Gravity as a Higgs Field. I.the Geometric Equivalence Principle

Abstract

{\it If gravity is a metric field by Einstein, it is a Higgs field.} Gravitation theory meets spontaneous symmetry breaking in accordance with the Equivalence Principle reformulated in the spirit of Klein-Chern geometries of invariants. In gravitation theory, the structure group of the principal linear frame bundle $LX$ over a world manifold $X^4$ is reducible to the connected 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 global 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, Dirac fermion fields are described by spinor bundles associated with different reduced Lorentz subbundles of $LX$, and we have nonequivalent representations of cotangent vectors to $X^4$ by Dirac's matrices. It follows that a fermion field must be regarded only in a pair with a certain tetrad field. These pairs fail to be represented by sections of any product bundle $S\times\Si$, but sections of the composite spinor bundle $S\to\Si\to X^4$. They constitute the so-called fermion-gravitation complex where values of tetrad gravitational fields play the role of coordinate parameters, besides the familiar world coordinates. In Part 1 of the article, geometry of the fermion-gravitation complex is investigated. The goal is the total Dirac operator into which components of a connection on $S\to\Si$ along tetrad coordinate directions make contribution. The Part II will be devoted to dynamics of fermion-gravitation complex. It is a constraint system to describe which we use the covariant multisymplectic generalization of the Hamiltonian formalism when canonical momenta correspond to derivatives of fields with respect to all world coordinates, not only the time.