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Infinite Dimensional Analysis Quantum Probability and Related Topics

WHITE NOISE HEISENBERG EVOLUTION AND EVANS–HUDSON FLOWS

Luigi Accardi, Wided Ayed, Habib Ouerdiane

Published Mar 1, 2007• 4 citations
Publisher

Abstract

We study white noise Heisenberg equations giving rise to flows which are *-automorphisms of the observable algebra, but not necessarily inner automorphisms. We prove that the causally normally ordered form of these white noise Heisenberg equations are equivalent to Evans–Hudson flows. This gives in particular, the microscopic structure of the maps defining these flows, in terms of the original white noise derivations.

DOI
10.1142/s0219025707002658

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We study white noise Heisenberg equations giving rise to flows which are *-automorphisms of the observable algebra, but not necessarily inner automorphisms. We prove that the causally normally ordered form of these white noise Heisenberg equations are equivalent to Evans–Hudson flows. This gives in particular, the microscopic structure of the maps defining these flows, in terms of the original white noise derivations.
The main result of this paper is to extend to Hilbert module level the proof of the inclusion of (non-Hamiltonian) stochastic differential equations based on free noise into the class of Hamiltonian equations driven by free white noise. To achieve this goal, free white noise calculus is extended to a trivial Hilbert module. The white noise formulation of the Ito table is radically different from the usual Itô tables, both classical and quantum and, combined with the Accardi–Boukas approach to Ito algebra, allows to drastically simplify calculations. Infinitesimal generators of Hilbert module free flows are characterized in terms of stochastic derivations from an initial algebra into a white noise Itô algebra. We prove that any such derivation is the difference of a ⋆-homomorphism and a trivial embedding.
The Hudson–Parthasarathy equation and the Itô–Schrödinger equation (known also as the Belavkin equation) describe a Markov approximation of the dynamics of open quantum systems. The former is a stochastic version of the classical Heisenberg equation, while the latter is a stochastic version of the classical Schrödinger equation (but this analogy is not complete). Two versions of stochastic Heisenberg equations are considered, one of which uses a white noise operator constructed from (non–self-adjoint) birth and death operators and the other uses a white noise operator constructed from (self-adjoint) coordinate and momentum operators.
Main purpose of this paper is to formulate canonical quantum continuity equations for white noise operators and to find their explicit unique solutions. By applying the continuity equations for white noise functionals and canonical topological isomorphisms between the spaces of white noise operators and the spaces of two-variable white noise functionals, we formulate canonical quantum continuity equations for white noise operators, and then we induce time dependent quantum evolution systems which are equivalent to the quantum continuity equations. Then we find explicit forms, in terms of the quantum analogues of generalized Fourier-Mehler transforms, of the unique solutions of the quantum continuity equations by solving the quantum evolution systems for white noise operators.