A vector wave equation for neutrinos

The Cartan map gives an isomorphism between spinors and isotropic vectors. Isotropic vectors F=E+iH satisfy the condition F ⋅ F=0. We show that via the Cartan map, the particle current for neutrinos is given by j0=‖E‖, j=E×H/‖E‖, and the neutrino wave equation becomes D0F=iD×F−(DF) ⋅ v, where v=j/j0=E×H/E2=velocity field, D0=i(h/2)(∂/∂t)−V0,D=−i (h/2)∇−V, where h=Planck’s constant and V=(V0,V)=external potential. This wave equation preserves the isotropic condition, and like the equivalent Dirac equation, causes j to be the conserved current. We show that the isotropic restriction on the vector field F accounts for the observable properties of a neutrino in an external field, in particular, for the observed spectrum of the energy, momentum, angular momentum, spin, velocity, and position operators.