Gravity as a Higgs Field. II.Fermion-Gravitation Complex

Abstract

Gravitation theory meets spontaneous symmetry breaking when the structure group of the principal linear frame bundle $LX$ over a world manifold $X^4$ is reducible to the Lorentz group $SO(3,1)$. The physical underlying reason of this reduction is Dirac fermion matter possessing only exact Lorentz symmetries. The associated Higgs field is a tetrad gravitational field $h$ represented by a section of the quotient $\Si$ of $LX$ by $SO(3,1)$. The feature of gravity as a Higgs field issues from the fact that, in the presence of different tetrad fields, there are nonequivalent representations of cotangent vectors to $X^4$ by Dirac's matrices. It follows that fermion fields must be regarded only in a pair with a certain tetrad field. These pairs constitute the so-called fermion-gravitation complex and are represented by sections of the composite spinor bundle $S\to\Si\to X^4$ where values of tetrad gravitational fields play the role of coordinate parameters, besides familiar world coordinates. In Part I of the work [gr-qc:9405013], geometry of this composite spinor bundle has been investigated. This Part is devoted to dynamics of the fermion-gravitation complex. It is a constraint system to describe which we use the covariant multimomentum Hamiltonian formalism when canonical momenta correspond to derivatives of fields with respect to all world coordinates, not only time. On the constraint space, the canonical momenta of tetrad gravitational fields are equal to zero, otherwise in the presence of fermion fields.