1 The path integral on the lattice
1.1 Hilbert space and propagation in Euclidean time
1.1.1 Hilbert spaces
1.1.2 Remarks on Hilbert spaces in particle physics
1.1.3 Euclidean correlators
1.2 The path integral for a quantum mechanical system
1.3 The path integral for a scalar field theory
1.3.1 The Klein-Gordon field
1.3.2 Lattice regularization of the Klein-Gordon Hamiltonian
1.3.3 The Euclidean time transporter for the free case
1.3.4 Treating the interaction term with the Trotter formula.
1.3.5 Path integral representation for the partition function..
1.3.6 Including operators in the path integral
1.4 Quantization with the path integral
1.4.1 Different discretizations of the Euclidean action
1.4.2 The path integral as a quantization prescription
1.4.3 The relation to statistical mechanics
References
2 QCD on the lattice - a first look
2.1 The QCD action in the continuum
2.1.1 Quark and gluon fields
2.1.2 The fermionic part of the QCD action
2.1.3 Gauge invaxiance of the fermion action
2.1.4 The gluon action
2.1.5 Color components of the gauge field
2.2 Naive discretization of fermions
2.2.1 Discretization of free fermions
2.2.2 Introduction of the gauge fields as link variables
2.2.3 Relating the link variables to the continuum gauge fields
2.3 The Wilson gauge action
2.3.1 Gauge-invariant objects built with link variables
2.3.2 The gauge action
2.4 Formal expression for the QCD lattice path integral
2.4.1 The QCD lattice path integral
References
3 Pure gauge theory on the lattice
3.1 Haar measure
3.1.1 Gauge field measure and gauge invariance
3.1.2 Group integration measure
3.1.3 A few integrals for SU(3)
3.2 Gauge invariance and gauge fixing
3.2.1 Maximal trees
3.2.2 Other gauges
3.2.3 Gauge invariance of observables
3.3 Wilson and Polyakov loops
3.3.1 Definition of the Wilson loop
3.3.2 Temporal gauge
3.3.3 Physical interpretation of the Wilson loop
3.3.4 Wilson line and the quark-antiquark pair
3.3.5 Polyakov loop
3.4 The static quark potential
3.4.1 Strong coupling expansion of the Wilson loop
3.4.2 The Coulomb part of the static quark potential
3.4.3 Physical implications of the static QCD potential
3.5 Setting the scale with the static potential
3.5.1 Discussion of numerical data for the static potential .
3.5.2 The Sommer parameter and the lattice spacing
3.5.3 Renormalization group and the running coupling
3.5.4 The true continuum limit
3.6 Lattice gauge theory with other gauge groups
References
4 Numerical simulation of pure gauge theory
4.1 The Monte Carlo method
4.1.1 Simple sampling and importance sampling
4.1.2 Markov chains
4.1.3 Metropolis algorithm - general idea
4.1.4 Metropolis algorithm for Wilson's gauge action
4.2 Implementation of Monte Carlo algorithms for SU(3)
4.2.1 Representation of the link variables
4.2.2 Boundary conditions
4.2.3 Generating a candidate link for the Metropolis update
4.2.4 A few remarks on random numbers
4.3 More Monte Carlo algorithms
5 Fermions on the lattice
6 Hadron spectroscopy
7 Chiral symmetry on the lattice
8 Dynamical fermions
9 Symanzik improvement and RG actions
10 More about lattice fermions
11 Hadron structure
12 Temperature and chemical potential
A Appendix
Index
