This will be an abbreviated version of the course, with the
selection of topics to be made according to the wishes of the
students. The complete course is described below.
The aim of the course is to describe non-perturbative approaches to non-Abelian gauge theories. Our main physical problem is quark confinement. The lattice provides a cutoff for field theory which permits many manipulations beyond the reach of Feynman-diagram perturbation theory. We will exploit exact transformations among different field theories and use topological methods and duality to study the physics of these theories.
After introducing the lattice formulation of QCD, we will develop techniques by using simpler field theories as laboratories: the Ising model, the XY (or planar Heisenberg) model, the Ising gauge theory (due to Wegner), the Abelian gauge theory (related to QED). The problem of taking the continuum limit will bring us to the study of critical points and the renormalization group.
The lattice has also provided the easiest approach to numerical calculation in field theory. We will describe the basics of Monte Carlo methods and show how the qualitative ideas presented above find quantitative realization in these calculations.
The foremost practical problem standing in the way of the
calculation of hadron properties is also the most interesting open
how to put fermion field theories on the lattice. We will review
long-standing problem and current attempts to solve it.