We introduce a finite dimensional matrix model approximation to the algebra of functions on a disc based on noncommutative geometry. The algebra is a subalgebra of the one characterizing the noncommutative plane with a * product and depends on two parameters N and θ. It is composed of functions which decay exponentially outside a disc. In the limit in which the size of the matrices goes to infinity and the noncommutativity parameter goes to zero the disc becomes sharper. We introduce a Laplacian defined on the whole algebra and calculate its eigenvalues. We also calculate the two-points correlation function for a free massless theory (Green's function). In both cases the agreement with the exact result on the disc is very good already for relatively small matrices. This opens up the possibility for the study of field theories on the disc with nonperturbative methods. The model contains edge states, a fact studied in a similar matrix model independently introduced by Balachandran, Gupta and Kürkçüoǧlu.
A detailed description of the infinite-dimensional Lie algebra of ⋆-gauge transformations in noncommutative Yang-Mills theory is presented. Various descriptions of this algebra are given in terms of inner automorphisms of the underlying deformed algebra of functions on spacetime, of deformed symplectic diffeomorphisms, of the infinite unitary Lie algebra u(∞), and of the C * -algebra of compact operators on a quantum mechanical Hilbert space. The spacetime and string interpretations are also elucidated.For simplicity, we will assume in this paper that the (Euclidean) spacetime dimension d is even and that θ ij is of maximal rank. Otherwise, the algebra A θ has a non-trivial centre which we can quotient out to effectively induce a non-degenerate deformation matrix. Geometrically, this operation corresponds to the identification of an ordinary, commutative subspace of noncommutative R d .The deformation is described by the Groenewold-Moyal ⋆-product [24,25]
We introduce the notion of generalized Weyl system, and use it to define * -products which generalize the commutation relations of Lie algebras. In particular we study in a comparative way various * -products which generalize the κ-Minkowski commutation relation.
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