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

I. Bars, D. Minic

Abstract

We discuss the quantum mechanics of a particle in a magnetic field when its position x^μ is restricted to a periodic lattice, while its momentum p^μ 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.

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

Abstract

We discuss the quantum mechanics of a particle in a magnetic field when its position x^μ is restricted to a periodic lattice, while its momentum p^μ 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.

Paper Structure

This paper contains 10 sections, 65 equations.

Table of Contents

  1. Introduction and results
  2. Non-commutative geometry on the lattice
  3. Two-torus
  4. $d$- dimensions
  5. Map $\left( \hat{\Delta}\left( x\right) \right) _{J}^{J^{\prime }}$ from position lattice to matrix
  6. Diamond product
  7. Discrete non-commutative gauge theory
  8. Non-commutative U(1) gauge theory
  9. Non-commutative U$\left( M\right)$ gauge theory
  10. Outlook