Space spinors

Abstract

Just as Maxwell’s electromagnetic field equations govern the evolution of electric and magnetic spatial vectors if some choice of time function has been made, so also the neutrino equation and Dirac equation may be understood as governing the evolutions of certain spatial quantities. In this space-plus-time view of the spinor field equations, it is accurate and natural to regard a two-component are written in 3-plus-1 form for both the spinor fields and the corresponding null vector fields. A spatial null vector is of the form M=E+iB, with E⋅E−B⋅B=0=E⋅B, so it is also of the correct algebraic form for describing a null electromagnetic field. The time derivative of a squared neutrino field Ma, however, is -i curl Ma+〈M〉cDaMc, compared with simply -i curl Ma for a source-free Maxwell field. Here 〈M〉c is the real spatial unit vector in the neutrino propagation direction E×B, and Da is the spatial covariant derivative.