1．INTRODUCTION 1
2．THE PATH INTEGRAL APPROACH TO QUANTIZATION 7
2.1 The Path Integral Method in Quantum Mechanics 8
2.2 Path Integral Representation of Bosonic Green Functions in Field theory 15
2.3 The Transfer Matrix 22
2.4 Path Integral Representation of Fermionic Green Functions 23
2.5 Discretizing Space-Time.The Lattice as a Regulator of a Quantum Field Theory 33
3．THE FREE SCALAR FIELD ON THE LATTICE 36
4．FERMIONS ON THE LATTICE 43
4.1 The Doubling Problem 43
4.2 A Closer Look at Fermion Doubling 48
4.3 Wilson Fermions 56
4.4 Staggered Fermions 57
4.5 Technical Details of the Staggered Fermion Formulation 61
4.6 Staggered Fermions in Momentum Space 69
4.7 Ginsparg-Wilson Fermions.The Overlap Operator 73
5．ABELIAN GAUGE FIELDS ON THE LATTICE AND COMPACT QED 77
5.1 Preliminaries 77
5.2 Lattice Formulation of QED 80
6．NON ABELIAN GAUGE FIELDS ON THE LATTICE COMPACT QCD 87
7．THE WILSON LOOP AND THE STATIC QUARK-ANTIQUARK POTENTIAL 95
7.1 A Look at Non-Relativistic Quantum Mechanics 96
7.2 The Wilson Loop and the Static q？-Potential in QED 97
7.3 The Wilson Loop in QCD 105
8．THE Q？-POTENTIAL IN SOME SIMPLE MODELS 109
8.1 The Potential in Quenched QED 109
8.2 The Potential in Quenched Compact QED2 114
9．THE CONTINUUM LIMIT OF LATTICE QCD 119
9.1 Critical Behaviour of Lattice QCD and the Continuum Limit 119
9.2 Dependence of the Coupling Constant on the Lattice Spacing and the Renormalization Group β-Function 122
10．LATTICE SUM RULES 130
10.1 Energy Sum Rule for the Harmonic Oscillator 130
10.2 The SU（N） Gauge Action on an Anisotropic Lattice 136
10.3 Sum Rules for the Static q？-Potential 138
10.4 Determination of the Electric,Magnetic and Anomalous Contribution to the ？Potential 146
10.5 Sum Rules for the Glueball Mass 148
11．THE STRONG COUPLING EXPANSION 151
11.1 The q？-Potential to Leading Order in Strong Coupling 151
11.2 Beyond the Leading Approximation 154
11.3 The Lattice Hamiltonian in the Strong Coupling Limit and the String Picture of Confinement 158
12．THE HOPPING PARAMETER EXPANSION 170
12.1 Path Integral Representation of Correlation Functions in Terms of Bosonic Variables 171
12.2 Hopping Parameter Expansion of the Fermion Propagator in an External Field 174
12.3 Hopping Parameter Expansion of the Effective Action 179
12.4 The HPE and the Pauli Exclusion Principle 183
13．WEAK COUPLING EXPANSION（Ⅰ）.THE Ф3-THEORY 192
13.1 Introduction 192
13.2 Weak Coupling Expansion of Correlation Functions in the φ3-Theory 195
13.3 The Power Counting Theorem of Reisz 201
14．WEAK COUPLING EXPANSION（Ⅱ）.LATTICE QED 209
14.1 The Gauge Fixed Lattice Action 209
14.2 Lattice Feynman Rules 216
14.3 Renormalization of the Axial Vector Current in One-Loop Order 222
14.4 The ABJ Anomaly 234
15．WEAK COUPLING EXPANSION（Ⅲ）.LATTICE QCD 242
15.1 The Link Integration Measure 243
15.2 Gauge Fixing and the Faddeev-Popov Determinant 247
15.3 The Gauge Field Action 252
15.4 Propagators and Vertices 257
15.5 Relation between ∧L and the ∧-Parameter of Continuum QCD 272
15.6 Universality of the Axial Anomaly in Lattice QCD 275
16．MONTE CARLO METHODS 284
16.1 Introduction 284
16.2 Construction Principles for Algorithms.Markov Chains 286
