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Physical Review A

From Feynman histories to observables

D. Sokolovski

Published Mar 1, 1998• 13 citations
Publisher

Abstract

We analyze a quantum observable $f$ given by the time average of a dynamical variable F along a Feynman path. We show that $f$ can be described by a (generalized) wave function satisfying the Schr\"odinger-like equation and measured by a meter similar to that of von Neumann. A connection with operators is established. Examples include the particle's momentum, energy and the fractional traversal time.

DOI
10.1103/physreva.57.r1469

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We analyze a quantum observable fff given by the time average of a dynamical variable F along a Feynman path. We show that fff can be described by a (generalized) wave function satisfying the Schr"odinger-like equation and measured by a meter similar to that of von Neumann. A connection with operators is established. Examples include the particle's momentum, energy and the fractional traversal time.
We analyze the connection between the path integral and operator approaches to the quantum measurement problem. In general, an act of measurement is shown to destroy interference between components of the wave function related to the particle's histories. Classically, there exist a class of meters suitable for measuring the value of a given dynamical variable F. Quantally, different meters produce different results. The standard von Neumann measurement is one particular case. Rearranging Feynman paths according to the value of (time average of) F defines a different type of meter. The two methods disagree if the duration of the measurement is very short. Possible ways to measure the particle's momentum are studied in detail. The semiclassical limit of a measurement is analyzed.
We demonstrate how a general von Neumann--like measurement can be analyzed in terms of histories (paths) constructed for the measured variable AAA. The Schr\"odinger state of a system in a Hilbert space of arbitrary dimensionality is decomposed into a set of substates, each of which corresponds to a particular detailed history of the system. The coherence between the substates may then be destroyed by meter(s) to a degree determined by the nature and the accuracy of the measurement(s) which may be of von Neumann, finite-time, or continuous type. The cases of a particle described by Feynman paths in the coordinate space and a qubit in a two-dimensional Hilbert space are studied in some detail.
We propose a general theoretical approach to quantum measurements based on the path (histories) summation technique. For a given dynamical variable A, the Schrödinger state of a system in a Hilbert space of arbitrary dimensionality is decomposed into a set of substates, each of which corresponds to a particular detailed history of the system. The coherence between the substates may then be destroyed by meter(s) to a degree determined by the nature and the accuracy of the measurement(s) which may be of von Neumann, finite-time or continuous type. Transformations between the histories obtained for non-commuting variables and construction of simultaneous histories for non-commuting observables are discussed. Important cases of a particle described by Feynman paths in the coordinate space and a qubit in a two dimensional Hilbert space are studied in some detail.
We analyze nonrelativistic quantum measurement of the time average of the particle's coordinate, $X\ensuremath{\equiv}{t}^{\ensuremath{-}1}{\ensuremath{\int}}_{0}^{t}{x(t}^{\ensuremath{'}}{)dt}^{\ensuremath{'}}.$ The measurement amplitude is constructed by restricting the Feynman path integral to paths with the required value of X. The resulting decomposition of the Schr\"odinger wave function determines the type of meter needed to measure X. We show that such meter can be realized as a magnetic moment traveling with the particle in a magnetic field whose magnitude linearly changes with x. Weak and strong measurement regimes are discussed.
In Feynman's space-time approach to nonrelativistic quantum mechanics, the wave function may be considered to be the sum of path integrals over Brownian-motion trajectories. This formalism is entirely equivalent to the Schr\"odinger-equation treatment with its intrinsic indeterminacy. Pursuing the analogy with Brownian motion further, one may introduce a hidden variable corresponding to particle velocity in order to represent the wave function as a sum of path integrals taken over phase-space trajectories. By means of a hidden parameter corresponding to the time scale of fictitious interactions of the particle with the vacuum, one can define generalized propagators which over very short times produce wave functions which are localizable in phase space and hence are deterministic. For long times the conventional Feynman formalism is recovered. This short-time causality does not violate the von Neumann injunction against hidden variables since the new velocity variable turns out to be non-Hermitian over the long time scales normally considered. In contrast to the models proposed by Bohm and others, the present Brownian model is linear and preserves the usual statistical interpretation of the wave function for sufficiently long times.
The quantum-mechanical traversal- (tunneling) time problem is reformulated in terms of a square-integrable amplitude distribution (wave function) whose evolution is governed by a generalized Schr\"odinger equation. The method uses an analogy between quantum mechanics and classical Brownian motion. Brownian and quantum traversal-time distributions are evaluated by the direct integration of corresponding equations of motion. The general relation between Feynman and Schr\"odinger quantum mechanics is discussed.