Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- Preface -- Part 1 Path Integrals -- 1 Operator calculus -- 1.1 Free propagator -- Remark on operator notations -- 1.2 Euclidean formulation -- Remark on Euclidean Gamma-matrices -- 1.3 Path-ordering of operators -- 1.4 Feynman disentangling -- Remark on definition of the measure -- 1.5 Calculation of the Gaussian path integral -- Remark on mathematical structure -- 1.6 Transition amplitudes -- Remark on parametric invariant representation -- Remark on discretized path-ordered exponential -- 1.7 Propagators in external field -- Remark on analogy with statistical mechanics -- 2 Second quantization -- 2.1 Integration over fields -- Remark on Minkowski-space formulation -- 2.2 Grassmann variables -- 2.3 Perturbation theory -- 2.4 Schwinger-Dyson equations -- 2.5 Commutator terms -- 2.6 Schwinger-Dyson equations (continued) -- Remark on connected correlators -- Remark on the LSZ reduction formula -- 2.7 Regularization -- 3 Quantum anomalies from path integral -- 3.1 QED via path integral -- 3.2 Chiral Ward identity -- Remark on Gamma in d dimensions -- Remark on gauge-fixing -- 3.3 Chiral anomaly -- Remark on regularization of the measure -- Remark on regularized Schwinger-Dyson equations -- 3.4 Chiral anomaly (calculation) -- Remark on the non-Abelian chiral anomaly -- 3.5 Scale anomaly -- Remark on the non-Abelian scale anomaly -- 4 Instantons in quantum mechanics -- 4.1 Double-well potential -- 4.2 The instanton solution -- 4.3 Instanton contribution to path integral -- 4.4 Symmetry restoration by instantons -- Remark on description of instantons -- Remark on instantons in Yang-Mills theory -- Bibliography to Part 1 -- Reference guide -- References -- Part 2 Lattice Gauge Theories -- 5 Observables in gauge theories -- 5.1 Gauge invariance -- 5.2 Phase factors (definition).

5.3 Phase factors (properties) -- Remark on an analogy with differential geometry -- A historical remark -- 5.4 Aharonov-Bohm effect -- Remark on quantum vs. classical observables -- 6 Gauge fields on a lattice -- 6.1 Sites, links, plaquettes and all that -- 6.2 Lattice formulation -- Remark on the naive continuum limit -- Remark on ambiguities of the lattice action -- 6.3 The Haar measure -- Remark on the lattice quantization -- 6.4 Wilson loops -- Remark on mass renormalization -- 6.5 Strong-coupling expansion -- 6.6 Area law and confinement -- Remark on the perimeter law -- Remark on the Creutz ratio -- 6.7 Asymptotic scaling -- Remark on dimensional transmutation -- Remark on second-order phase transition -- 7 Lattice methods -- 7.1 Phase transitions -- Remark on confinement in 4… -- 7.2 Mean-field method -- 7.3 Mean-field method (variational) -- Remark on the criterion for phase transition -- 7.4 Lattice renormalization group -- 7.5 Monte Carlo method -- Heat bath algorithm -- Metropolis algorithm -- 7.6 Some Monte Carlo results -- 8 Fermions on a lattice -- 8.1 Chiral fermions -- 8.2 Fermion doubling -- Remark on the Nielsen-Ninomiya theorem -- 8.3 Kogut-Susskind fermions -- Remark on four generations -- 8.4 Wilson fermions -- 8.5 Quark condensate -- Remark on Monte Carlo simulations with fermions -- 9 Finite temperatures -- 9.1 Feynman-Kac formula -- Remark on thermal density matrix -- 9.2 QCD at finite temperature -- 9.3 Confinement criterion at finite temperature -- Remark on high temperatures -- 9.4 Deconfining transition -- Remark on the deconfining phase transition in the early universe -- 9.5 Restoration of chiral symmetry -- Bibliography to Part 2 -- Reference guide -- References -- Part 3 1/N Expansion -- 10 O(N) vector models -- 10.1 Four-Fermi theory -- Remark on auxiliary fields -- 10.2 Bubble graphs as the zeroth order in 1/N.

