Flux-Across-Surfaces Theorem for a Dirac Particle

TL;DR

This paper extends flux across surfaces theorems to relativistic spin-1/2 particles described by the Dirac equation, showing that crossing probabilities converge to momentum direction probabilities at large distances.

Contribution

It generalizes non-relativistic flux across surfaces theorems to the relativistic Dirac particle case, providing a rigorous proof of asymptotic crossing probabilities.

Findings

Probability of crossing a surface converges to momentum direction probability.

Results apply to particles with external static potentials.

Generalizes classical flux theorems to relativistic quantum mechanics.

Abstract

We consider the asymptotic evolution of a relativistic spin-1/2-particle. i.e. a particle whose wavefunction satisfies the Dirac equation with external static potential. We prove that the probability for the particle crossing a (detector) surface converges to the probability, that the direction of the momentum of the particle lies within the solid angle defined by the (detector) surface, as the distance of the surface goes to infinity. This generalizes earlier non relativistic results, known as flux across surfaces theorems, to the relativistic regime.