Relativistic Bohmain Mechanics

TL;DR

This paper extends Bohmian mechanics to the relativistic domain using the relativistic Schrödinger equation, deriving new guidance equations and exploring the implications of relaxing the quantum equilibrium hypothesis.

Contribution

It introduces a relativistic formulation of Bohmian mechanics that does not inherently assume quantum equilibrium, allowing for testable predictions beyond standard relativistic quantum mechanics.

Findings

Derived relativistic guidance equations and quantum potential.

Showed that quantum equilibrium is not intrinsic in the relativistic extension.

Provided example calculations illustrating the theory's predictions.

Abstract

In this paper we generalize the ideas of de Broglie and Bohm to the relativistic case which is based on the relativistic Schr\"odinger equation. In this regard, the relativistic forms of the guidance equation and quantum potential are derived. In our formulation of Relativistic Bohmian Mechanic, the quantum equilibrium hypothesis $(\rho=|\psi|^{2})$ and the probabilistic interpretation of the wave function are not an intrinsic feature of the theory as well as expected from the theoretical structure of Bohmian mechanics but still we can extract the statistical predictions of this theory. In fact with assuming the quantum equilibrium hypothesis in non-relativistic case and then acceleration to particles by an external field we go to the relativistic regime. So in this case the quantum equilibrium would not be established and theory will have testable predictions that can be experienced, or this will be compared to the result of relativistic quantum mechanics. In some simple cases such calculations have been done.