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Journal of the Korean Mathematical Society

Stochastic Mehler kernels via oscillatory path integrals

Aubrey Truman, Tomasz Zastawniak

Published Jan 1, 2001• 4 citations
OpenAlex

Abstract

Abstract. The configuration space and phase space oscillatory path integrals are computed in the case of the stochastic Schrödinger equation for the harmonic oscillator with a stochastic term of the form (Kψt)(x) ◦ dWt, where K is either the position operator or the momentum operator, and Wt is the Wiener process. In this way formulae are derived for the stochastic analogues of the Mehler kernel. 1.

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arXiv:quant-ph/9703031•Mar 18, 1997•0 citations
Abstract. The configuration space and phase space oscillatory path integrals are computed in the case of the stochastic Schrödinger equation for the harmonic oscillator with a stochastic term of the form (Kψt)(x) ◦ dWt, where K is either the position operator or the momentum operator, and Wt is the Wiener process. In this way formulae are derived for the stochastic analogues of the Mehler kernel. 1.
In this paper we prove a path integral representation in configuration space for the solution of a stochastic Schrödinger equation driven by a continuous semimartingale. We follow the approach of path integration established by Itô Albeverio and Høegh-Krohn. The equation we consider is the classical Schrödinger equation plus the stochastic term K(t, x)Ψt(x) o dMt, where Mt is a continuous semimartingale. As an application of our result we solve the equation for a stochastic quantum harmonic oscillator. We solve the path integral representation twice, first by discretising the path integral and also by a change of variable. We finally give the solution to the corresponding stochastic heat equation.
We develop a stochastic integration theory with respect to creation, annihilation and gauge operators on the full Fock space. This is done by using a kernel representation for a large class of bounded operators on the full Fock space. It is shown that the kernels form a Banach algebra. Having established the definition of processes and stochastic integrals we go on to prove an Ito formula and use this for examining stochastic evolutions and constructing dilations of special completely positive semigroups. Explicit solutions of the corresponding stochastic differential equations are given.
A class of functions ℋnF was chosen, which are boundary values of holomorphic functions. It was shown that if the potential V∈ℋnF and the initial wave function ψ∈ℋnF then the solution of the Schrödinger equation is ψt∈ℋnF. The unitary evolution ψ → ψt can be expressed by an integral with respect to the Wiener measure. To each ψt∈ℋnF is associated a complex Markov process qt fulfilling stochastic Hamilton equations. The time evolution of ψ as well as multitime correlation functions of arbitrary observables in quantum mechanics take the form of expectation values with respect to qt. Some statistical and ergodic aspects of quantum mechanics resulting from the stochastic interpretation of Feynman’s sum over trajectories are discussed.
A generalized Feynman-Kac formula based on the Wiener measure is presented. Within the setting of a quantum particle in an electromagnetic field it yields the standard Feynman-Kac formula for the corresponding Schrödinger semigroup. In this case rigorous criteria for its validity are compiled. Finally, phase-space path-integral representations for more general quantum Hamiltonians are derived. These representations rely on a generalized Lie-Trotter formula which takes care of the operator-ordering multiplicity, but in general is not related to a path measure.