Operators and Field Equations in the Electroweak Sector of Particle Physics

This paper derives a linear, first-order, partial differential field equation (a Dirac-like equation) in the geometric calculus of the geometric algebra 𝒢_4,1 that has free plane-wave solutions distinct from one another that correspond to the left and right chiral states of the electron and the neutrino. Besides the usual spacetime dependence of plane waves, the solutions have a multivector structure yielding a ladder of states with raising and lowering operators appropriate to electroweak theory and having an SU(2)_L relationship among the chiral electron and neutrino states. The required Dirac-like equation in 𝒢_4,1 results from a systematic review of Dirac-like equations (i.e., first-order field equations whose solutions also satisfy the Klein–Gordon equation) in geometric algebras of lower dimension.