Dirac Fields on Spacelike Hypersurfaces, Their Rest-Frame Description and Dirac Observables

TL;DR

This paper reformulates Dirac and electromagnetic fields on spacelike hypersurfaces in Minkowski spacetime, focusing on their rest-frame description and highlighting geometric inconsistencies in fermion spin structure representation.

Contribution

It introduces a reformulation of Dirac fields on spacelike hypersurfaces and reveals a fundamental geometric inconsistency in describing fermion spin trajectories.

Findings

Fermion description is incomplete without a bosonic spin carrier.

Reformulation on Wigner hyperplanes clarifies rest-frame dynamics.

Identifies a geometric inconsistency in fermion spin structure.

Abstract

Grassmann-valued Dirac fields together with the electromagnetic field (the pseudoclassical basis of QED) are reformulated on spacelike hypersurfaces in Minkowski spacetime and then restricted to Wigner hyperplanes to get their description in the rest-frame Wigner-covariant instant form of dynamics. The canonical reduction to the Wigner-covariant Coulomb gauge is done in the rest frame. It is shown, on the basis of a geometric incosistency, that the description of fermions is incomplete, because there is no bosonic carrier of the spin structure describing the trajectory of the electric current in Minkowski spacetime, as it was already emphasized in connection with the first quantization of spinning particles in a previous paper.