Probability Density in Relativistic Quantum Mechanics

TL;DR

This paper evaluates the validity of Dirac and Foldy-Wouthuysen probability densities in relativistic quantum mechanics, concluding that only the Dirac density satisfies all physical criteria such as Lorentz invariance and covariance.

Contribution

It provides a critical analysis of probability densities for spin-1/2 particles, establishing the Dirac density as the physically consistent choice in relativistic settings.

Findings

Dirac density satisfies Lorentz invariance and covariance.

Foldy-Wouthuysen density is less consistent with relativistic criteria.

Both densities are valid within a few Compton wavelengths.

Abstract

In the realm of relativistic quantum mechanics, we address a fundamental question: Which one, between the Dirac or the Foldy-Wouthuysen density, accurately provide a probability density for finding a massive particle with spin $1/2$ at a certain position and time. Recently, concerns about the Dirac density's validity have arisen due to the Zitterbewegung phenomenon, characterized by a peculiar fast-oscillating solution of the coordinate operator that disrupts the classical relation among velocity, momentum, and energy. To explore this, we applied Newton and Wigner's method to define proper position operators and their eigenstates in both representations, identifying 'localized states' orthogonal to their spatially displaced counterparts. Our analysis shows that both densities could represent the probability of locating a particle within a few Compton wavelengths. However, a critical analysis of Lorentz transformation properties reveals that only the Dirac density meets all essential physical criteria for a relativistic probability density. These criteria include covariance of the position eigenstate, adherence to a continuity equation, and Lorentz invariance of the probability of finding a particle. Our results provide a clear and consistent interpretation of the probability density for a massive spin-$1/2$ particle in relativistic quantum mechanics.