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Journal of Physics A Mathematical and General

Quantum stochastics: the passage from a relativistic to a non-relativistic path integral

Ted Jacobson A, Lawrence S. Schulman

Published Feb 1, 1984• 132 citations
Publisher

Abstract

Feynman's path integral for the one-dimensional Dirac particle uses paths for which Delta x approximately Delta t. For non-relativistic path integrals, typical paths satisfy ( Delta x) 2 approximately Delta t (as in Brownian motion). The authors demonstrate the consistency of these two stochastic schemes and show how the relativistic formalism contains within itself the scale of the transition regime, namely Delta x approximately h(cross)/mc, the Compton wavelength.

DOI
10.1088/0305-4470/17/2/023

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Feynman's path integral for the one-dimensional Dirac particle uses paths for which Delta x approximately Delta t. For non-relativistic path integrals, typical paths satisfy ( Delta x) 2 approximately Delta t (as in Brownian motion). The authors demonstrate the consistency of these two stochastic schemes and show how the relativistic formalism contains within itself the scale of the transition regime, namely Delta x approximately h(cross)/mc, the Compton wavelength.
The Feynman path integral formalism for non-relativistic quantum mechanics is revisited. A comparison is made with cases of light propagation (Huygens' principle) and Brownian motion. The difficulties for a physical model applying Feynman's formalism are pointed out. A reformulation is proposed, where the transition probability of a particle from one space-time point to another one is the sum of probabilities of the possible paths. As an application, Born approximation for scattering is derived within the formalism, which suggests an interpretation involving the stochastic motion of a particle rather than the square of a wavelike amplitude.
The analogy between quantum 'paths' and random walks of a Brownian particle is extended to the relativistic regime. This yields a coherent picture of the fine structure of quantum paths: these can be viewed as analogies of the trajectory of a random walker. The fractal dimension of quantum paths is df=2 at scales large compared with the Compton wavelength Lambda but small compared with the de Broglie wavelength lambda and becomes df=1 at scales smaller than Lambda . The transition between the two regimes is exhibited explicitly and is analogous to the cross-over from ballistic to diffusive motion of Brownian particles. The analogue of the collisions in Brownian motion is electron-positron pair creations and annihilations, and the unit thermodynamic free energy kT corresponds to the rest mass mc2 of the particle.
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 formalism for non-relativistic quantum mechanics is revisited. A comparison is made with the cases of light progagation (Huygens principle) and Brownian motion. The difficulties for a physical model behind Feynman formalism are pointed out. It is proposed a reformulation where the transition probability from one space-time point to another one is the sum of probabilities of the possible paths. The Born approximation for scattering is derived within the formalism, which suggests an interpretation in terms of particles, without the need of Born assumption that the modulus squared of the wavefunction is a probability density.
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.
Path Integrals in Classical Theory Brownian motion: Introduction to the concept of path integration Wiener path integrals and stochastic processes Path Integrals in Quantum Mechanics Feynman path integrals Path integrals in Hamiltonian formalism Quantizations, the operator ordering problem and path integrals Path integrals and quantizations in spaces with topologiecal constraints Path integrals in curved spaces, space time transformations and the Coulomb problem Path integrals over anticommuting variables for fermions and generalizations Appendices Bibliography Index