Gauge Theory of Gravity

The gauge theory of gravity is an approach to combining gravity with other fundamental interactions that are successfully described in the framework of the gauge theory .

Content

History

The first gauge model of gravity was proposed by R. Utiyama in 1956, two years after the birth of the gauge theory itself. ^[1] However, initial attempts to construct a gauge theory of gravity by analogy with the Yang – Mills gauge theory of internal symmetries encountered the problem of describing general covariant transformations and a pseudo-Riemannian metric (tetrad field) in the framework of such a gauge model.

To solve this problem, it was proposed to represent the tetrad field as a calibration field of the translation group. ^[2] In this case, the generators of general covariant transformations were considered as generators of the translation translation group and the tetrad field (coreper field) was identified with the translational part of the affine connection on the space-time manifold $X$ ${\ displaystyle X}$ . Any such connection is the sum $K=\Gamma +\Theta$ ${\ displaystyle K = \ Gamma + \ Theta}$ general linear connectivity $\Gamma$ ${\ displaystyle \ Gamma}$ on $X$ ${\ displaystyle X}$ and solder form $\Theta =\Theta _{\mu }^{a}dx^{\mu }\otimes \vartheta _{a}$ ${\ displaystyle \ Theta = \ Theta _ {\ mu} ^ {a} dx ^ {\ mu} \ otimes \ vartheta _ {a}}$ where $\vartheta _{a}=\vartheta _{a}^{\lambda }\partial _{\lambda }$ ${\ displaystyle \ vartheta _ {a} = \ vartheta _ {a} ^ {\ lambda} \ partial _ {\ lambda}}$ - non-holonomic benchmark.

There are various physical interpretations of the translation part $\Theta$ ${\ displaystyle \ Theta}$ affine connection. In the gauge theory of dislocations, the field $\Theta$ ${\ displaystyle \ Theta}$ describes distortion. ^[3] In a different interpretation, if the linear frame $\vartheta _{a}$ ${\ displaystyle \ vartheta _ {a}}$ given decomposition $\theta =\vartheta ^{a}\otimes \vartheta _{a}$ ${\ displaystyle \ theta = \ vartheta ^ {a} \ otimes \ vartheta _ {a}}$ gives rise to a number of authors to consider coreper $\vartheta ^{a}$ ${\ displaystyle \ vartheta ^ {a}}$ just like the calibration field of translations. ^[four]

General covariant transformations

The difficulty in constructing the gauge theory of gravity by analogy with the Yang - Mills theory is due to the fact that the gauge transformations of these two theories belong to different classes. In the case of internal symmetries, the gauge transformations are vertical automorphisms of the principal bundle $P\to X$ ${\ displaystyle P \ to X}$ leaving motionless its base $X$ ${\ displaystyle X}$ . At the same time, the theory of gravity is built on the main bundle $F X$ ${\ displaystyle FX}$ tangent frames to $X$ ${\ displaystyle X}$ . It belongs to the category of natural bundles. $T\to X$ ${\ displaystyle T \ to X}$ for which base diffeomorphisms $X$ ${\ displaystyle X}$ canonically continue to automorphisms $T$ ${\ displaystyle T}$ . ^[5] These automorphisms are called general covariant transformations . General covariant transformations are enough to formulate both the general theory of relativity and the affine-metric theory of gravity as a gauge theory. ^[6]

In the gauge theory on natural bundles, gauge fields are linear connections on a space-time manifold $X$ ${\ displaystyle X}$ defined as connections on the main frame $F X$ ${\ displaystyle FX}$ , and the metric (tetrad) field plays the role of the Higgs field responsible for the spontaneous violation of the general covariant transformations. ^[7]

Pseudo-Riemannian metric and Higgs fields

Spontaneous symmetry breaking is a quantum effect when the vacuum is not invariant with respect to a certain group of transformations. In the classical gauge theory, spontaneous symmetry breaking occurs when a structural group $G$ ${\ displaystyle G}$ main bundle $P\to X$ ${\ displaystyle P \ to X}$ reduced to its closed subgroup $H$ ${\ displaystyle H}$ , i.e. there is a main subbundle of the bundle $P$ ${\ displaystyle P}$ with structural group $H$ ${\ displaystyle H}$ . ^[8] In this case, there is a one-to-one correspondence between the reduced subbundles $P$ ${\ displaystyle P}$ with structural group $H$ ${\ displaystyle H}$ and global sections of quotient bundles $P/H\to X$ ${\ displaystyle P / H \ to X}$ . These sections describe classical Higgs fields .

