Classical field theory admits a comprehensive mathematical formulation in the geometric terms of smooth fibre bundles. For instance, Yang – Mills gauge theory is theory of principal connections on principal bundles.
Gauge gravitation theory as particular classical field theory also is formulated in the terms of fibre bundles.
Studying gauge gravitation theory, one believes reasonable to require that it incorporates Einstein's General Relativity and, therefore, it should be based on Relativity and Equivalence Principles reformulated in the fibre bundle terms.
In these terms, Relativity Principle states that gauge symmetries of classical gravitation theory are general covariant transformations. It should be emphasized that these gauge symmetries differ from gauge symmetries of the above mentioned Yang – Mills gauge theory which constitute a gauge group of vertical automorphisms of a principal bundles. Fibre bundles possessing general covariant transformations constitute the category of so called natural bundles.
Let Y->X be a smooth fibre bundle. Any automorphism of Y, by definition, is projected onto a diffeomorphism of its base X. The converse is not true. A fibre bundle Y->X is called the natural bundle if there exists a monomorphism of the group of diffeomorphisms of X to the group of bundle automorphisms of Y->X, called general covariant transformations of Y.
The tangent bundle TX of X exemplifies a natural bundle. Any diffeomorphism f of X gives rise to the tangent automorphisms Tf of TX which is a general covariant transformation of TX.The associated principal bundle is a fibre bundle LX of frames in the tangent spaces to X also is a natural bundle. Moreover, all fibre bundles associated with LX are natural bundles. Principal connections on LX yield linear connections on the tangent bundle TX and other associated bundles. They are called the world connections.
Following Relativity Principle, one thus should develop gravitation theory as gauge theory of principal connections on a principal frame bundle LX over a four-dimensional manifold X, called the world manifold. A key point however is that this gauge theory also is characterized by spontaneous symmetry breaking in accordance with geometric Equivalence Principle.
Though spontaneous symmetry breaking is quantum effect, spontaneous symmetry breaking in classical gauge theory on a principal bundle P->X with a structure Lie group G is characterized as a reduction of this structure group to its closed Lie subgroup H. By virtue of the well-known theorem, such a reduction takes place if and only if there exists a global sections h of the quotient bundle P/H->X which are treated as a classical Higgs field.
There are different formulations of Equivalence Principle in gravitation theory. In particular, one separates weakest, weak, middle-strong and strong Equivalence Principles. All of them are based on the empirical equality of inertial mass, gravitational active and passive charges. The weakest Equivalence Principle is restricted to the motion law of a probe point mass in a uniform gravitational field. Its localization is the weak Equivalence Principle that states the existence of a desired local inertial frame at a given world point. This is the case of equations depending on a gravitational field and its first order derivatives, e.g., the equations of mechanics of probe point masses, and the equations of electromagnetic and Dirac fermion fields. The middle-strong Equivalence Principle is concerned with any matter, except a gravitational field, while the strong one is applied to all physical laws.
The above mentioned variants of Equivalence Principle aim to guarantee the transition of General Relativity to Special Relativity in a certain reference frame. However, only the particular weakest and weak Equivalence Principles are true. To overcome this difficulty, Equivalence Principle can be formulated in geometric terms as follows. In the spirit of Felix Klein's Erlanger program, Special Relativity can be characterized as the Klein geometry of Lorentz group invariants. Then geometric Equivalence Principle is formulated to require the existence of Lorentz invariants on a world manifold X. This requirement holds if the tangent bundle of X admits an atlas with Lorentz transition functions, i.e., a structure group of the associated frame bundle LX of linear tangent frames in is reduced to the Lorentz group SO(1,3). By virtue of the above mentioned theorem, this reduction takes place if and only if the quotient bundle LX/SO(1,3) possesses a global section, which is a pseudo-Riemannian metric on X.
Thus geometric Equivalence Principle provides the necessary and sufficient conditions of the existence of a pseudo-Riemannian metric, i.e., a gravitational field on a world manifold. Based on geometric Equivalence Principle, gravitation theory is formulated as gauge theory where a gravitational field is described as a classical Higgs field responsible for spontaneous breakdown of world gauge symmetries which are general covariant transformations.
The character of gravity as a Higgs field responsible for spontaneous breaking of general covariant transformations is displayed as follows. Given different gravitational fields, the representations of holonomic coframes dx by Dirac matrices acting on Dirac spinor fields are nonequivalent. Consequently, Dirac operators in the presence of different gravitational fields fails to be equivalent, too.
It follows that, since the Dirac operators in the presence of different gravitational fields are nonequivalent, Dirac spinor fields fail to be considered, e.g., in the case of a superposition of different gravitational fields. Therefore, quantization of a metric gravitational field fails to satisfy the superposition principle, and one can suppose that a metric gravitational field as a Higgs field is non-quantized in principle.
WikipediA: Gauge gravitation theory
G.Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys.8 (2011) 1869-1895.
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