Remark on scale invariance at the fixed point -- Remark on conformal invariance at fixed point -- Remark on broken scale invariance -- 10.3 Functional methods for… -- Remark on the effective action -- 10.4 Nonlinear sigma model -- 10.5 Large-N factorization in vector models -- 11 Multicolor QCD -- 11.1 Index or ribbon graphs -- Remark on the U(N ) gauge group -- 11.2 Planar and nonplanar graphs -- 11.3 Planar and nonplanar graphs (the boundaries) -- Remark on oriented Riemann surfaces -- Remark on cyclic-ordered Green functions -- Remark on generating functionals for planar graphs -- 11.4 Topological expansion and quark loops -- Remark on the order of gauge action -- 11.5 't Hooft versus Veneziano limits -- Remark on asymptotic freedom in the Veneziano limit -- Remark on phenomenology of multicolor QCD -- 11.6 Large-N factorization -- Remark on factorization beyond perturbation theory -- 11.7 The master field -- Remark on noncommutative probability theory -- 11.8 1/N as semiclassical expansion -- Remark on the large-N limit as statistical averaging -- 12 QCD in loop space -- 12.1 Observables in terms of Wilson loops -- Remark on renormalization of Wilson loops -- 12.2 Schwinger-Dyson equations for Wilson loop -- 12.3 Path and area derivatives -- Remark on Bianchi identity for Stokes functionals -- Remark on the regularized length -- Remark on the relation with the variational derivative -- 12.4 Loop equations -- 12.5 Relation to planar diagrams -- 12.6 Loop-space Laplacian and regularization -- Remark on functional Laplacian -- 12.7 Survey of nonperturbative solutions -- 12.8 Wilson loops in QCD -- Remark on the string representation -- 12.9 Gross-Witten transition in lattice QCD -- 13 Matrix models -- 13.1 Hermitian one-matrix model -- Remark on discretization of random surfaces -- 13.2 Hermitian one-matrix model (solution at N = …).

13.3 The loop equation -- Remark on the Virasoro constraints -- 13.4 Solution in 1/N -- Remark on the iterative solution -- 13.5 Continuum limit -- Remark on the KdV hierarchy -- Remark on the ontinuum Wilson loop -- Remark on the ontinuum Virasoro onstraints -- Remark on the Kontsevich matrix model -- Remark on 2D topologi al gravity -- 13.6 Hermitian multimatrix models -- Remark on the d = 1 barrier -- Remark on the Kazakov-Migdal model -- Bibliography to Part 3 -- Reference guide -- References -- Part 4 Reduced Models -- 14 Eguchi-Kawai model -- 14.1 Reduction of the scalar field (lattice) -- Remark on large but finite N -- 14.2 Reduction of the scalar field (continuum) -- Remark on higher genera -- 14.3 Reduction of the Yang-Mills field -- 14.4 The continuum Eguchi-Kawai model -- 14.5 R symmetry in perturbation theory -- Remark on supersymmetric case -- 14.6 Quenched Eguchi-Kawai model -- Remark on the quenched Eguchi-Kawai model on the lattice -- 15 Twisted reduced models -- 15.1 Twisting prescription -- 15.2 Twisted reduced model for scalars -- Remark on twisted versus quenched models at large but finite N -- Remark on mapping between matrices and fields -- Remark on SU… -- 15.3 Twisted Eguchi-Kawai model -- Remark on twisted boundary conditions -- Remark on U(N ) gauge fields -- 15.4 Twisting prescription in the continuum -- Remark on the number of states in Hilbert space -- 15.5 Continuum version of TEK -- Remark on TEK with fundamental matter -- 16 Noncommutative gauge theories -- 16.1 The noncommutative space -- Remark on the Weyl transformation -- Remark on the Moyal bracket -- Remark on nonlocality of the star product -- Remark on the double scaling limit -- 16.2 The U(1) gauge theory -- Remark on the Lorentz invariance -- Remark on the U(n) gauge theory -- 16.3 One-loop renormalization -- Remark on the UV/IR mixing.

16.4 Noncommutative quantum electrodynamics -- Remark on large but finite Lambda -- Remark on phenomenology in NCQED -- 16.5 Wilson loops and observables -- Remark on the definition of open Wilson loops -- 16.6 Compactification to tori -- Remark on finite Heisenberg-Weyl group -- Remark on Wilson loops on the lattice -- 16.7 Morita equivalence -- Remark on constraint TEK -- Remark on fundamental matter -- Remark on classical solutions -- Bibliography to Part 4 -- Reference guide -- References -- Index.