ИСТИНА

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 $LX\to X$ 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 $\Sigma=LX/L$ and, thus, it is treated as a classical Higgs field responsible for the reduction of a structure group $GL(4,\mathbb R)$ of $LX$ 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 $dx^\mu\mapsto h^\mu_a\gamma^a$ of holonomic coframes $\{dx^\mu\}$ 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 $S\to \Sigma \to X$ where $S\to\Sigma$ is a spinor bundle associated with a $SO(1,3)$-principal bundle $LX\to \Sigma$. A key point is that, given a global section $h$ of $\Sigma\to X$, the pull-back bundle $h^*S$ of $S\to\Sigma$ describes Dirac spinor fields in the presence of a gravitational field $h$. At the same time, $S\to X$ 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.