Quantization of stationary Gaussian random processes and their generalizations

Manuel J. Miranda, Paolo Bocchini

Jiushu Shao

John Edward Gough

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.

Ji‐Suo Wang, Xiang‐Guo Meng, Hong-Yi Fan

By extending the usual Wigner operator to the s-parameterized one, we find that in the process of the generalized Weyl quantization the s parameter plays the role of correlation between two quadratures Q and P . This can be exposed by comparing the normally ordered form of Ω s with the standard form of the Gaussian bivariate normal distribution of random variables in statistics. Three different expressions of Ω s and the quantization scheme with use of it are presented.