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. It should not be confused with 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, and not gravity itself.

The first gauge model of gravity was suggested by R. Utiyama 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 ${\displaystyle X}$. Any such connection is a sum ${\displaystyle K=\Gamma +\Theta }$ of a linear world connection ${\displaystyle \Gamma }$ and a soldering form ${\displaystyle \Theta =\Theta _{\mu }^{a}dx^{\mu }\otimes \vartheta _{a}}$ where ${\displaystyle \vartheta _{a}=\vartheta _{a}^{\lambda }\partial _{\lambda }}$ is a non-holonomic frame. For instance, if ${\displaystyle K}$ is the Cartan connection, then ${\displaystyle \Theta =\theta =dx^{\mu }\otimes \partial _{\mu }}$ is the canonical soldering form on ${\displaystyle X}$. There are different physical interpretations of the translation part ${\displaystyle \Theta }$ of affine connections. In gauge theory of dislocations, a field ${\displaystyle \Theta }$ describes a distortion. At the same time, given a linear frame ${\displaystyle \vartheta _{a}}$, the decomposition ${\displaystyle \theta =\vartheta ^{a}\otimes \vartheta _{a}}$ motivates many authors to treat a coframe ${\displaystyle \vartheta ^{a}}$ as a translation gauge field.

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