A relativistic formulation of the de la Peña-Cetto stochastic quantum mechanics

Abstract

A covariant generalization of a non-relativistic stochastic quantum mechanics introduced by de la Peña 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 coupling of the particle with the stochastic forces. Then, particular 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 ρ(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 ψ(x) which exactly satisfies the Klein-Gordon (KG) equation. These results for the zeroth order solution reproduce the ones formerly and independently derived in the literature. It is also
argued that when the KG solution is either of positive or negative energy, the total number of particles conserves in the random motion. Other solutions for the joint distribution function in lowest order, satisfying the positive condition are also presented here. The are consistent with the assumed lack of stochastic forces implied by the zeroth order equations. It is also argued that such joint distributions, after considering the action of the stochastic forces, might furnish an explanation of the quantum mechanical properties, as associated to ensembles of particles in which the vacuum makes such particles behave in a similar way as Couder’s droplets moving over oscillating liquid surfaces. Some remarks on the solutions of the positive joint distribution problem proposed in the Olavos’s analysis are also presented.