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^[1] just two years after birth of the gauge theory itself.^[2] 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.^[3] 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 $X$ . Any such connection is a sum $K=\Gamma +\Theta$ of a linear world connection $\Gamma$ and a soldering form $\Theta =\Theta _{\mu }^{a}dx^{\mu }\otimes \vartheta _{a}$ where $\vartheta _{a}=\vartheta _{a}^{\lambda }\partial _{\lambda }$ is a non-holonomic frame. For instance, if $K$ is the Cartan connection, then $\Theta =\theta =dx^{\mu }\otimes \partial _{\mu }$ is the canonical soldering form on $X$ . There are different physical interpretations of the translation part $\Theta$ of affine connections. In gauge theory of dislocations, a field $\Theta$ describes a distortion.^[4] At the same time, given a linear frame $\vartheta _{a}$ , the decomposition $\theta =\vartheta ^{a}\otimes \vartheta _{a}$ motivates many authors to treat a coframe $\vartheta ^{a}$ as a translation gauge field.^[5]

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 $P\to X$ leaving its base $X$ fixed. On the other hand, gravitation theory is built on the principal bundle $FX$ of the tangent frames to $X$ . It belongs to the category of natural bundles $T\to X$ for which diffeomorphisms of the base $X$ canonically give rise to automorphisms of $T$ .^[6] 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 $X$ , defined as principal connections on the linear frame bundle $FX$ , and a metric (tetrad) gravitational field plays the role of a Higgs field responsible for spontaneous symmetry breaking of general covariant transformations.^[7]

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 $G$ of a principal bundle $P\to X$ is reducible to a closed subgroup $H$ , i.e., there exists a principal subbundle of $P$ with the structure group $H$ .^[8] By virtue of the well-known theorem, there exists one-to-one correspondence between the reduced principal subbundles of $P$ with the structure group $H$ and the global sections of the quotient bundle $P/H\to X$ . 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 $GL(4,{\mathbb R})$ , of which the Lorentz group is a Cartan subgroup.^[9] 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 of that the structure group $GL(4,{\mathbb R})$ of the linear frame bundle $FX$ is reduced to the Lorentz group. Then the very definition of a pseudo-Riemannian metric on a manifold $X$ as a global section of the quotient bundle $FX/O(1,3)\to X$ 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 $SL(2,{\mathbb C})$ of the restricted Lorentz group, $SO^{+}(1,3)$ .^[10]

Notes

↑ R Utiyama, Invariant theoretical interpretation of interaction, Physical Review 101 (1956) 1597. doi:10.1103/PhysRev.101.1597
↑ Blagojević, Milutin; Hehl, Friedrich W. (2013). Gauge Theories of Gravitation: A Reader with Commentaries. World Scientific. ISBN 978-184-8167-26-1.
↑ F.Hehl, J. McCrea, E. Mielke, Y. Ne'eman, Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilaton invariance, Physics Reports 258 (1995) 1. doi:10.1016/0370-1573(94)00111-F
↑ C.Malyshev, The dislocation stress functions from the double curl $T(3)$ -gauge equations: Linearity and look beyond, Annals of Physics 286 (2000) 249. doi:10.1006/aphy.2000.6088
↑ M. Blagojević, Gravitation and Gauge Symmetries (IOP Publishing, Bristol, 2002).
↑ I. Kolář, P. W. Michor, J. Slovák, Natural Operations in Differential Geometry (Springer-Verlag, Berlin, Heidelberg, 1993).
↑ D.Ivanenko, G.Sardanashvily, The gauge treatment of gravity, Physics Reports 94 (1983) 1. doi:10.1016/0370-1573(83)90046-7
↑ L. Nikolova, V. Rizov, Geometrical approach to the reduction of gauge theories with spontaneous broken symmetries, Reports on Mathematical Physics 20 (1984) 287. doi:10.1016/0034-4877(84)90039-9
↑ M. Leclerk, The Higgs sector of gravitational gauge theories, Annals of Physics 321 (2006) 708. doi:10.1016/j.aop.2005.08.009
↑ G. Sardanashvily, O.Zakharov, Gauge Gravitation Theory (World Scientific, Singapore, 1992).

References

I. Kirsch, A Higgs mechanism for gravity, Phys. Rev. D72 (2005) 024001; arXiv: hep-th/0503024.
G. Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1869-1895; arXiv: 1110.1176.
Yu. Obukhov, Poincaré gauge gravity: selected topics, Int. J. Geom. Methods Mod. Phys. 3 (2006) 95-138; arXiv: gr-qc/0601090.

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Gauge gravitation theory

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References