1．INTRODUCTION

2．THE PATH INTEGRAL APPROACH TO QUANTIZATION

2.1 The Path Integral Method in Quantum Mechanics

2.2 Path Integral Representation of Bosonic Green Functions in Field theory

2.3 The Transfer Matrix

2.4 Path Integral Representation of Fermionic Green Functions

2.5 Discretizing Space-Time.The Lattice as a Regulator of a Quantum Field Theory

3．THE FREE SCALAR FIELD ON THE LATTICE

4．FERMIONS ON THE LATTICE

4.1 The Doubling Problem

4.2 A Closer Look at Fermion Doubling

4.3 Wilson Fermions

4.4 Staggered Fermions

4.5 Technical Details of the Staggered Fermion Formulation

4.6 Staggered Fermions in Momentum Space

4.7 Ginsparg-Wilson Fermions.The Overlap Operator

5．ABELIAN GAUGE FIELDS ON THE LATTICE AND COMPACT QED

5.1 Preliminaries

5.2 Lattice Formulation of QED

6．NON ABELIAN GAUGE FIELDS ON THE LATTICE COMPACT QCD

7．THE WILSON LOOP AND THE STATIC QUARK-ANTIQUARK POTENTIAL

7.1 A Look at Non-Relativistic Quantum Mechanics

7.2 The Wilson Loop and the Static q？-Potential in QED

7.3 The Wilson Loop in QCD

8．THE Q？-POTENTIAL IN SOME SIMPLE MODELS

8.1 The Potential in Quenched QED

8.2 The Potential in Quenched Compact QED2

9．THE CONTINUUM LIMIT OF LATTICE QCD

9.1 Critical Behaviour of Lattice QCD and the Continuum Limit

9.2 Dependence of the Coupling Constant on the Lattice Spacing and the Renormalization Group β-Function

10．LATTICE SUM RULES

10.1 Energy Sum Rule for the Harmonic Oscillator

10.2 The SU（N） Gauge Action on an Anisotropic Lattice

10.3 Sum Rules for the Static q？-Potential

10.4 Determination of the Electric,Magnetic and Anomalous Contribution to the ？Potential

10.5 Sum Rules for the Glueball Mass

11．THE STRONG COUPLING EXPANSION

11.1 The q？-Potential to Leading Order in Strong Coupling

11.2 Beyond the Leading Approximation

11.3 The Lattice Hamiltonian in the Strong Coupling Limit and the String Picture of Confinement

12．THE HOPPING PARAMETER EXPANSION

12.1 Path Integral Representation of Correlation Functions in Terms of Bosonic Variables

12.2 Hopping Parameter Expansion of the Fermion Propagator in an External Field

12.3 Hopping Parameter Expansion of the Effective Action

12.4 The HPE and the Pauli Exclusion Principle

13．WEAK COUPLING EXPANSION（Ⅰ）.THE Ф3-THEORY

13.1 Introduction

13.2 Weak Coupling Expansion of Correlation Functions in the φ3-Theory

13.3 The Power Counting Theorem of Reisz

14．WEAK COUPLING EXPANSION（Ⅱ）.LATTICE QED

14.1 The Gauge Fixed Lattice Action

14.2 Lattice Feynman Rules

14.3 Renormalization of the Axial Vector Current in One-Loop Order

14.4 The ABJ Anomaly

15．WEAK COUPLING EXPANSION（Ⅲ）.LATTICE QCD

15.1 The Link Integration Measure

15.2 Gauge Fixing and the Faddeev-Popov Determinant

15.3 The Gauge Field Action

15.4 Propagators and Vertices

15.5 Relation between ∧L and the ∧-Parameter of Continuum QCD

15.6 Universality of the Axial Anomaly in Lattice QCD

16．MONTE CARLO METHODS

16.1 Introduction

16.2 Construction Principles for Algorithms.Markov Chains

16.3 The Metropolis Method

16.4 The Langevin Algorithm

16.5 The Moleeular Dynamics Method

16.6 The Hybrid Algorithm

16.7 The Hybrid Monte Carlo Algorithm

16.8 The Pseudofermion Method

16.9 Application of the Hybrid Monte Carlo Algorithm to Systems with Fermions

17．SOME RESULTS OF MONTE CARLO CALCULATIONS

17.1 The String Tension and the q？ Potential in the SU（3）Gauge Theory

17.2 The q？-Potential in Full QCD

17.3 Chiral Symmetry Breaking

17.4 Glueballs

17.5 Hadron Mass Spectrum

17.6 Instantons

17.7 Flux Tubes in q？and qqq-Systems

17.8 The Dual Superconductor Picture of Confinement

17.9 Center Vortices and Confinement

17.10 Calorons

18．PATH-INTEGRAL REPRESENTATON OF THE THERMODYNAMICAL PARTITION FUNCTION FOR SOME SOLVABLE BOSONIC AND FERMIONIC SYSTEMS

18.1 Introduction

18.2 Path-Integral Representation of the Partition Function in Quantum Mechanics

18.3 Sum Rule for the Mean Energy

18.4 Test of the Energy Sum Rule.The Harmonic Oscillator

18.5 The Free Relativistic Boson Gas in the Path Integral Approach

18.6 The Photon Gas in the Path Integral Approach

18.7 Functional Methods for Fermions.Basics

18.8 Path Integral Representation of the Partition Function for a Fermionic System valid for Arbitrary Time-Step

18.9 A Modified Fermion Action Leading to Fermion Doubling

18.10 The Free Dirac Gas.Continuum Approach

18.11 Dirac Gas of Wilson Fermions on the Lattice

19．FINITE TEMPERATURE PERTURBATION THEORY OFF AND ON THE LATTICE

19.1 Feynman Rules For Thermal Green Functions in the λφ4Theory

19.2 Generation of a Dynamical Mass at T≠0

19.3 Perturbative Expansion of the Thermodynamical Potential

19.4 Feynman Rules for QED and QCD at Non-Vanishing Temperature and Chemical Potential in the Continuum

19.5 Temporal Structure of the Fermion Propagator at T≠0 and μ≠0 in the Continuum

19.6 The Electric Screening Mass in Continuum QED in One-Loop Order

19.7 The Electric Screening Mass in Continuum QCD in One-Loop Order

19.8 Lattice Feynman Rules for QED and QCD at T≠0 and μ≠0

19.9 Particle-Antiparticle Spectrum of the Fermion Propagator at T≠0 andμ≠0.Naive vs.Wilson Fermions

19.10 The Electric Screening Mass for Wilson Fermions in Lattice QED to One-Loop Order

19.11 The Electric Screening Mass for Wilson Fermions in Lattice QCD to One-Loop Order

19.12 The Infrared Problem

20．NON-PERTURBATIVE QCD AT FINITE TEMPERATURE

20.1 Thermodynamics on the Lattice

20.2 The Wilson Line or Polyakov Loop

20.3 Spontaneous Breakdown of the Center Symmetry and the Deconfinement Phase Transition

20.4 How to Determine the Transition Temperature

20.5 A Two-Dimensional Model.Test of Theoretical Concepts

20.6 Monte Carlo Study of the Deconfinement Phase Transition in the Pure SU（3） Gauge Theory

20.7 The Chiral Phase Transition

20.8 Some Monte Carlo Results on the High Temperature Phase of QCD

20.9 Some Possible Signatures for Plasma Formation

Appendix A

Appendix B

Appendix C

Appendix D

Appendix E

Appendix F

Appendix G

References

Index