Hamilton Formalism in Non-Commutative Geometry

Abstract

We study the Hamilton formalism for Connes-Lott models, i.e., for Yang-Mills theory in non-commutative geometry. The starting point is an associative $\ast$-algebra $\cA$ which is of the form $\cA=C(I,\cAs)$ where $\cAs$ is itself a associative $\ast$-algebra. With an appropriate choice of a k-cycle over $\cA$ it is possible to identify the time-like part of the generalized differential algebra constructed out of $\cA$. We define the non-commutative analogue of integration on space-like surfaces via the Dixmier trace restricted to the representation of the space-like part $\cAs$ of the algebra. Due to this restriction it possible to define the Lagrange function resp. Hamilton function also for Minkowskian space-time. We identify the phase-space and give a definition of the Poisson bracket for Yang-Mills theory in non-commutative geometry. This general formalism is applied to a model on a two-point space and to a model on Minkowski space-time $\times$ two-point space.

35p

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Cited by in corpus (10)

• An Introduction to Noncommutative Spaces and their Geometry
• The Standard Model as a noncommutative geometry: the low energy regime
• Dixmier traces and some applications to noncommutative geometry
• Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes
• Gravity from Dirac Eigenvalues
• Observables of the Euclidean Supergravity
• Euclidean Supergravity in Terms of Dirac Eigenvalues
• Local and Global Properties of the World
• Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras
• Metrics of nonpositive curvature on graph-manifolds and electromagnetic fields on graphs