quantum Mechanics of a Particle.- 12.2 Gauge Invariance and Operator
Ordering.- 12.3 Correlation Functions.- 12.4 Fields.- 12.5 Gaussian Integrals
and Free Fields.- 12.6 Perturbation Theory.- 13 Functional Methods.- 13.1
Generating Functionals.- 13.2 Ward Identities.- 14 Path Integrals for
Fermions.- 14.1 Berezin Integrals.- 14.2 Fermionic Coherent States.- 14.3
Superconductors.- 15 Lattice Field Theory.- 15.1 Boson Fields.- 15.2 Random
Walks.- 15.3 Interactions and Bose Condensation.- 15.4 Lattice Fermions.- 16
The Renormalization Group.- 16.1 Transfer Matrices.- 16.2 Block Spins and
Renormalization Group.- 17 Fields and Renormalization.- 17.1 The Free-Field
Fixed Point.- 17.2 The Gaussian Model.- 17.3 General Method.- 17.4 Nonlinear
? Model.- 17.5 Renormalizing ??4.- 18 Large N Expansions.- 18.1O(N) Linear
?-Model.- 18.2 Large N Expansions.- 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.
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