A search for the minimal unified field theory. I. Parallel transport of   Dirac field
  Two real vector fields are revealed as spin connections of the spinor field, which is introduced as a representation of the local Lorentz group by Dirac spinors. One of these fields is identified as the Maxwell field. Another one is the axial field, which is sufficient to describe parity nonconservation effects in atoms and it is sensitive to the second internal degree of freedom of the Dirac spinor field. The polarization associated with this degree of freedom is a distinctive characteristic of the Dirac spinor field in spatially closed localized states. The short-distance behavior of the axial field can lead to a not self-adjoint Dirac Hamiltonian and trigger a ``falling onto a centre'' phenomenon. In the second part of this work, the axial field will be linked to the gravity. 
Alexander Makhlin