Noncommutative Lattices as Finite Approximations and Their Noncommutative Geometries

Details Noncommutative Lattices as Finite Approximations and Their Noncommutative Geometries Noncommutative Lattices as Finite Approximations and Their Noncommutative Geometries

1995-10-30

arxiv.org/abs/hep-th/9510217v2

J.Geom.Phys. 18 (1996) 163-194

Physics

High Energy Physics

High Energy Physics - Theory

37 pages, 12 eps figures, uses psbox.tex To appear in J. Geom. and Physics, a problem due to the mailer hopefully solved

Scientific paper

10.1016/0393-0440(95)00006-2

Lattice discretizations of continuous manifolds are common tools used in a variety of physical contexts. Conventional discrete approximations, however, cannot capture all aspects of the original manifold, notably its topology. In this paper we discuss an approximation scheme due to Sorkin which correctly reproduces important topological aspects of continuum physics. The approximating topological spaces are partially ordered sets (posets), the partial order encoding the topology. Now, the topology of a manifold $M$ can be reconstructed from the commutative C*-algebra $\cc(M)$ of continuous functions defined on it. In turn, this algebra is generated by continuous probability densities in ordinary quantum physics on $M$. The latter also serve to specify the domains of observables like the Hamiltonian. For a poset, the role of this algebra is assumed by a noncommutative C$^*$-algebra $\ca $. This fact makes any poset a genuine `noncommutative' (`quantum') space, in the sense that the algebra of its `continuous functions' is a noncommutative C$^*$-algebra. We therefore also have a remarkable connection between finite approximations to quantum physics and noncommutative geometries. We use this connection to develop various approximation methods for doing quantum physics using $\ca$.

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