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Journal of Mathematical Physics

Generalized stochastic processes and continual observations in quantum mechanics

G. Lupieri

Published Sep 1, 1983• 26 citations
Publisher

Abstract

We give here a mathematically rigorous form to an earlier work by Barchielli, Lanz, and Prosperi, in which it was found that a generalized stochastic process describes the results of continual observations of the position of a quantum particle. With the help of Albeverio and Ho/egh-Krohn’s theory of Feynman path integrals, we define the characteristic functional of this process and demonstrate that it possesses the necessary properties of normalization, continuity, and positive definiteness. An explicit calculation of the Feynman path integral which defines the functional allows an analysis of the process to be made.

DOI
10.1063/1.525981

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We give here a mathematically rigorous form to an earlier work by Barchielli, Lanz, and Prosperi, in which it was found that a generalized stochastic process describes the results of continual observations of the position of a quantum particle. With the help of Albeverio and Ho/egh-Krohn’s theory of Feynman path integrals, we define the characteristic functional of this process and demonstrate that it possesses the necessary properties of normalization, continuity, and positive definiteness. An explicit calculation of the Feynman path integral which defines the functional allows an analysis of the process to be made.
It has been stated in the physics literature that the probability amplitude for a quantum system after a continuous position measurement can be expressed by the restricted or the weighted Feynman path integral containing information about the result and the resolution of its measurement. In the present paper it is proved that the restricted Feynman path integrals can be defined in a rigorous sense. More specifically, we consider a quantum system consisting of some particles and simultaneously perform a continuous measurement of the positions of all particles in the time interval [0,T][0,T][0,T]. Then, it is shown rigorously that the restricted Feynman path integral after the measurement can be defined in the L2L^2L2 space and satisfies the corresponding non-self-adjoint Schroeodinger equation. In addition, our results are generalized to a quantum spin system where a continuous position measurement for all spin components of all particles is simultaneously performed, though the results obtained are inadequate for general use.
Abstract Feynman developed the path integral approach to quantum mechanics as an alternative to wave or matrix mechanics that has found fruitful implementation in quantum mechanics, quantum electrodynamics, and statistical physics. Despite its many successes, the usual Feynman path integral formalism suffers from a lack of mathematical rigor. Questions of mathematical rigor still exist, but Mark Kac provided a rigorous path integral approach to quantum mechanics. The Kac formulation for quantum systems is given in terms of Wiener measure for a path integral and employs the Brownian motion distribution. In this paper, we review the Feynman–Kac approach and other stochastic processes related to Brownian motion. We also give an account of the successes and difficulties we faced during our journey through numerical path integral problems involved in physics.
"Feynman path integrals are ubiquitous in quantum physics, even if a large part of the scientific community still considers them as a heuristic tool that lacks a sound mathematical definition. Our book aims to refute this prejudice, providing an extensive and self-contained description of the mathematical theory of Feynman path integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. This second edition presents a detailed discussion of the general theory of complex integration on infinite dimensional spaces, providing on one hand a unified view of the various existing approaches to the mathematical construction of Feynman path integrals and on the other hand a connection with the classical theory of stochastic processes. Moreover, new chapters containing recent applications to several dynamical systems have been added. This book bridges between the realms of stochastic analysis and the theory of Feynman path integration. It is accessible to both mathematicians and physicists"--
By using path integrals, the stochastic process associated to the time evolution of the quantum probability density is formally rewritten in terms of a stochastic differential equation, given by Newton's equation of motion with an additional multiplicative stochastic force. However, the term playing the role of the stochastic force is defined by a non-positive-definite probability functional, providing a clear example of the negative (or "extended") probabilities characteristic of quantum mechanics.
In the late forties Richard Feynman (1948) proposed a path-integral form of quantum mechanics that proved to be equivalent to the conventional Schrödinger-equation form but possessed a much more heuristic force (see Feynman and Hibbs (1965) for a detailed exposition of Feynman’s theory). Our consideration of continuous measurements in the following chapters will be based on Feynman path integrals. A theory of quantum continuous measurements will be formulated directly in terms of path integrals instead of a limiting process of the type used in Chapter 2 .