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Infinite Dimensional Analysis Quantum Probability and Related Topics

Anticipating quantum stochastic integrals

Un Cig Ji

Published May 23, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0219025724400071

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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.
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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.
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.
Abstract Quantum stochastic differential equations are shown to naturally arise because of the mathematical background of the correctness of the Schrödinger equation, which describes the interaction of an open quantum system with broadband quantized fields in the Markov approximation. The main classical and quantum stochastic processes appear in the notation in the Markov approximation of the operator of the electric broadband quantized field strength and the effective Hamiltonian of the interaction of a quantum system with it obtained by the methods of algebraic resonance perturbation theory. All basic stochastic quantum processes, including the counting and two-quantum Lévy processes, are constructed based on quantum white noise, which differs significantly from the classical case. At the same time, the apparatus of stochastic differential equations demonstrates consistency and parallelism.