Non - Commutative Geometry on a Discrete Periodic Lattice and Gauge Theory ∗
نویسنده
چکیده
We discuss the quantum mechanics of a particle in a magnetic field when its position x is restricted to a periodic lattice, while its momentum p 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 is the number of lattice points. Research partially supported by the U.S. Department of Energy under grant number DE-FG03-84ER40168. e-mail: [email protected], [email protected] 1
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