A gauge model with spinor group for a description of local interaction of a fermion with electromagnetic and gravitational fields

Abstract

We suggest model equations, which, from some point of view, describe local interaction of three physical fields: a field of matter, an electromagnetic field and a gravitational field. A base of the model is a field of matter described by the wave function of fermion satisfying the equation similar to Dirac equation for electron. Electromagnetic and gravitational fields appear as the gauge fields for this equation. We have found the connection between these fields and the curvature tensor of Riemannian manifold. We present a main Lagrangian from which the equations of the model are deduced. The covariance of the model equations under changes of coordinates is considered. We develop mathematical techniques needed for the model connected with an exterior algebra of Euclidean or Riemannian space. The exterior algebra is considered as a bialgebra with two operations of multiplications -- an exterior multiplication and Clifford multiplication. We define a structure of Euclidean or Riemannian space on the exterior algebra, which leads to the notions of Spin-isometric change of coordinates and Spin-isometric manifold used in the model.In the revised paper we correct an error with the formula $G_{ij}=-U^{-1}D_{ij}U/2$, (now U=1).