Non-Commutative Geometry on a Discrete Periodic Lattice and Gauge Theory
arXiv:hep-th/9910091 · doi:10.1103/PhysRevD.62.105018
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.
arXiv commentlatex, 20 pages, new paragraph added at end of section 2.1
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