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Contents

Preface

1 Introduction

1.1 Historical remarks

1.2 Path integral in quantum mechanics

1.2.1 Feynman path integral

1.2.2 Euclidean path integral

1.3 Euclidean quantum field theory

1.3.1 Scalar fields

1.3.2 Schwinger functions

1.3.3 Wick rotation

1.3.4 Free field

1.3.5 Reflection positivity

1.4 Euclidean functional integrals

1.4.1 Gaussian integrals

1.4.2 Euclidean free field

1.4.3 Functional integral for the Euclidean free field

1.4.4 Functional integral for the interacting field

1.4.5 Perturbation theory

1.5 Quantum field theory on a lattice

1.5.1 Lattice regularization

1.5.2 Transfer matrix

1.5.3 Reflection positivity

1.6 Continuum limit and critical behaviour

1.6.1 Lattice field theory and statistical systems

1.6.2 Renormalization and critical behaviour

1.6.3 Universality

1.7 Renormalization group equations

1.7.1 Renormalization group equations for the bare theory

1.7.2 Callan-Symanzik equations

1.7.3 Renormalization group equations for a massless theory

1.7.4 Fixed points

1.8 Thermodynamics of quantum fields

1.8.1 Field theory at finite physical temperature

1.8.2 Anisotropic lattice regularization

2 Scalar fields

2.1 ø4 model on the lattice

2.1.1 Green's functions

2.1.2 Particle states

2.1.3 Renormalized quantities

2.2 Perturbation theory

2.2.1 Free field theory

2.2.2 Perturbation theory in the symmetric phase

2.2.3 Perturbation theory in the phase with broken symmetry

2.3 Hopping parameter expansions

2.4 Lüscher-Weisz solution and triviality of the continuum limit

2.4.1 Triviality of four-dimensional ø4 theory

2.4.2 Infinite bare coupling limit

2.5 Finite-volume effects

2.5.1 Perturbative finite-volume effects

2.5.2 Tunneling

2.6 TV-component model

3 Gauge fields

3.1 Continuum gauge fields

3.1.1 SU(N) gauge fields

3.1.2 Abelian gauge fields

3.2 Lattice gauge fields and Wilson's action

3.2.1 Lattice gauge fields

3.2.2 Wilson's action

3.2.3 Functional integral

3.2.4 Observables

3.2.5 Gauge fixing

3.2.6 Transfer matrix

3.2.7 Group characters

3.2.8 Reflection positivity

3.2.9 Other actions

3.3 Perturbation theory

3.3.1 Feynman rules

3.3.2 Renormalization group

3.3.3 A-parameters

3.4 Strong-coupling expansion

3.4.1 High-temperature expansions

3.4.2 Strong-coupling graphs

3.4.3 Moments and cumulants

3.4.4 Cluster expansion for the free energy

3.4.5 Results from the strong-coupling expansion

3.5 Static quark potential

3.5.1 Wilson loop criterion

3.5.2 Strong-coupling expansion

3.5.3 Roughening transition

3.5.4 Numerical investigations

3.6 Glueball spectrum

3.6.1 Glueball states

3.6.2 Strong-coupling expansion

3.6.3 Monte Carlo results

3.7 Phase structure of lattice gauge theory

3.7.1 General results

3.7.2 Abelian gauge groups

3.7.3 Non-Abelian gauge groups

4 Fermion fields

4.1 Fermionic variables

4.1.1 Creation and annihilation operators

4.1.2 A simple example

4.1.3 Grassmann variables

4.1.4 Polymer representation

4.2 Wilson fermions

4.2.1 Hamiltonian formulation

4.2.2 Euclidean formulation: Wilson action

4.2.3 Reflection positivity of the Wilson action

4.2.4 Wilson fermion propagator and Green functions

4.2.5 Timeslices of Wilson fermions

4.3 Kogut-Susskind staggered fermions

4.3.1 From naive to staggered fermions

4.3.2 Flavours of staggered fermions

4.3.3 Connection to the Dirac-Kahler equation

4.4 Nielsen-Ninomiya theorem and mirror fermions

4.4.1 Doublers in lattice fermion propagators

4.4.2 Doublers and the axial anomaly

4.5 QED on the lattice

4.5.1 Lattice actions

4.5.2 Analytic results

4.5.3 Non-perturbative studies

5 Quantum chromodynamics

5.1 Lattice action and continuum limi

5.1.1 Lattice actions

5.1.2 Quenched approximation

5.1.3 Hopping parameter expansion

5.1.4 Strong gauge coupling limit

5.1.5 Lattice perturbation theory

5.1.6 Continuum limit

5.2 Hadron spectrum

5.2.1 Hadronic two-point functions

5.2.2 Hadron sources

5.2.3 Heavy quark systems

5.3 Broken chiral symmetry on the lattice

5.3.1 Ward-Takahashi identities

5.3.2 PCAC and the quark mass

5.3.3 Current algebra

5.3.4 Scalar densities

5.3.5 The U(1) problem

5.3.6 Current matrix elements

5.4 Hadron thermodynamics

5.4.1 Hadron thermodynamics on the lattice

5.4.2 Deconfinement and chiral symmetry restoration

5.4.3 Non-zero quark number density

6 Higgs and Yukawa models

6.1 Lattice Higgs models

6.1.1 Lattice actions

6.1.2 Lattice perturbation theory

6.1.3 Phase structure and symmetry restoration

6.1.4 Triviality upper bound

6.1.5 Weak gauge coupling limit

6.2 Lattice Yukawa models

6.2.1 Lattice actions

6.2.2 The Golterman-Petcher theorem

6.2.3 Numerical simulations, phase structure

6.2.4 Vacuum stability lower bound

7 Simulation algorithms

7.1 Numerical simulation and Markov processes

7.1.1 Updating processes

7.1.2 Updating with constraints

7.1.3 Error estimates

7.1.4 Improved estimators

7.2 Metropolis algorithms

7.3 Heatbath algorithms

7.3.1 Heatbath in lattice gauge theories

7.4 Fermions in numerical simulations

7.4.1 Fermion matrix inversion

7.5 Fermion algorithms based on differential equations

7.5.1 Classical dynamics algorithms

7.5.2 Langevin algorithms

7.6 Hybrid Monte Carlo algorithms

7.6.1 HMC for scalar fields

7.6.2 HMC for gauge and fermion fields

7.7 Cluster algorithms

8 Appendix

8.1 Notation conventions and basic formulas

8.1.1 Pauli matrices

8.1.2 Dirac matrices

8.1.3 Lie algebra generators

8.1.4 Continuum gauge fields

8.1.5 Lattice notations

8.1.6 Free fields

8.1.7 Reduction formulas

References

Index