Lattice Gauge Theories

The following sections are included:

• DEDICATION

• PREFACE

• PREFACE TO THE FIRST EDITION

• CONTENTS

It is generally accepted that quantum field theory is the appropriate framework for describing the strong, electromagnetic and weak interactions between elementary particles. As for the electromagnetic interactions, it has been known for a long time that they are described by a quantum gauge field theory. But that the principle of gauge invariance also plays a fundamental role in the construction of a theory for the strong and weak interactions has been recognized only much later. The unification of the weak and electromagnetic interactions by Glashow, Salam and Weinberg was a major breakthrough in our understanding of elementary particle physics. For the first time one had been able to construct a renormalizable quantum field theory describing simultaneously the weak and electromagnetic interactions of hadrons and leptons. The “electro-weak” theory of Glashow, Salam and Weinberg is based on a non-abelian SU(2) × U(1) gauge symmetry, which is broken down spontaneously to the U(1) symmetry of the electromagnetic interactions. This breaking manifests itself in the fact that, in contrast to the massless photon, the particles mediating the weak interactions, i.e., the W⁺, W⁻ and Z⁰ vector bosons, become massive. In fact they are very massive, which reflects the fact that the weak interactions are very short ranged. The detection of these particles constituted one of the most beautiful tests of the Glashow-Salam-Weinberg theory…

The following sections are included:

• The Path Integral Method in Quantum Mechanics

• Path Integral Representation of Bosonic Green Functions in Field Theory

• The Transfer Matrix

• Path Integral Representation of Fermionic Green Functions

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

Consider the classical field equation

The following sections are included:

• The Doubling Problem

• A Closer Look at Fermion Doubling

• Wilson Fermions

• Staggered Fermions

• Technical Details of the Staggered Fermion Formulation

• Staggered Fermions in Momentum Space

The following sections are included:

• Preliminaries

• Lattice Formulation of QED

The lattice gauge theory we discussed in chapter 5 can be easily extended to the case where the abelian group U(1) is replaced by a non-abelian unitary group. Thus suppose that instead of a single free Dirac field we have N such fields ψ^a (a = 1, …, N) of mass M₀…

The following sections are included:

• A Look at Non-Relativistic Quantum Mechanics

• The Wilson Loop and the Static -Potential in QED

• The Wilson Loop in QCD

The following sections are included:

• The Potential in Quenched QED

• The Potential in Quenched Compact QED₂

The following sections are included:

• Critical Behaviour of Lattice QCD and the Continuum Limit

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

The following sections are included:

• Energy Sum Rule for the Harmonic Oscillator

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

• Sum Rules for the Static -Potential

• Sum Rules for the Glueball Mass

The following sections are included:

• The -Potential to Leading Order in Strong Coupling

• Beyond the Leading Approximation

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

The following sections are included:

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

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

• Hopping Parameter Expansion of the Effective Action

• The HPE and the Pauli Exclusion Principle

The following sections are included:

• Introduction

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

• The Power Counting Theorem of Reisz

• Theorem

The following sections are included:

• The Gauge Fixed Lattice Action

• Lattice Feynman Rules

The following sections are included:

• The Link Integration Measure

• Gauge Fixing and the Faddeev-Popov Determinant

• The Gauge Field Action

• Propagators and Vertices

• Relation between Λ_L and the Λ-Parameter of Continuum QCD

The following sections are included:

• Introduction

• Construction Principles for Algorithms. Markov Chains

• The Metropolis Method

• The Langevin Algorithm

• The Molecular Dynamics Method

• The Hybrid Algorithm

• The Hybrid Monte Carlo Algorithm

• The Pseudofermion Method

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

The following sections are included:

• The String Tension and the -Potential in the SU(3) Gauge Theory

• The -Potential in Full QCD

• Chiral Symmetry Breaking

• Glueballs

• Hadron Mass Spectrum

• The Dual Superconductor Picture of Confinement

The following sections are included:

• Introduction

• Path Integral Representation of the Partition Function in Quantum Mechanics

• Sum Rule for the Mean Energy

• Test of the Energy Sum Rule. The Harmonic Oscillator

• The Free Relativistic Boson Gas in the Path Integral Approach

• The Photon Gas in the Path Integral Approach

• Functional Methods for Fermions. Basics

• Path Integral Representation of the Partition Function for a Fermionic System Valid for Arbitrary Time-Step

• A Modified Fermion Action Leading to Fermion Doubling

• The Free Dirac Gas. Continuum Approach

• Dirac Gas of Wilson Fermions on the Lattice

The following sections are included:

• Feynman Rules For Thermal Green Functions in the λϕ⁴ Theory

• Generation of a Dynamical Mass at T ≠ 0

• Perturbative Expansion of the Thermodynamical Potential

• Feynman Rules for QED and QCD at non-vanishing Temperature and Chemical Potential in the Continuum

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

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

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

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

• Particle-Antiparticle Spectrum of the Fermion Propagator at T ≠ 0 and μ ≠ 0. Naive vs. Wilson Fermions

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

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

• The Infrared Problem

The following sections are included:

• Thermodynamics on the Lattice

• The Wilson Line or Polyakov Loop

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

• How to Determine the Transition Temperature

• A Two-Dimensional Model. Test of Theoretical Concepts

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

• The Chiral Phase Transition

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

• Some Possible Signatures for Plasma Formation

The following sections are included:

• APPENDIX A

• APPENDIX B

• APPENDIX C

• APPENDIX D

• APPENDIX E: The Electric Screening Mass for Naive Fermions in Lattice QED to One-Loop Order

• REFERENCES

• INDEX