16.3 The Metropolis Method 291
16.4 The Langevin Algorithm 293
16.5 The Moleeular Dynamics Method 295
16.6 The Hybrid Algorithm 301
16.7 The Hybrid Monte Carlo Algorithm 304
16.8 The Pseudofermion Method 307
16.9 Application of the Hybrid Monte Carlo Algorithm to Systems with Fermions 313
17．SOME RESULTS OF MONTE CARLO CALCULATIONS 317
17.1 The String Tension and the q？ Potential in the SU（3）Gauge Theory 317
17.2 The q？-Potential in Full QCD 324
17.3 Chiral Symmetry Breaking 326
17.4 Glueballs 330
17.5 Hadron Mass Spectrum 336
17.6 Instantons 345
17.7 Flux Tubes in q？and qqq-Systems 359
17.8 The Dual Superconductor Picture of Confinement 363
17.9 Center Vortices and Confinement 373
17.10 Calorons 382
18．PATH-INTEGRAL REPRESENTATON OF THE THERMODYNAMICAL PARTITION FUNCTION FOR SOME SOLVABLE BOSONIC AND FERMIONIC SYSTEMS 402
18.1 Introduction 402
18.2 Path-Integral Representation of the Partition Function in Quantum Mechanics 403
18.3 Sum Rule for the Mean Energy 405
18.4 Test of the Energy Sum Rule.The Harmonic Oscillator 408
18.5 The Free Relativistic Boson Gas in the Path Integral Approach 413
18.6 The Photon Gas in the Path Integral Approach 417
18.7 Functional Methods for Fermions.Basics 420
18.8 Path Integral Representation of the Partition Function for a Fermionic System valid for Arbitrary Time-Step 424
18.9 A Modified Fermion Action Leading to Fermion Doubling 429
18.10 The Free Dirac Gas.Continuum Approach 432
18.11 Dirac Gas of Wilson Fermions on the Lattice 436
19．FINITE TEMPERATURE PERTURBATION THEORY OFF AND ON THE LATTICE 443
19.1 Feynman Rules For Thermal Green Functions in the λφ4Theory 443
19.2 Generation of a Dynamical Mass at T≠0 452
19.3 Perturbative Expansion of the Thermodynamical Potential 453
19.4 Feynman Rules for QED and QCD at Non-Vanishing Temperature and Chemical Potential in the Continuum 459
19.5 Temporal Structure of the Fermion Propagator at T≠0 and μ≠0 in the Continuum 464
19.6 The Electric Screening Mass in Continuum QED in One-Loop Order 467
19.7 The Electric Screening Mass in Continuum QCD in One-Loop Order 471
19.8 Lattice Feynman Rules for QED and QCD at T≠0 and μ≠0 474
19.9 Particle-Antiparticle Spectrum of the Fermion Propagator at T≠0 andμ≠0.Naive vs.Wilson Fermions 479
19.10 The Electric Screening Mass for Wilson Fermions in Lattice QED to One-Loop Order 483
19.11 The Electric Screening Mass for Wilson Fermions in Lattice QCD to One-Loop Order 491
19.12 The Infrared Problem 501
20．NON-PERTURBATIVE QCD AT FINITE TEMPERATURE 504
20.1 Thermodynamics on the Lattice 504
20.2 The Wilson Line or Polyakov Loop 509
20.3 Spontaneous Breakdown of the Center Symmetry and the Deconfinement Phase Transition 514
20.4 How to Determine the Transition Temperature 515
20.5 A Two-Dimensional Model.Test of Theoretical Concepts 517
20.6 Monte Carlo Study of the Deconfinement Phase Transition in the Pure SU（3） Gauge Theory 531
20.7 The Chiral Phase Transition 539
20.8 Some Monte Carlo Results on the High Temperature Phase of QCD 543
20.9 Some Possible Signatures for Plasma Formation 551
Appendix A 561
Appendix B 571
Appendix C 573
Appendix D 576
Appendix E 579
Appendix F 581
Appendix G 583
References 590
Index 605