Spinor Representation of Lie Algebra for Complete Linear Group

Spinor representation of the group GL(4,R) is needed for correct description of the Fermi fields on Riemann space, such as the space-time of general relativity. It is used for two purposes: to define the connectivity and covariant derivative of spinor field and to define the Lie derivative. Recent publications [1–3] have reminded of this problem. The important problem for definition of Fermi fields on Riemann space is that transformation properties of Dirac equation correspond [4] to Spin(3, 1) representation of Lorentz group SO(3, 1) only, not the full linear group GL(4,R). Covariant derivative definition and field equations based on it can be defined by the field of orthonormal basis – tetrad description of curve geometry. In such way the tetrad connectivity is a member of Lorentz group and generates the spinor connectivity as standard Spin(3, 1) representation. Spinor representation of group GL(4,R) is needed for investigation of the spinor field symmetry as realization of the space-time symmetry. In the case when the space-time symmetry subgroup G is different from SO(3, 1), the subgroup spinor representation of that symmetry cannot be realized as Spin(3, 1) subgroup and, one needs the spinor representation of G. As example we can take the standard model of Universe and its G(6) group of symmetry. It contains the subgroup G(3) of isotropy – subgroup of Lorentz group SO(3, 1), and subgroup G(3) of translations. The latter is not a part of the Lorentz group and we can describe translation properties of spinor field (i.e. electron) through spinor representation of group GL(4,R) only. Here we give results of investigations in special construction for the spinor field on the spacetime of general relativity. This is an extension of our investigation [5] of spinor representation for full linear group GL(4,R).