Functional quantization of stationary Gaussian and non-Gaussian random processes

A. I. Oksak

Marek Bozejko, Burkhard Kümmerer, Roland Speicher

. We examine, for \\Gamma1 ! q ! 1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) X t = a t + a t -- where the a t fulfill the q-commutation relations asa t \\Gamma qa t as = c(s; t) \\Delta 1 for some covariance function c(\\Delta; \\Delta) -- equipped with the vacuum expectation state. We show that there is a q- analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possess a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB]. Introduction What we are going to call q-Gaussian processes was essentially introduced in a remarkable paper by Frisch ...

Hiroshi Nakazawa

The structure of the projection-operator theory of Mori for quantum Brownian motion is reconsidered, emphasizing the nature of underlying quantum processes. Quantum-mechanical assumptions of the theory are extracted to sets of axioms. The Hilbert space scrV formed with relevant observables for quantum generalized Langevin equations (QGLE's) includes, in general, elements that cannot be operators on state vectors of the original system. The definition of Liouville operator L, as a mapping from one operator to another, thus requires its reconstruction in scrV. The work shows that the mentioned axioms not only ensure a unified reconstruction of L (hence of QGLE's) in systems in stationary states that may be nonequilibrium as well as equilibrium, but also extend intrinsically the domain of applicability of QGLE's. Illustrations of the setting for these extended QGLE's are presented by concrete examples of Hilbert spaces formed with quantum-mechanical Gaussian and non-Gaussian random variables, showing specifically the significance of a response-function-type inner product that assigns \ensuremath{\delta} correlation to the quantized Gaussian white noise which is an operator-valued distributional process with Kubo-Martin-Schwinger property.

Kunio Yasue

Don H. Johnson, P.S.V. Subba Rao

The dependence structure and the amplitude distribution of stationary random sequences are linked, with specification of one placing constraints on the other. Time-reversible processes can be Gaussian or non-Gaussian, but all Gaussian processes must be time reversible. We examine the thermodynamics of measurement, showing that while ‘‘information” can be extracted from a system without altering system entropy, mat measurement techniques irreversibly alter thermodynamic state with a consequent entropy increase. Because of the second law of thermodynamics, such entropy changes cannot be undone and measurements reflecting thermodynamic state cannot be time reversible. We conclude that physical measurements are not time reversible, implying that onZy non-time-reversible processes model physically relevant signals. Consequently, Gaussian processes would seem to be imprecise representations of physical measurements.