Time-Orthogonal Unitary Dilations and Noncommutative Feynman-Kac Formulae. II

Abstract

In a previous paper [6] it was shown that a certain two-parameter dilation of a given strongly continuous self-adjoint contraction semigroup, called the time-orthogonal unitary dilation, gives rise to noncommutative Feynman-Kac formulae through the mechanism of Boson second quantisation in Fock space. This paper explores the modifications of this theory which arise firstly by using Fermion rather than Boson second quantisation, and secondly by using Boson second quantisation based on extremal universally invariant states of the CCR algebra. In the second case it is found that the programme is successful if and only if the infinitesimal generator of the original semigroup is of trace class.