John Bell’s Observations on the Chiral Anomaly and Some Properties of Its Descendants

John Bell’s emphasis of the essential ambiguities in anomaly calculations is recalled. Some descendants of the anomaly are reviewed. 1 John Bell and the Chiral Anomaly I expect that most everyone in this audience knows that John Bell codiscovered the mechanism of anomalous symmetry breaking in quantum field theory. Indeed, our paper on this subject [1] is his (and my) most-cited work. The symmetry breaking in question is a quantum phenomenon that violates the correspondence principle; it arises from the necessary infinities of quantum field theory. Over the years it has become evident that theoretical/mathematical physicists are not the only ones to acknowledge this effect. Nature makes fundamental use of the anomaly in at least two ways: the neutral pion’s decay into two photons is controlled by the anomaly [1, 2] and elementary fermions (quarks and leptons) arrange themselves in patterns such that the anomaly cancels in those channels to which gauge bosons – photon, W, Z – couple [3]. (There are also phenomenological applications of the anomaly to collective, as opposed to fundamental, physics – for example, to edge states in the quantum Hall effect.) Beyond physics, in mathematics one finds closely related structures, such as Atiyah-Singer index theory, zero modes of Dirac operators, Chern-Pontryagin characteristics of gauge fields, Chern-Simons secondary characteristics. The mathematical ideas were developed at nearly the same time as the physical ones, and this unexpected conjunction between physics and mathematics seeded joint activity that flourishes to this day.