Superluminality and a Curious Phenomenon in the Relativistic Quantum Hamilton-Jacobi Equation

TL;DR

This paper investigates the relativistic quantum Hamilton-Jacobi equation and finds that averaging oscillating solutions can lead to superluminal velocities, raising questions about causality in quantum theory.

Contribution

It introduces an averaging method for oscillating solutions in the relativistic quantum Hamilton-Jacobi equation, revealing conditions under which superluminal speeds can occur.

Findings

Averaging oscillating solutions resolves the classical limit of particle speed.

Superluminal solutions depend on integration constants.

The approach highlights potential superluminal phenomena in relativistic quantum mechanics.

Abstract

A basic problem in the relativistic quantum Hamilton-Jacobi theory is to understand whether it may admit superluminal solutions. Here we consider the averaging of the speed on a period of the oscillating term which is similar to Dirac's averaging of the oscillating part of the free electron's speed. Such an averaging solves the problem of getting the $\hbar\to 0$ limit of the speed of the free particle, and leads to solutions that, depending on the integration constants, may be superluminal.