Misplaced Pages

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Gravitational anomaly" – news · newspapers · books · scholar · JSTOR (November 2019) (Learn how and when to remove this message)

This article provides insufficient context for those unfamiliar with the subject. Please help improve the article by providing more context for the reader. (November 2019) (Learn how and when to remove this message)

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 ${\displaystyle n=1+D/2}$ where ${\displaystyle D}$ is the spacetime dimension.

Gravitational anomalies

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

$${\displaystyle Z=e^{-W}=\int d{\bar {\psi }}d\psi \;\;e^{-\int d^{4}xe{\mathcal {L}}_{\psi }},}$$

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

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

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
• Gravitational instanton

References

• Luis Álvarez-Gaumé; Edward Witten (1984). "Gravitational Anomalies". Nucl. Phys. B. 234 (2): 269. Bibcode:1984NuPhB.234..269A. doi:10.1016/0550-3213(84)90066-X.
• Witten, Edward (1985). "Global gravitational anomalies". Commun. Math. Phys. 100 (2): 197–229. Bibcode:1985CMaPh.100..197W. doi:10.1007/BF01212448. S2CID 9145165.

External links

v t e Quantum gravity
Central concepts	AdS/CFT correspondence Batalin–Vilkovisky formalism CA-duality Causal patch Faddeev–Popov ghost Gravitational anomaly Graviton Holographic principle IR/UV mixing Planck units Quantum foam Ryu–Takayanagi conjecture Trans-Planckian problem Weinberg–Witten theorem
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 Bekenstein bound Bousso's holographic bound Cosmic censorship hypothesis 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 Dual graviton Group field theory Noncommutative geometry Spin foam Superfluid vacuum theory Twistor theory
Applications	Quantum cosmology Eternal inflation FRW/CFT duality Multiverse
See also: Template:Quantum mechanics topics

This quantum mechanics-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Anomalies (physics)
• Quantum gravity
• Quantum physics stubs