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Journal of Mathematical Physics

Quantum continuity equations and quantum evolution systems for white noise operators

Un Cig Ji

Published Jul 1, 2024• 1 citation
Publisher

Abstract

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.

DOI
10.1063/5.0176568

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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.
Based on white noise theory, we first discuss the Gaussian generalized Mehler semigroups (on white noise functionals), along with their fundamental properties: invariant measures, invariant white noise operators, and mean ergodicity. Then, utilizing canonical topological isomorphisms between the spaces of two-variable white noise functionals and the spaces of white noise operators, we formulate a quantum analogue of the Gaussian generalized Mehler semigroups and discuss their fundamental properties. By establishing the notion of the positivity of white noise operators, we introduce the concept of quantum dynamical (Markov) semigroups acting on the space of white noise operators. Finally, we discuss the Gaussianity of the quantum dynamical semigroups induced by the Gaussian generalized Mehler semigroups. The construction of the quantum dynamical semigroups leads to the general notion of quantum dynamical semigroups on generalized operators.
We study an implementation problem for quadratic functions of annihilation and creation operators on a boson field in terms of quantum white noise calculus. The implementation problem is shown to be equivalent to a linear differential equation for white noise operators containing quantum white noise derivatives. The solution is explicitly obtained and turns out to form a class of white noise operators including generalized Fourier–Gauss and Fourier–Mehler transforms, Bogoliubov transform, and a quantum extension of the Girsanov transform.
Based on the quantum white noise theory, we formulate new types of anticipating quantum stochastic integrals by combining the Hitsuda–Skorokhod quantum stochastic integrals and the interactions between the integrands and the integrators. For our purpose, we prove various versions of analytic characterization theorems of symbols of white noise operators as more general cases. Also, several types of quantum (stochastic) gradients are formulated as continuous linear operators which are related to the quantum white noise derivatives and reflecting the interactions with the external noises. Finally, we formulate further systematical study of anticipating quantum stochastic integrals.
Quantum evolution was recently shown to be equivalent to the mathematical expectation of the stochastic motion described by the Schr\"odinger equation with momentum-coupled white noise [J. Shao, J. Chem. Phys. 159, 024128 (2023)]. The stochastic formulation provides a convenient means of evaluating quantum propagators because the involved kinetic and potential contributions are disentangled. With the help of the auxiliary functional averaging technique, we apply this scheme to the derivation of the quantum propagators for systems of exponential, linear, harmonic, and $\ensuremath{\delta}$-function potentials. Notably, our work presents a truly workable expression of the quantum propagator for the exponential potential problem. To demonstrate the features and advantages of this propagator, we use it to calculate the wave-packet dynamics and compare it to the split-operator method.