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

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

1999-10-12

arxiv.org/abs/hep-th/9910091v3

Phys.Rev. D62 (2000) 105018

Physics

High Energy Physics

High Energy Physics - Theory

latex, 20 pages, new paragraph added at end of section 2.1

Scientific paper

10.1103/PhysRevD.62.105018

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.

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