Classical
gravitation theory on a world manifold

*X*is formulated as gauge theory on natural bundles over*X*which admit general covariant transformations as the canonical functorial lift of diffeomorphisms of their base*X*. Natural bundles are exemplified by a principal linear frame bundle*LX -> X*and the associated, (e.g., tensor) bundles. This is metric-affine gravitation theory whose dynamic variables are general linear connections (principal connections on*LX*) and a metric (tetrad) gravitational field. The latter is represented by a global section of the quotient bundle*W=LX/SO(3,1)*and, thus, it is treated as a classical Higgs field responsible for the reduction of a structure group*GL(4,*of**R**)*LX*to a Lorentz group*SO(1,3)*. The underlying physical reason of this reduction is both the geometric Equivalence Principle and the existence of Dirac spinor fields. Herewith, a structure Lorentz group of*LX*always is reducible to its maximal compact subgroup*SO(3)*that provides a world manifold*X*with a space-time structure. The physical nature of gravity as a Higgs field is characterized by the fact that, given different tetrad gravitational fields*h*, the representations of holonomic coframes*{dx}*on a world manifold*X*by Dirac's gamma-matrices are non-equivalent. Consequently, the Dirac operators in the presence of different gravitational fields fails to be equivalent, too. To solve this problem, we describe Dirac spinor fields in terms of a composite spinor bundle*S -> W -> X*where*S -> W*is a spinor bundle associated with a*SO(1,3)*-principal bundle*LX -> W*. A key point is that, given a global section*h*of*W -> X*, the pull-back bundle*h*S*of*S -> W*describes Dirac spinor fields in the presence of a gravitational field*h*. At the same time,*W -> X*is a natural bundle which admits general covariant transformations. As a result, we obtain a total Lagrangian of a metric-affine gravity and Dirac spinor fields, whose gauge invariance under general covariant transformations implies an energy-momentum conservation law. Our physical conjecture is that a metric gravitational field as the Higgs one is non-quantized, but it is classical in principle.**References**

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*Int. J. Geom. Methods Mod. Phys.***8**(2011) 1869-1895**(#)**;*.***arXiv: 1110.1176**
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