On the Variational Principle in the Cartan–Weyl Space

Abstract

In [1, 2] the point of view was expressed that a gauge approach to the theory of gravitation should be added to the existing geometrical approach. On the basis of the gauge theory of the Poincaré-Weyl group it was shown in [1, 2] that the Cartan-Weyl space 4 CW is the geometric background in the modern theory of gravitation. In these same works it was shown that within the framework of the gauge procedure the scalar Dirac field  arises in a natural way from geometrical requirements as an important addition to the metric tensor, having as fundamental a geometrical status as the metric tensor itself. The Dirac field  was introduced in a well-known work by Dirac, and also by Deser [3]. In the construction of the theory of gravitation in post-Riemannian spaces characterized by the 2-forms of curvature a b R and torsion a T and the 1-form of nonmetricity ab Q , the choice of variational procedure and the basis for using this procedure to derive the gravitational field equations play an important role. In the present work we generalize the variational formalism in exterior forms, developed in [3, 4] for the Cartan-Weyl space 4 CW , to the presence of the Dirac scalar field, where the additional condition (Q is the Weyl 1-form)