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Intheoretical physics, agravitational anomaly is an example of agauge anomaly: it is an effect ofquantum mechanics — usually aone-loop diagram—that invalidates thegeneral covariance of a theory ofgeneral relativity combined with some other fields.^{[citation needed]} The adjective "gravitational" is derived from the symmetry of a gravitational theory, namely from general covariance. A gravitational anomaly is generally synonymous withdiffeomorphism anomaly, sincegeneral covariance is symmetry under coordinate reparametrization; i.e.diffeomorphism.

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

The anomaly usually appears as aFeynman diagram with achiralfermion running in the loop (a polygon) withn externalgravitons attached to the loop where${\displaystyle n=1+D/2}$ where${\displaystyle D}$ is thespacetime dimension.

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

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

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

$${\displaystyle \delta W[e_{\mu }^a]=\int d^{4}xe⟨T_a^{\mu }⟩\delta e_{\mu }^a}$$

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

Lorentz anomaly

$${\displaystyle {\delta }_{\alpha }W=\int d^{4}xe{\alpha }_{ab}⟨T^{ab}⟩,}$$

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

Einstein anomaly

$${\displaystyle {\delta }_{\xi }W=-\int d^{4}xe{\xi }^{\nu }\left({\mathrm{\nabla }}_{\nu }⟨T_{\nu }^{\mu }⟩-{\omega }_{ab\nu }⟨T^{ab}⟩\right),}$$

this is related to the non-conservation of the energy-momentum tensor, i.e.${\displaystyle {\mathrm{\nabla }}_{\mu }⟨T^{\mu \nu }⟩\ne 0}$.

Weyl anomaly

$${\displaystyle {\delta }_{\sigma }W=\int d^{4}xe\sigma ⟨T_{\mu }^{\mu }⟩,}$$

which indicates that the trace is non-zero.

• Mixed anomaly
• Green–Schwarz mechanism
• Gravitational instanton

• 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.

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