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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Stochastic mechanics of spin-½ particles

Francesco Guerra, R. Marra

Published Dec 15, 1984• 8 citations
Publisher

Abstract

Following the strategy of stochastic quantization, based on variational principles for processes taking values on a discrete configuration space, we give a detailed and self-contained description of the stochastic processes associated to a two-level quantum system, of interest for the physical description of spin-\textonehalf{} particles and Fermi fields. In particular we investigate the class of processes related to unperturbated quantum evolution and those simulating quantum-measurement procedures leading to mixture formation.

DOI
10.1103/physrevd.30.2579

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Following the strategy of stochastic quantization, based on variational principles for processes taking values on a discrete configuration space, we give a detailed and self-contained description of the stochastic processes associated to a two-level quantum system, of interest for the physical description of spin-\textonehalf{} particles and Fermi fields. In particular we investigate the class of processes related to unperturbated quantum evolution and those simulating quantum-measurement procedures leading to mixture formation.
We consider stochastic variational principles for random processes taking values on a discrete configuration space. For a suitable time-reversal-invariant choice of the stochastic action, the resulting programming equations can be related to the Schr\"odinger equation for a discrete system. If the discrete system is considered as an approximation of a continuous system, then the limit reproduces Nelson's stochastic mechanics and allows one to derive the assumptions on the random noise acting on the system. In fact, the variational principle also gives information about the osmotic behavior of the process. Finally, we show that there are also critical processes whose behavior can be interpreted as a model for quantum measurement, because they relax in time to mixtures of processes. Therefore, stochastic variational principles can provide a very simple conceptual model simulating quantum behavior, both from the point of view of unperturbed time evolution and of the measurement phenomena leading to wave-function collapse.
Among the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum mechanical systems with kinetic equations of motion in configuration space based on conservative diffusion processes. This leads to the representation of physical observables through stochastic processes instead of self-adjoint operators. The mathematical foundations of this approach were laid by Edward Nelson in 1966. It allows a different perspective on quantum phenomena without necessarily using the wave-function. This article recaps the development of stochastic mechanics with a focus on variational and extremal principles. Furthermore, based on recent developments of optimal control theory, the derivation of generalized canonical equations of motion for quantum systems within the stochastic picture are discussed. These so-called quantum Hamilton equations add another layer to the different formalisms from classical mechanics that find their counterpart in quantum mechanics.
In the general framework of stochastic control theory we introduce a suitable form of stochastic action associated to the controlled process. A variational principle gives all the main features of Nelson's stochastic mechanics. In particular, we derive the expression for the current velocity field as the gradient of the phase action. Moreover, the stochastic corrections to the Hamilton-Jacobi equation are in agreement with the quantum-mechanical form of the Madelung fluid (equivalent to the Schr\"odinger equation). Therefore, stochastic control theory can provide a very simple model simulating quantum-mechanical behavior.