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arXiv:quant-ph/9505012 · Journal of Mathematical Physics

Feynman–Kac kernels in Markovian representations of the Schrödinger interpolating dynamics

Piotr Garbaczewski, Robert Olkiewicz

Published Feb 1, 1996• 28 citations
arXivHTMLPDF

Abstract

Probabilistic solutions of the so-called Schrödinger boundary data problem provide for a unique Markovian interpolation between any two strictly positive probability densities designed to form the input–output statistics data for the process taking place in a finite-time interval. The key issue is to select the jointly continuous in all variables positive Feynman–Kac kernel, appropriate for the phenomenological (physical) situation. We extend the existing formulations of the problem to cases when the kernel is not a fundamental solution of a parabolic equation, and prove the existence of a continuous Markovian interpolation in this case. Next, we analyze the compatibility of this stochastic evolution with the original parabolic dynamics, which is assumed to be governed by the temporally adjoint pair of (parabolic) partial differential equations, and prove that the pertinent random motion is a diffusion process. In particular, in conjunction with Born’s statistical interpretation postulate in quantum theory, we consider stochastic processes which are compatible with the Schrödinger picture quantum evolution.

DOI
10.1063/1.531412

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Probabilistic solutions of the so-called Schrödinger boundary data problem provide for a unique Markovian interpolation between any two strictly positive probability densities designed to form the input–output statistics data for the process taking place in a finite-time interval. The key issue is to select the jointly continuous in all variables positive Feynman–Kac kernel, appropriate for the phenomenological (physical) situation. We extend the existing formulations of the problem to cases when the kernel is not a fundamental solution of a parabolic equation, and prove the existence of a continuous Markovian interpolation in this case. Next, we analyze the compatibility of this stochastic evolution with the original parabolic dynamics, which is assumed to be governed by the temporally adjoint pair of (parabolic) partial differential equations, and prove that the pertinent random motion is a diffusion process. In particular, in conjunction with Born’s statistical interpretation postulate in quantum theory, we consider stochastic processes which are compatible with the Schrödinger picture quantum evolution.
Abstract: Probablistic solutions of the so called Schrödinger boundary data problem provide for a unique Markovian interpolation between any two strictly positive probability densities designed to form the input-output statistics data for a certain dynamical process taking place in a finite-time interval. The key problem is to select the jointly continuous in all variables positive semigroup kernel, appropriate for the phenomenological (physical) situation. The issue of deriving a microscopic dynamics from the input-output statistics data (analyzed in terms of densities) was addressed, as the Schrödinger problem of a probabilistic interpolation, in a number of publications [1] – [6]. We shall consider Markovian propagation scenarios so remaining within the well established framework, where for any two Borel sets A, B ⊂ R on which the strictly positive boundary densities ρ(x, 0) and ρ(x, T) are defined, the transition probability m(A, B) from the set A to the set B in the time interval T> 0 has a bi-variate density given in a specific factorized form: m(x, y) = f(x)k(x, 0, y, T)g(y), with marginals: ∫
The existing formulations of the Schrödinger interpolating dynamics, which is constrained by the prescribed input-output statistics data, utilize strictly positive Feynman-Kac kernels. This implies that the related Markov diffusion processes admit vanishing probability densities only at the boundaries of the spatial volume confining the process. We extend the framework to encompass singular potentials and associated nonnegative Feynman-Kac-type kernels. It allows to deal with general nonnegative solutions of the Schrödinger boundary data problem. The resulting stochastic processes are capable of both developing and destroying nodes (zeros) of probability densities in the course of their evolution.
We invert the well-developed strategy of studying dynamics in terms of probability densities and investigate the problem of the most likely microscopic propagation scenario, which is consistent with the given priori (possibly phenomenological) input-output statistics data for the process taking place in a finite-time interval. A solution of this so-called Schr\"odinger problem is known to provide an adequate probabilistic framework for the measure preserving dynamics which is Markovian. We pay particular attention to the subclass of nonstationary solutions, determined by unitary-wave-packet evolution (Schr\"odinger wave mechanics). The existence of the pertinent Markovian diffusion is known on general grounds, but no explicit demonstration (through detailed computational arguments) until now was available even in the simplest cases. We give a definitive probabilistic description of the free quantum dynamics as a stochastic process solving Schr\"odinger's interpolation problem. The Markov diffusion arised as a particular case singled out by a suitable Feynman-Kac semigroup.
The main purpose of the paper is an essentially probabilistic analysis of relativistic quantum mechanics. It is based on the assumption that whenever probability distributions arise, there exists a stochastic process that is either responsible for temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schrödinger problem of probabilistic evolution, which provides for a unique Markov stochastic interpolation between any given pair of boundary probability densities for a process covering a fixed, finite duration of time, provided we have decided a priori what kind of primordial dynamical semigroup transition mechanism is involved. In the nonrelativistic theory, including quantum mechanics, Feyman-Kac-like kernels are the building blocks for suitable transition probability densities of the process. In the standard "free" case (Feynman-Kac potential equal to zero) the familiar Wiener noise is recovered. In the framework of the Schrödinger problem, the "free noise" can also be extended to any infinitely divisible probability law, as covered by the Lévy-Khintchine formula. Since the relativistic Hamiltonians ∣∇∣|\nabla |∣∇∣
Recent developments in quantum physics make heavy use of so-called “quantum trajectories.” Mathematically, this theory gives rise to “stochastic Schrödinger equations,” that is, perturbation of Schrödinger-type equations under the form of stochastic differential equations. But such equations are in general not of the usual type as considered in the literature. They pose a serious problem in terms of justifying the existence and uniqueness of a solution, justifying the physical pertinence of the equations. In this article we concentrate on a particular case: the diffusive case, for a two-level system. We prove existence and uniqueness of the associated stochastic Schrödinger equation. We physically justify the equations by proving that they are a continuous-time limit of a concrete physical procedure for obtaining a quantum trajectory.