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Europhysics Letters (EPL)

A Relativistic Version of Nelson's Stochastic Mechanics

Tomasz Zastawniak

Published Sep 1, 1990• 24 citations
Publisher

Abstract

Markov diffusions corresponding to solutions of the Klein-Gordon equation are introduced in a similar way as in Nelson's stochastic mechanics. A relativistic version of the Nelson-Newton law for Markov diffusions is established. Stochastic generalizations of the proper time and of the relativistic action functional are proposed and a stochastic version of the relativistic principle of least action is stated. The formalism can be regarded as a quantization procedure leading from classical special relativity to the Klein-Gordon equation.

DOI
10.1209/0295-5075/13/1/003

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Markov diffusions corresponding to solutions of the Klein-Gordon equation are introduced in a similar way as in Nelson's stochastic mechanics. A relativistic version of the Nelson-Newton law for Markov diffusions is established. Stochastic generalizations of the proper time and of the relativistic action functional are proposed and a stochastic version of the relativistic principle of least action is stated. The formalism can be regarded as a quantization procedure leading from classical special relativity to the Klein-Gordon equation.
A stochastic interpretation of Klein-Gordon equation is proposed. The starting point is the construction of relativistically covariant diffusions satisfying a property similar but not identical to the Markov property. It is then shown that the continuity equation associated with these processes together with the relativistic version of mean Newton equation are equivalent to the Klein-Gordon equation On propose une interpretation stochastique de l'equation de Klein Gordon. Le point de depart est la construction d'une diffusion relativiste covariante qui satisfait une propriete semblable (mais pas identique) a la propriete de Markov. On montre ensuite que l'equation de continuite associee a ce processus plus la version relativiste de l'equation de Newton en moyenne sont equivalentes a l'equation de Klein-Gordon
Given a positive energy solution of the Klein–Gordon equation, the motion of the free, spinless, relativistic particle is described in a fixed Lorentz frame by a Markov diffusion process with nonconstant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation (dτ)2=−(1/c2)dXν dXν. A random time-change transformation provides the bridge between the t and the τ domain. In the τ domain, we obtain an M4-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein–Gordon solution is an invariant, nonintegrable density for this Markov process. It satisfies a relativistically covariant continuity equation.
Several years ago Nelson succeeded in deriving the nonrelativistic Schrödinger equation within a stochastic model which included Newton’s second law as the fundamental dynamical rule. Unfortunately, the relativistic extension of Nelson’s work is not so straightforward as might at first be supposed. This paper examines the difficulties inherent in such a relativization and proposes supplemental axioms which resolve those difficulties. A stochastic derivation of the Klein–Gordon equation is then presented.
The stochastic theory of relativistic quantum mechanics presented here is modelled on the one that has been proposed previously and that was claimed to be a promising substitute to the orthodox theory in the non-relativistic domain. So it also relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. Along the way, we introduce ad hoc hypotheses for the sole reason that they facilitate and even make possible the further development of the theory. These and the so-called Nelson equations are used to determine the drift vectors and the diffusion tensor characteristic of such a process. It is then possible to write down the Fokker-Planck and Kolmogorov equations that can be used to determine the normal and conditional probability densities. This logical sequence of steps leads to the solution of specific problems and we apply it to the special case of the free particle.
The difficulties of extending to relativity the results obtained in a preceding paper are set forth; in particular the relativistic behaviour of the time and length minimal constants brings us to a well-known problem. The proposed views attempt to deal with these equations. They give a relativistic generalization of the classical diffusion theory, from which the Klein-Gordon equation is deduced. Consistent results are obtained by a relativistic survey of Dirac's fluid model. Subsequent investigations have been carried out concerning the character of relativity and the limitation of its validity in microphysics.