Gravitational anomaly

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. The adjective "gravitational" is derived from the symmetry of a gravitational theory, namely from general covariance. A gravitational anomaly is generally synonmous with diffeomorphism anomaly, since general covariance is symmetry under coordinate reparametrization; i.e. diffeomorphism.

General covariance is the basis of general relativity, the current 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. Field-theoretic pure gravitational anomalies occur only in even spacetime dimensions.^[1] However, diffeomorphism anomalies can occur in the case of an odd-dimensional spacetime manifold with boundary.^[1]

Gravitational anomalies

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

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

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

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

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

Lorentz anomaly

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

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

Einstein anomaly

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

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

Weyl anomaly

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

which indicates that the trace is non-zero.

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.

1 2 Larsson, Thomas A. (19 April 2004). "A diffeomorphism anomaly in every dimension". Retrieved 13 October 2014. Field-theoretic pure gravitational anomalies only exist in 4k+2 dimensions. However, canonical quantization of non-field-theoretic systems may give rise to diffeomorphism anomalies in any number of dimensions. I present a simple example, where a higher-dimensional generalization of the Virasoro algebra arises upon quantization. [...] Nevertheless, the fact that diffeomorphism anomalies in every dimension are possible outside a field-theoretical framework is quite striking and apparently not widely known.

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Gravitational anomaly

Gravitational anomalies

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