Asymptotic velocities in quantum and Bohmian mechanics

TL;DR

This paper explores the relationship between quantum and Bohmian asymptotic velocities, demonstrating their distribution equivalence under certain conditions and proposing a covariant relativistic Bohmian framework.

Contribution

It establishes the equivalence of asymptotic velocity distributions in quantum and Bohmian mechanics and introduces a covariant formulation for relativistic Bohmian mechanics.

Findings

Distribution of Bohmian asymptotic velocities matches quantum predictions.

In the relativistic case, the velocity distribution is covariant and foliation-independent.

Proposes a mathematical framework for a covariant relativistic Bohmian mechanics.

Abstract

In this paper the relations between the asymptotic velocity operators of a quantum system and the asymptotic velocities of the associated Bohmian trajectories are studied. In particular it is proved that, under suitable conditions of asymptotic regularity, the probability distribution of the asymptotic velocities of the Bohmian trajectories is equal to the one derived from the asymptotic velocity operators of the associated quantum system. It is also shown that in the relativistic case the distribution of the asymptotic velocities of the Bohmian trajectories is covariant, or equivalently, it does not depend on a preferred foliation (it is well known that this is not the case for the structure of the Bohmian trajectories or for their spatial distribution at a finite time). This result allows us to develop a covariant formulation of relativistic Bohmian mechanics; such a formulation is proposed here merely as a mathematical possibility, while its empirical adequacy will be discussed elsewhere.