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,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 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
WikipediA:
Gauge gravitation theory.
D. Ivanenko,
G. Sardanashvily: The gauge treatment of gravity, Phys. Rep. 94 (1983)
1-45 (#).
G. Sardanashvily:
Classical gauge gravitation theory, Int. J.
Geom. Methods Mod. Phys. 8
(2011) 1869-1895 (#); arXiv: 1110.1176.
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