- Art Gallery -

In theoretical physics, a gravitational anomaly is an example of a gauge anomaly: it is an effect of quantum mechanics — usually a one-loop diagram—that invalidates the general covariance of a theory of general relativity combined with some other fields. The adjective "gravitational" is derived from the symmetry of a gravitational theory, namely from general covariance. A gravitational anomaly is generally synonymous with diffeomorphism anomaly, since general covariance is symmetry under coordinate reparametrization; i.e. diffeomorphism.

General covariance is the basis of general relativity, the classical theory of gravitation. Moreover, it is necessary for the consistency of any theory of quantum gravity, since it is required in order to cancel unphysical degrees of freedom with a negative norm, namely gravitons polarized along the time direction. Therefore, all gravitational anomalies must cancel out.

The anomaly usually appears as a Feynman diagram with a chiral fermion running in the loop (a polygon) with n external gravitons attached to the loop where n=1+D/2 where D is the spacetime dimension.

Gravitational anomalies

Consider a classical gravitational field represented by the vielbein \( {\displaystyle e_{\;\mu }^{a}} \) and a quantized Fermi field \( \psi \) . The generating functional for this quantum field is

\( {\displaystyle Z[e_{\;\mu }^{a}]=e^{-W[e_{\;\mu }^{a}]}=\int d{\bar {\psi }}d\psi \;\;e^{-\int d^{4}xe{\mathcal {L}}_{\psi }},} \)

where W is the quantum action and the e {\displaystyle e} e factor before the Lagrangian is the vielbein determinant, the variation of the quantum action renders

\( {\displaystyle \delta W[e_{\;\mu }^{a}]=\int d^{4}x\;e\langle T_{\;a}^{\mu }\rangle \delta e_{\;\mu }^{a}} \)

in which we denote a mean value with respect to the path integral by the bracket \( {\displaystyle \langle \;\;\;\rangle } \). Let us label the Lorentz, Einstein and Weyl transformations respectively by their parameters \( {\displaystyle \alpha ,\,\xi ,\,\sigma } \); they spawn the following anomalies:

Lorentz anomaly

\( {\displaystyle \delta _{\alpha }W=\int d^{4}xe\,\alpha _{ab}\langle T^{ab}\rangle }, \)

which readily indicates that the energy-momentum tensor has an anti-symmetric part.

Einstein anomaly

\( {\displaystyle \delta _{\xi }W=-\int d^{4}xe\,\xi ^{\nu }\left(\nabla _{\nu }\langle T_{\;\nu }^{\mu }\rangle -\omega _{ab\nu }\langle T^{ab}\rangle \right)}, \)

this is related to the non-conservation of the energy-momentum tensor, i.e. \( {\displaystyle \nabla _{\mu }\langle T^{\mu \nu }\rangle \neq 0}. \)

Weyl anomaly

\( {\displaystyle \delta _{\sigma }W=\int d^{4}xe\,\sigma \langle T_{\;\mu }^{\mu }\rangle }, \)

which indicates that the trace is non-zero.
See also

Mixed anomaly
Green–Schwarz mechanism

References

Alvarez-Gaumé, Luis; Edward Witten (1984). "Gravitational Anomalies". Nucl. Phys. B. 234 (2): 269. Bibcode:1984NuPhB.234..269A. doi:10.1016/0550-3213(84)90066-X.

External links

vte

Quantum gravity
Central concepts

AdS/CFT correspondence Ryu-Takayanagi Conjecture Causal patch Gravitational anomaly Graviton Holographic principle IR/UV mixing Planck scale Quantum foam Trans-Planckian problem Weinberg–Witten theorem Faddeev-Popov ghost

Toy models

2+1D topological gravity CGHS model Jackiw–Teitelboim gravity Liouville gravity RST model Topological quantum field theory

Quantum field theory in curved spacetime

Bunch–Davies vacuum Hawking radiation Semiclassical gravity Unruh effect

Black holes

Black hole complementarity Black hole information paradox Black-hole thermodynamics Bousso's holographic bound ER=EPR Firewall (physics) Gravitational singularity

Approaches
String theory

Bosonic string theory M-theory Supergravity Superstring theory

Canonical quantum gravity

Loop quantum gravity Wheeler–DeWitt equation

Euclidean quantum gravity

Hartle–Hawking state

Others

Causal dynamical triangulation Causal sets Noncommutative geometry Spin foam Group field theory Superfluid vacuum theory Twistor theory Dual graviton

Applications

Quantum cosmology
Eternal inflation Multiverse FRW/CFT duality

Physics Encyclopedia

World

Index

Hellenica World - Scientific Library

Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License