Communications in Mathematical Physics

, Volume 83, Issue 2, pp 261–280 | Cite as

Time-orthogonal unitary dilations and noncommutative Feynman-Kac formulae

  • R. L. Hudson
  • P. D. F. Ion
  • K. R. Parthasarathy


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.


Neural Network Covariance Hilbert Space Nonlinear Dynamics Quantum Computing 
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Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • R. L. Hudson
    • 1
  • P. D. F. Ion
    • 2
  • K. R. Parthasarathy
    • 3
  1. 1.Mathematics DepartmentUniversity of NottinghamUniversity ParkEngland
  2. 2.Mathematical ReviewsUniversity of MichiganAnn ArborUSA
  3. 3.Indian Statistical InstituteNew DehliIndia

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