On the stochastic mechanics of the free relativistic particle

Abstract

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 non-constant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation $(dτ)^2=-\frac{1}{c^2}dX_νdX_ν$. A random time-change transformation provides the bridge between the $t$ and the $τ$ domain. In the $τ$ domain, we obtain an $\M^4$-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein-Gordon solution is an invariant, non integrable density for this Markov process. It satisfies a relativistically covariant continuity equation.