Spin-1/2 Maxwell Fields

TL;DR

This paper explores a novel spin-1/2 transformation framework for Maxwell fields, revealing new invariants and invariance properties, and connects Maxwell and Dirac equations through complex Klein-Gordon fields.

Contribution

It introduces spin-1/2 transformation rules for Maxwell fields, deriving new invariants and linking Maxwell equations to Dirac equations via Klein-Gordon fields.

Findings

Spin-1/2 transformations yield new invariants for Maxwell fields.

Maxwell and Dirac equations are connected through complex Klein-Gordon fields.

Charge and gauge invariance are lost in the spin-1/2 formulation.

Abstract

Requiring covariance of Maxwell's equations without {\it a priori} imposing charge invariance allows for both spin-1 and spin-1/2 transformations of the complete Maxwell field and current. The spin-1/2 case yields new transformation rules, with new invariants, for all traditional Maxwell field and source quantities. The accompanying spin-1/2 representations of the Lorentz group employ the Minkowski metric, and consequently the primary spin-1/2 Maxwell invariants are also spin-1 invariants; for example, $\Phi^2 - {\bf A}^2$, ${\bf E}^2 - {\bf B}^2 + 2i {\bf E} \bm{\cdot} {\bf B} - ({\partial}_{0}{\Phi} + {\bm{\nabla \cdot}}{\bf A})^2$. The associated Maxwell Lagrangian density is also the same for both spin-1 and spin-1/2 fields. However, in the spin-1/2 case, standard field and source quantities are complex and both charge and gauge invariance are lost. Requiring the potentials to satisfy the Klein-Gordon equation equates the Maxwell and field-potential equations with two Dirac equations of the Klein-Gordon mass, and thus one complex Klein-Gordon Maxwell field describes either two real vector fields or two Dirac fields, all of the same mass.