Probability density of relativistic spinless particles

TL;DR

This paper introduces a new probability density for relativistic spinless particles based on a conserved current for the Klein-Gordon equation, addressing localization issues in relativistic quantum theory.

Contribution

It derives a novel conserved current whose first component acts as a probability density, overcoming previous localization challenges in relativistic quantum mechanics.

Findings

New conserved current for Klein-Gordon equation

Probability density reduces to ||^2 in non-relativistic limit

Significant deviation from ||^2 when momentum uncertainty exceeds m_0c

Abstract

In this paper, a new conserved current for Klein-Gordon equation is derived. It is shown, for $1+1$-dimensions, the first component of this current is non-negative and reduces to $|\phi|^2$ in non-relativistic limit. Therefore, it can be interpreted as the probability density of spinless particles. In addition, main issues pertaining to localization in relativistic quantum theory are discussed, with a demonstration on how this definition of probability density can overcome such obstacles. Our numerical study indicates that the probability density deviates significantly from $|\phi|^2$ only when the uncertainty in momentum is greater than $m_0c$.