Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1994-09-30
J.Geom.Phys. 18 (1996) 349-380
Physics
High Energy Physics
High Energy Physics - Theory
35p
Scientific paper
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 $*$-algebra $\cA$ which is of the form $\cA=C(I,\cAs)$ where $\cAs$ is itself a associative $*$-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.
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