Chiral symmetry on the lattice

After a brief introduction to the basic ideas of lattice gauge theories and discussion of the fermion doubling problem we study chiral symmetry for Hamiltonian lattice fermions formulated in terms of Shockley surface states bound to a wall in an extra spatial dimension.; For hadronic physics this provides a natural formulation of QCD in the chiral limit without requiring a precise tuning of parameters. We illustrate the chiral anomaly as a flow of states in the extra dimension. Two alternatives for extending this picture to a chiral coupling of gauge fields to such fermions are discussed: one with explicit breaking of gauge symmetry and one with heavy mirror fermions. In the context of gauge noninvariant formulation we describe a nonstandard definition of winding number on the lattice.