Anomalies

I have been thinking a lot about anomalies of late. The somewhat naive slogan given in almost every quantum field theory textbook states: a quantum anomaly is a symptom of the failure to preserve classical symmetry in the process of quantisation and regularisation. The prototypical example used to introduce this view is the chiral (or axial) anomaly from triangle Feynman diagrams. The simplest example of such a calculation would be to consider something like massive spinor electrodynamics with the Lagrangian $\mathcal{L} = \bar{\psi}\gamma^{\mu}(i\partial_{\mu} - e A_{\mu})\psi - m\bar{\psi}\psi$ , and then to study the Ward-Takahashi identities. (From the view of classical field theory, a basic statement of the Noether theorem is how symmetries are related with conservation laws. When we quantise, the conservation laws of the continuous symmetries become embodied in what are called Ward-Takahashi identities). The vector current can be defined as $V_{\mu}(x) = \bar{\psi}(x)\gamma_{\mu}\psi(x)$ and similarly the axialvector current $A_{\mu}(x) = \bar{\psi}(x)\gamma_{\mu}\gamma_5\psi(x)$ . Classically, one will find that the vector current is conserved $\partial^{\mu}V_{\mu} = 0$ and, for massless fermions $m=0$ , the same is true for the axialvector current $\partial^{\mu}A_{\mu} = 0$ . However, in the case of massive fermions $m\neq0$ , the conservation law for the axialvector current becomes $\partial^{\mu}A_{\mu} = 2mP$ , where $P$ is a pseudoscalar current. At the end – depending on the regularisation method used (as the calculations involve divergent integrals) and other conventions, so for the sake of this example let’s use Pauli-Villars – the anomalous Ward identity will generally have the structure

$\displaystyle (k_1 + k_2)_{\mu}T^{\mu \nu \rho}(k_1,k_2,m) = 2imT_{\nu \rho}(m) + \frac{e^2}{2\pi^2}\epsilon_{\nu \rho \alpha \beta} k_{1\alpha}k_{2\beta},$

where the second term on the right-hand side of the equality is the anomaly. This is also known as the Adler-Bell-Jackiw anomaly (often in the literature ABJ anomalies, chiral anomalies, and also triangle anomalies are used interchangeably). One nice observation is that the chiral anomalies will always have an epsilon structure; so when faced with calculating a bunch of regulated integrals this is a quick way to see which will contribute to the structure of the anomalous Ward identity.

In general, I’ve learned that I don’t like the above definition of anomalies according to its interpretation as a violation of symmetries when moving from the classical to the quantum. (I also don’t like how such calculations as those described above seem riddled with ambiguities at various stages, with a number of operating assumptions appearing to me to be less than satisfactory). I mean, it is one layer of the overall picture; but to take this view ignores the fact that there are also examples of purely classical anomalies. So on one level, yes, anomalies are due to the fact that quantum field theories require regularisation and there is difficulty in satisfying all of the symmetries of the classical theory. This view is then deepened when we think of the chiral anomalies by way of the path integral using Fujikawa’s approach. In this approach we see not only the connection between the anomaly and the violation of symmetry, as above; more deeply, anomalies are found to arise when symmetries of the action are not symmetries of the functional measure in the path integral. In fact, there is an even stronger statement to be made: anomalies arise due to the path integral measure $\mathcal{D}\psi \mathcal{D}\bar{\psi}$ being ill-defined! The general idea is that under symmetry transformation the anomalous part of the measure arises from the Jacobian giving $\mathcal{D}\psi^{\prime} \mathcal{D}\bar{\psi}^{\prime} = \mathcal{D}\psi \mathcal{D}\bar{\psi} \det C^{-2} = \mathcal{D}\psi \mathcal{D}\bar{\psi} \exp(-2 Tr \ln C)$ . This is UV divergent and thus the path integral measure requires regularisation! But there is no regularised functional measure in the path integral that preserves all of the symmetries. So, again, we see connections between preservation of the underlying group symmetries, regularisation, and general anomalous structure.

Learning the Fujikawa approach and then thinking of its generalisation has felt like peeling another layer off the onion. This also happens to be the subject of some ongoing work in the context of $\eta$ regularisation that I am growing increasingly excited about. But I’m not sure how many more layers there are to peel; for example, I know from string theory there are topological connections. According to this comment by David Bar Moshe, there are also geometric interpretations which, on the mathematics side, relate to various obstructions when we try to life the action to adequate geometrical objects (for instance, from classical to quantum). At the heart of it, at least from what I presently understand, is again the issue of the realisation of the relevant symmetry group. Still, there are more layers of the onion to peel. Another account describes anomalies as arising from the requirement of an ordering prescription of quantum operators – you may have guessed, there are then issues is respecting the underlying symmetries for a given ordering prescription. There is also the Hamiltonian view regarding generators of the symmetries, and finally the account concerning central charges in the conservation algebra.

All of these different accounts can make for some confusion; and it doesn’t help, at least from my reading, that there seem to be some contradictions in the literature. But no matter how one slices the onion, the study of anomalies seems no less captivating.

If one wants to read about the historical development of anomalies there are two terrific sources I’d recommend as a start: the book by Fujikawa and Suzuki, Path Integrals and Quantum Anomalies, and the textbook by Bastianelli and van Nieuwenhuizen, Path Integrals and Anomalies in Curved Space.

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