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      Discrete differential calculus, graphs, topologies and gauge theory

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          Abstract

          Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic exploration of differential algebras on a given set. Associated with a differential algebra is a (di)graph where two vertices are connected by at most two (antiparallel) arrows. The interpretation of such a graph as a `Hasse diagram' determining a (locally finite) topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure. It is shown that field theories, and in particular gauge theories, can be formulated on a discrete set in close analogy with the continuum case. The framework presented generalizes ordinary lattice theory which is recovered from an oriented (hypercubic) lattice graph. It also includes, e.g., the two-point space used by Connes and Lott (and others) in models of elementary particle physics. The formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an `internal' discrete space ({\`a} la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, also a `symmetric lattice' is studied which (in a certain continuum limit) turns out to be related to a `noncommutative differential calculus' on manifolds.

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          Finitary substitute for continuous topology

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            Differential calculus and gauge theory on finite sets

            We develop differential calculus and gauge theory on a finite set G. An elegant formulation is obtained when G is supplied with a group structure and in particular for a cyclic group. Connes' two-point model (which is an essential ingredient of his reformulation of the standard model of elementary particle physics) is recovered in our approach. Reductions of the universal differential calculus to `lower-dimensional' differential calculi are considered. The `complete reduction' leads to a differential calculus on a periodic lattice which is related to q-calculus.
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              Zur Frage nach Vermeidung der unendlichen Selbstr�ckwirkung des Elektrons

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                Author and article information

                Journal
                19 April 1994
                1994-09-23
                Article
                10.1063/1.530638
                hep-th/9404112
                87ccb26e-c55a-4b7e-b2ca-6a3d630bd288
                History
                Custom metadata
                GOET-TP 1/94
                J.Math.Phys. 35 (1994) 6703-6735
                36 pages, revised version, appendix added
                hep-th gr-qc math.QA

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