WKB analysis of relativistic Stern-Gerlach measurements

TL;DR

This paper derives a relativistic Stern-Gerlach spin operator from first principles using WKB approximation, revealing its momentum dependence and implications for quantum tomography.

Contribution

It provides the first-principles derivation of a relativistic Stern-Gerlach operator based on electromagnetic field transformations and WKB analysis of the Dirac equation.

Findings

The derived operator correctly splits spin eigenstates in a Stern-Gerlach setup.

The operator depends on particle momentum, unlike other common operators.

Implications for quantum tomography are discussed.

Abstract

Spin is an important quantum degree of freedom in relativistic quantum information theory. This paper provides a first-principles derivation of the observable corresponding to a Stern-Gerlach measurement with relativistic particle velocity. The specific mathematical form of the Stern-Gerlach operator is established using the transformation properties of the electromagnetic field. To confirm that this is indeed the correct operator we provide a detailed analysis of the Stern-Gerlach measurement process. We do this by applying a WKB approximation to the minimally coupled Dirac equation describing an interaction between a massive fermion and an electromagnetic field. Making use of the superposition principle we show that the +1 and -1 spin eigenstates of the proposed spin operator are split into separate packets due to the inhomogeneity of the Stern-Gerlach magnetic field. The operator we obtain is dependent on the momentum between particle and Stern-Gerlach apparatus, and is mathematically distinct from two other commonly used operators. The consequences for quantum tomography are considered.