%0 Journal Article
%T Non-Commutative Geometry on a Discrete Periodic Lattice and Gauge Theory
%A I. Bars
%A D. Minic
%J Physics
%D 1999
%I arXiv
%R 10.1103/PhysRevD.62.105018
%X We discuss the quantum mechanics of a particle in a magnetic field when its position x^{\mu} is restricted to a periodic lattice, while its momentum p^{\mu} is restricted to a periodic dual lattice. Through these considerations we define non-commutative geometry on the lattice. This leads to a deformation of the algebra of functions on the lattice, such that their product involves a ``diamond'' product, which becomes the star product in the continuum limit. We apply these results to construct non-commutative U(1) and U(M) gauge theories, and show that they are equivalent to a pure U(NM) matrix theory, where N^{2} is the number of lattice points.
%U http://arxiv.org/abs/hep-th/9910091v3