Editing Gravitational anomaly

{{Short description|Breakdown of general covariance at the quantum level}}
{{Distinguish|Gravity anomaly}}
{{More citations needed|date=November 2019}}
{{Context|date=November 2019}}
[[File:Triangle diagram.svg|thumb|right|Anomalies in the usual 4 spacetime dimensions arise from triangle Feynman diagrams]]
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.{{Citation needed|date=September 2019}} 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 [[graviton]]s polarized along the time direction. Therefore, all gravitational anomalies must cancel out.

The anomaly usually appears as a [[Feynman diagram]] with a [[chirality (physics)|chiral]] [[fermion]] running in the loop (a polygon) with ''n'' external [[graviton]]s attached to the loop where <math>n=1+D/2</math> where <math>D</math> is the [[spacetime]] dimension.

== Gravitational anomalies ==
Consider a classical gravitational field represented by the vielbein <math>e^a_{\;\mu}</math> and a quantized Fermi field <math>\psi</math>. The generating functional for this quantum field is

<math display="block">Z[e^a_{\;\mu}]=e^{-W[e^a_{\;\mu}]}=\int d\bar{\psi}d\psi\;\; e^{-\int d^4x e \mathcal{L}_{\psi}},</math>

where <math>W</math> is the quantum action and the <math>e</math> factor before the Lagrangian is the vielbein determinant, the variation of the quantum action renders

<math display="block">\delta W[e^a_{\;\mu}]=\int d^4x \; e \langle T^\mu_{\;a}\rangle \delta e^a_{\;\mu}</math>

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

'''Lorentz anomaly'''

<math display="block">\delta_\alpha W=\int d^4x e \, \alpha_{ab}\langle T^{ab} \rangle,</math>

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

'''Einstein anomaly'''

<math display="block">\delta_\xi W=-\int d^4x e \, \xi^\nu \left(\nabla_\nu\langle T^\mu_{\;\nu}\rangle-\omega_{ab\nu}\langle T^{ab}\rangle\right),</math>

this is related to the non-conservation of the energy-momentum tensor, i.e. <math>\nabla_\mu\langle T^{\mu\nu}\rangle \neq 0</math>.

'''Weyl anomaly'''

<math display="block">\delta_\sigma W=\int d^4x e \, \sigma\langle T^\mu_{\;\mu}\rangle,</math>

which indicates that the trace is non-zero.

== See also ==
* [[Mixed anomaly]]
* [[Green–Schwarz mechanism]]
*[[Gravitational instanton]]

== References ==
* {{cite journal | author1=[[Luis Álvarez-Gaumé]] |author2=[[Edward Witten]]  | title = Gravitational Anomalies | journal = Nucl. Phys. B | volume = 234 | issue = 2 | pages = 269 | date = 1984 | doi = 10.1016/0550-3213(84)90066-X |bibcode = 1984NuPhB.234..269A }}
* {{Cite journal|last=Witten|first=Edward|date=1985|title=Global gravitational anomalies|url=https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-100/issue-2/Global-gravitational-anomalies/cmp/1103943444.full|journal=Commun. Math. Phys.|volume=100|issue=2|pages=197–229|doi=10.1007/BF01212448|bibcode=1985CMaPh.100..197W|s2cid=9145165}}

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==External links==
{{quantum gravity}}

[[Category:Anomalies (physics)]]
[[Category:Quantum gravity|Anomaly]]
{{quantum-stub}}