Lattice Gauge Fields and Noncommutative Geometry

Abstract

Conventional approaches to lattice gauge theories do not properly consider the topology of spacetime or of its fields. In this paper, we develop a formulation which tries to remedy this defect. It starts from a cubical decomposition of the supporting manifold (compactified spacetime or spatial slice) interpreting it as a finite topological approximation in the sense of Sorkin. This finite space is entirely described by the algebra of cochains with the cup product. The methods of Connes and Lott are then used to develop gauge theories on this algebra and to derive Wilson's actions for the gauge and Dirac fields therefrom which can now be given geometrical meaning. We also describe very natural candidates for the QCD theta term and Chern-Simons action suggested by this algebraic formulation. Some of these formulations are simpler than currently available alternatives. The paper treats both the functional integral and Hamiltonian approaches.

34 pages, Latex, Minor corrections, Section 6 corrected

References in corpus (12)

• Discrete differential calculus, graphs, topologies and gauge theory
• Differential calculus and gauge theory on finite sets
• TOPOLOGY CHANGE AND QUANTUM PHYSICS
• Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice
• Noncommutative Lattices as Finite Approximations and Their Noncommutative Geometries
• Dimension on Discrete Spaces
• Lattices and Their Continuum Limits
• Noncommutative Lattices and Their Continuum Limits
• Finite Approximations to Quantum Physics: Quantum Points and their Bundles
• Topologimeter and the problem of physical interpretation of topology lattice
• Noncommutative Differential Geometry and Connections on Simplicial Manifolds
• Self-Intersection Numbers and Random Surfaces on the Lattice

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• Projective Systems of Noncommutative Lattices as a Pregeometric Substratum