A relativistic formulation of the de la Pe\~na-Cetto stochastic quantum mechanics

TL;DR

This paper develops a covariant, Lorentz-invariant formulation of de la Peña and Cetto's stochastic quantum mechanics, linking it to the Klein-Gordon equation and exploring particle conservation and droplet analogies.

Contribution

It introduces a covariant generalization of stochastic quantum mechanics that avoids non-covariant time parameters and connects solutions to the Klein-Gordon equation.

Findings

Derives covariant iterative equations for joint distribution function Q(x,p)

Shows that zeroth-order solutions relate to Klein-Gordon equation solutions

Suggests a link between stochastic quantum mechanics and droplet experiments

Abstract

A covariant generalization of a non-relativistic stochastic quantum mechanics introduced by de la Pe\~na and Cetto is formulated. The analysis is done in space-time and avoids the use of a non-covariant time evolution parameter in order to search for Lorentz invariance. The covariant form of the set of iterative equations for the joint coordinate and momentum distribution function $Q(x,p)$ is derived and expanded in power series of the charge of the particle. Then, solutions of the zeroth order in the charge of the iterative equations for $Q(x,p)$ are considered. For them, it follows that the space-time probability density $\rho(x)$ and the function $S(x)$ which gradient defines the mean value of the momentum at the space time point $x$, define a complex function $\psi(x)$ which exactly satisfies the Klein-Gordon (KG) equation. It is argued that when the KG solution is either of positive or negative energy, the total number of particles conserves in the random motion. It follows that the total number of particles conserves when $\psi(x)$ composed alternatively of pure positive or negative energy modes. In addition, solutions for the joint distribution function in lowest order, satisfying the positive condition are also found. They suggests a possible link with the Couder's experiments on droplet movements over oscillating liquid surfaces.