Noncommutative geometry on a discrete periodic lattice and gauge theory

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 $\mathrm{U}\mathrm{}\left(M\right)\mathrm{}$ gauge theories, and show that they are equivalent to a pure $\mathrm{U}\mathrm{}\left(\mathrm{NM}\right)\mathrm{}$ matrix theory, where $N^2$ is the number of lattice points.