Initially, the idea to interpret the pseudo-Riemannian metric as a Higgs field arose when constructing the induced representations of a general linear group $GL(4,\mathbb {R} )$ ${\ displaystyle GL (4, \ mathbb {R})}$ by the Lorentz subgroup . ^{[9] The} geometric principle of equivalence , which postulates the existence of a reference frame in which the Lorentz invariants are preserved, suggests a reduction of the structural group $GL(4,\mathbb {R} )$ ${\ displaystyle GL (4, \ mathbb {R})}$ main stratification $F X$ ${\ displaystyle FX}$ to the Lorentz group . Then the very definition of a pseudo-Riemannian metric on a manifold $X$ ${\ displaystyle X}$ as a global section of factor bundles $FX/O(1,3)\to X$ ${\ displaystyle FX / O (1,3) \ to X}$ leads to its physical interpretation as a Higgs field.

Notes

↑ R. Utiyama Invariant theoretical interpretation of interaction, - Physical Review 101 (1956) 1597 (Russian translation in Sat. Elementary particles and compensating fields, edited by D. D. Ivanenko , - M .: Mir, 1964).
↑ 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.
↑ C. Malyshev The dislocation stress functions from the double curl $T(3)$ ${\ displaystyle T (3)}$ -gauge equations: Linearity and look beyond, - Annals of Physics 286 (2000) 249.
↑ M. Blagojević Gravitation and Gauge Symmetries, - IOP Publishing, Bristol, 2002.
↑ I. Kolář, PW Michor, J. Slovák Natural Operations in Differential Geometry, - Springer-Verlag, Berlin, Heidelberg, 1993.
↑ Ivanenko D.D. , Pronin P.I., Sardanashvili G.A. Gauge theory of gravity, - M .: Izd. Moscow State University, 1985.
↑ D. Ivanenko , G. Sardanashvily The gauge treatment of gravity, - Physics Reports 94 (1983) 1.
↑ L. Nikolova, V. Rizov Geometrical approach to the reduction of gauge theories with spontaneous broken symmetries, - Reports on Mathematical Physics 20 (1984) 287.
↑ M. Leclerk The Higgs sector of gravitational gauge theories, - Annals of Physics 321 (2006) 708.

Literature

D. D. Ivanenko , G. A. Sardanashvili . Gravity, 4th ed., - M .: Izd. LCI, 2010.
I. Kirsch A Higgs mechanism for gravity, - Phys. Rev. D72 (2005) 024001; arXiv: hep-th / 0503024 .
Yu. Obukhov Poincare gauge gravity: selected topics, - Int. J. Geom. Methods Mod. Phys. 3 (2006) 95-138; arXiv: gr-qc / 0601090 .
G. Sardanashvily Classical gauge gravitation theory, - Int. J. Geom. Methods Mod. Phys. 8 (2011) 1869-1895; arXiv: 1110.1176 .

[1] R. Utiyama Invariant theoretical interpretation of interaction, - Physical Review 101 (1956) 1597 (Russian translation in Sat. Elementary particles and compensating fields, edited by D. D. Ivanenko , - M .: Mir, 1964).

[2] 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.

[3] C. Malyshev The dislocation stress functions from the double curl $T(3)$ ${\ displaystyle T (3)}$ -gauge equations: Linearity and look beyond, - Annals of Physics 286 (2000) 249.

[4] M. Blagojević Gravitation and Gauge Symmetries, - IOP Publishing, Bristol, 2002.

[5] I. Kolář, PW Michor, J. Slovák Natural Operations in Differential Geometry, - Springer-Verlag, Berlin, Heidelberg, 1993.

[6] Ivanenko D.D. , Pronin P.I., Sardanashvili G.A. Gauge theory of gravity, - M .: Izd. Moscow State University, 1985.

[7] D. Ivanenko , G. Sardanashvily The gauge treatment of gravity, - Physics Reports 94 (1983) 1.

[8] L. Nikolova, V. Rizov Geometrical approach to the reduction of gauge theories with spontaneous broken symmetries, - Reports on Mathematical Physics 20 (1984) 287.

[9] M. Leclerk The Higgs sector of gravitational gauge theories, - Annals of Physics 321 (2006) 708.