Has it been observed in solid state physics?
At the Journal Club of Condensed Matter, Patrick Lee has a very nice and helpful commentary, Observation of the chiral anomaly in solids, that highlights a preprint, based on a recent talk by Phuan Ong at the APS March meeting.
Later I hope to write more about the solid state physics, earlier work of which I have mentioned before.
I just want to highlight a beautiful and succinct paragraph from the preprint that explains why the chiral anomaly is so important in quantum field theory.
A bit of topological physics fell into quantum field theory (QFT) in the late 1960s [8–11]. The charged pions π± are remarkably long-lived mesons (lifetimes τ ∼ 2.3 × 10−8 s) because, being the lightest hadron, they can only decay by the weak interaction into muons and neutrinos via the processes π− → μ− + ν ̄μ and π+ → μ+ + νμ. Mysteriously, the neutral pion π0 decays much more quickly (by a factor of 300 million) even though it is a member of the same isospin triplet. Instead of the slow leptonic channels, π0 decays by coupling to the electromagnetic field Fμν in the process π0 → 2γ. The relevant diagram (shown below), called the Adler-Bell-Jackiw anomaly [8, 9], is a triangular fermion loop that links the π0 (the axial current) to the 2 photons (vector currents) [10, 11].
A hint of the topological nature of the anomaly is that the one-loop diagram receives no further corrections to all orders of perturbation theory. Subsequent research revealed that the anomaly expresses the breaking of a classical symmetry by quantum fluctuations. In modern QFT, the anomaly plays the fundamental role of killing unviable gauge theories [10, 11]. A proposed chiral gauge theory must be anomaly free. Otherwise it is not renormalizable. Arguably the most important example of the anomaly-free rule is the Glashow- Salam-Weinberg electroweak theory, in which the 4 triangle anomalies linking the lepton and quark doublets with gauge bosons sum exactly to zero within each generation. This fortuitous cancellation has been called “magical” .
A measure of the importance of this idea of anomaly cancellation is the fact that much of the initial string theory hype in 1980s arose because it was found that certain string theories with certain symmetry groups did have the sought after anomaly cancellations. See here for more.
Addendum (23 June, 2015): Peter Woit just posted about a recent talk "Anomalies Revisited" by Ed Witten.
The fundamental issue is that these are theories where the path integral does not determine the phase of the partition function. Part of story here is the well-known story of anomalies, perturbative and global. One interesting point Witten makes is that vanishing of these anomalies is not sufficient to be able to consistently choose the phase of the partition function, and he gives a conjecture for a necessary condition that is stronger than anomaly cancellation.
Witten discusses the specific case of three-dimensional massless Majorana fermions that may be realised in a topological superconductor.