Gauge gravitation theory
In quantum field theory, gauge gravitation theory is the effort to extend Yang–Mills theory, which provides a universal description of the fundamental interactions, to describe gravity. "Gauge gravitation theory" should not be confused with the similarly-named gauge theory gravity, which is a formulation of (classical) gravitation in the language of geometric algebra. Nor should it be confused with Kaluza–Klein theory, where the gauge fields are used to describe particle fields, but not gravity itself. Overview. The first gauge model of gravity was suggested by Ryoyu Utiyama (1916–1990) in 1956 just two years after birth  of the gauge theory itself. However, the initial attempts to construct the gauge theory of gravity by analogy with the gauge models of internal symmetries encountered a problem of treating general covariant transformations and establishing the gauge status of a pseudo-Riemannian metric (a tetrad field). In order to overcome this drawback, representing tetrad fields as gauge fields of the translation group was attempted. Infinitesimal generators of general covariant transformations were considered as those of the translation gauge group, and a tetrad (coframe) field was identified with the translation part of an affine connection on a world manifold formula_1. Any  such connection is a sum formula_2 of a linear world connection formula_3 and a soldering form formula_4 where formula_5 is a non-holonomic frame. For instance, if formula_6 is the Cartan connection, then formula_7 is the canonical soldering form on formula_1. There are different physical interpretations of the translation part formula_9 of affine connections. In gauge theory of dislocations, a field formula_9 describes a distortion. At the same time, given a linear frame formula_11, the decomposition formula_12 motivates many authors to treat a coframe formula_13 as a translation gauge field. Difficulties of constructing gauge gravitation theory by analogy with the  Yang-Mills one result from the gauge transformations in these theories belonging to different classes. In the case of internal symmetries, the gauge transformations are just vertical automorphisms of a principal bundle formula_14 leaving its base formula_1 fixed. On the other hand, gravitation theory is built on the principal bundle formula_16 of the tangent frames to formula_1. It belongs to the category of natural bundles formula_18 for which diffeomorphisms of the base formula_1 canonically give rise to automorphisms of . These automorphisms are called general covariant transformations. General covariant transformations are sufficient in order to restate Einstein's general relativity and  metric-affine gravitation theory as the gauge ones. In terms of gauge theory on natural bundles, gauge fields are linear connections on a world manifold formula_1, defined as principal connections on the linear frame bundle formula_16, and a metric (tetrad) gravitational field plays the role of a Higgs field responsible for spontaneous symmetry breaking of general covariant transformations. Spontaneous symmetry breaking is a quantum effect when the vacuum is not invariant under the transformation group. In classical gauge theory, spontaneous symmetry breaking occurs if the structure group formula_22 of a principal bundle formula_14 is reducible to a closed subgroup formula_24 , i.e., there exists a principal subbundle of formula_25 with the structure group formula_24. By virtue of the well-known theorem, there exists one-to-one correspondence between the reduced principal subbundles of formula_25 with the structure group formula_24 and the global sections of the quotient bundle . These sections are treated as classical Higgs fields. The idea of the pseudo-Riemannian metric as a Higgs field appeared while constructing non-linear (induced) representations of the general linear group , of which the Lorentz group is a Cartan subgroup. The geometric equivalence principle postulating the existence of a reference frame in which Lorentz  invariants are defined on the whole world manifold is the theoretical justification for the reduction of the structure group of the linear frame bundle to the Lorentz group. Then the very definition of a pseudo-Riemannian metric on a manifold formula_1 as a global section of the quotient bundle leads to its physical interpretation as a Higgs field. The physical reason for world symmetry breaking is the existence of Dirac fermion matter, whose symmetry group is the universal two-sheeted covering of the restricted Lorentz group, .