Non-Commutative Geometry on a Discrete Periodic Lattice and Gauge Theory

TL;DR

This paper develops a framework for non-commutative geometry on a discrete periodic lattice and constructs corresponding gauge theories, revealing their equivalence to matrix models.

Contribution

It introduces a non-commutative geometric structure on a lattice and formulates gauge theories that correspond to matrix models, bridging discrete geometry and gauge theory.

Findings

Defined non-commutative geometry on a lattice using a diamond product.

Constructed non-commutative U(1) and U(M) gauge theories.

Showed equivalence to pure U(NM) matrix theory.

Abstract

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.