Time-orthogonal unitary dilations and noncommutative Feynman-Kac formulae

An analysis of Feynman-Kac formulae reveals that, typically, the unperturbed semigroup is expressed as the expectation of a random unitary evolution and the perturbed semigroup is the expectation of a perturbation of this evolution in which the latter perturbation is effected by a cocycle with certain covariance properties with respect to the group of translations and reflections of the line. We consider generalisations of the classical commutative formalism in which the probabilistic properties are described in terms of non-commutative probability theory based on von Neumann algebras. Examples of this type are generated, by means of second quantisation, from a unitary dilation of a given self-adjoint contraction semigroup, called the time orthogonal unitary dilation, whose key feature is that the dilation operators corresponding to disjoint time intervals act nontrivially only in mutually orthogonal supplementary Hilbert spaces.Parts of this work were carried out when the first author was a participant in the Sonderforschungsbereich 123 (Stochastische Mathematische Modelle) of the Deutsche-Forschungsgemeinschaft and when the third author was Visiting Fellow at the University of Warwick