Hilbert Space Representations of Probability Distributions

Stanley Gudder

Aubrey Wulfsohn

The article deals with non-commutative (boson) probability theory. The phase space considered is a linear symplectic space built up from a Hilbert space. Generating functionals for representations of the canonical commutation relations are related to characteristic functionals for random distributions on the underlying Hilbert space. An attempt is made to characterise those Levy-Khinchin characteristic functionals on phase space which are generating functionals by means of a generalised Heisenberg inequality.

Mathieu Beau, T. C. Dorlas

F. G. HALL, R. Collins

Beginning with an intuitive consideration of sequences of measurements, we define a time-ordered event space representing the collection of all imaginable outcomes for measurement sequences. We then postulate the generalized distributive relation on the event space and examine the class of measurements for which this relation can be experimentally validated. The generalized distributive relation is shown to lead to a σ-additive conditional probability on the event space and to a predictive and retrodictive formalism for stochastic processes. We then show that this formalism has a predictive and a retrodictive representation in a separable Hilbert space H, which has no counterpart in unitary quantum dynamics.

Forrest G. Hall

Heinz‐Peter Breuer, Francesco Petruccione

The stochastic dynamics of open quantum systems interacting with a zero temperature environment is investigated by employing a formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. It is demonstrated that the open system dynamics can consistently be described by a stochastic process on the reduced state space. The physical meaning of reduced probability distributions on projective Hilbert space is derived from a complete, orthogonal measurement of the environment. The elimination of the variables of the environment is shown to lead to a piecewise deterministic process in Hilbert space defined by a differential Chapman-Kolmogorov equation. A Hilbert space path integral representation of the stochastic process is constructed. The general theory is illustrated by means of three examples from quantum optics. For these examples the microscopic derivation of the stochastic process is given and the general solution of the differential Chapman-Kolmogorov equation is constructed by means of the path integral representation.