  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].
