Quantum diffusion on almost commutative spectral triples and spinor bundles

TL;DR

This paper explores quantum diffusion on almost commutative spectral triples by analyzing heat semigroups on spinor bundles, establishing their quantum dynamical properties and constructing associated quantum stochastic flows.

Contribution

It extends the characterization of almost commutative spectral triples to endomorphism algebras of Dirac bundles and demonstrates the quantum dynamical nature of related heat semigroups.

Findings

Heat semigroups are quantum dynamical semigroups.

Existence of covariant quantum stochastic flows on spinor bundles.

Connection between geometric Dirac operators and spectral triples.

Abstract

Based on the observation that Cacic [10]'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 by a perturbation, the question of complete positivity of the heat semigroups generated by connection laplacian and Dirac and Kostant's cubic Dirac laplacians is approached using spin geometry and C *-Dirichlet forms. The geometric heat semigroups for on endomorphosm algebras of spinor bundles are shown to be quantum dynamical semigroups and the existence of covariant quantum stochastic flows associated to the heat semigroups on spinor bundles over reductive homogeneous spaces is established using the construction of Sinha and Goswami [34].