  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.

Quantum Physics