Wilson--Yukawa Chiral Model on Lattice and Non-commutative Geometry

Based upon the mathematical formulas of Lattice gauge theory and non-commutative geometry differential calculus, we developed an approach of generalized gauge theory on a product of the spacetime lattice and the two discrete points(or a $Z_2$ discrete group). We introduce a differentiation for non-nearest-neighbour points and find that this differentiation may lead to the introduction of Wilson term in the free fermion Lagrangian on lattice. The Wilson-Yukawa chiral model on lattice is constructed by the generalized gauge theory and a toy model and Smit-Swift model are studied.