Stochastic Theory of Relativistic Quantum Mechanics

TL;DR

This paper develops a stochastic framework for relativistic quantum mechanics, modeling particle trajectories as diffusion processes and deriving equations to describe their probabilistic behavior, extending non-relativistic stochastic theories.

Contribution

It introduces a relativistic stochastic model where particle paths are diffusion processes, providing a new approach to relativistic quantum mechanics based on stochastic calculus.

Findings

Derived Fokker-Planck and Kolmogorov equations for the model

Applied the theory to the free particle case

Established a connection between stochastic processes and relativistic quantum trajectories

Abstract

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.