On Operator Fields in the Bundle of Dirac Spinors

The bundle of Dirac spinors is used for describing particles with half-integer spin in general relativity and in quantum field theory. It is a special four-dimensional complex vector-bundle over the space-time manifold M . Let’s remind that the space-time manifold M itself is a four-dimensional real manifold equipped with a Minkowski type metric g of the signature (+,−,−,−). Apart from g, the spacetime manifoldM is equipped with two other geometric structures — the orientation and the polarization. The orientation distinguishes right quadruples of tangent vectors from left ones, while the polarization distinguishes future and past half light cones in tangent spaces at each point of M . The bundle of Dirac spinors is denoted DM . It is equipped with four basic spin-tensorial fields in addition to g. They are presented in the following table.