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arXiv · hep-th/9910091

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

I. Bars·D. Minic
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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.

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I. Bars, D. Minic. 2000-01-05. Non-Commutative Geometry on a Discrete Periodic Lattice and Gauge Theory. https://doi.org/10.1103/physrevd.62.105018

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