Quantum diffusion on almost commutative spectral triples and spinor bundles

Abstract

Based on the observation that Ćaćić [13]'s characterization of almost commutative spectral triples as Clifford module bundles can be pushed to endomorphim algebras of Dirac bundles, with the geometric Dirac operator related to the Dirac operator of the spectral triple, the question of complete positivity of the Dirac heat semigroup is approached using spin geometry. The geometric heat semigroups for on endomorphosm algebras of spinor bundles and the almost commutative spectral triples are shown to be quantum dynamical semigroups. Such algebra bundles are also considered with the C * -bundle structure of Davies and Rothaus [25]; in this setting a version of Cipriani and Sauvageot [21]'s result on Clifford bundle to bundles defining almost commutative spectral triples is obtained. The existence of covariant quantum stochastic flows associated to the heat semigroups generated by canonical connection, Dirac, and Kostant's cubic Dirac laplacians on spinor bundles over reductive homogeneous spaces is established using the construction of Sinha and Goswami [48]. A candidate curvature operator on inductive limits of almost commutative spectral triples is offered; the operator is computed for a globally trivial Fock space bundle and the effect of potential for the spin connection is considered. * email. Acknowledgements: The author would like to thank Matilde Marcolli for useful discussions, comments and for supporting this and other projects. The author would also like to thank Juan Pablo Vigneaux for many thoughtful discussions.