Relativistic generalization of the Born rule

TL;DR

This paper derives a Lorentz-invariant generalization of the Born rule for relativistic quantum mechanics, addressing inconsistencies with the Salpeter equation and proposing a modified probability density.

Contribution

It introduces a new relativistic form of the Born rule compatible with Lorentz symmetry for the Salpeter equation, extending quantum measurement theory.

Findings

The standard Born rule is incompatible with Lorentz symmetry in this context.

A new relativistic probability density formula is proposed.

The modified rule is consistent with the Lorentz symmetry of the Salpeter equation.

Abstract

We have shown that the Born rule $(\rho=|\psi|^{2})$ is inconsistent with Lorentz symmetry of the Salpeter equation (square root Klein-Gordon equation). So we find relativistic modification of the Born rule as $\rho=\frac{1}{2mc^{2}}(|\sqrt{\widehat{H}-mc^{2}}\psi|^{2}+|\sqrt{\widehat{H}+mc^{2}}\psi|^{2})$ , which is consistent with Lorentz symmetry of this equation