String Theory Journal Club

The chiral anomaly is often presented either as a consequence of the non-invariance of the Fujikawa path-integral measure, or as the result of a subtle failure of naive dimensional regularisation in the one-loop triangle diagram. In this talk, I argue that this standard presentation can obscure the more elementary algebraic origin of the effect. The anomaly can be understood as a Schwinger term: a c-number extension in the equal-time current algebra of normal-ordered fermion bilinears.Starting from the algebra of fermionic creation and annihilation operators, I show how the choice of Dirac sea modifies the current algebra by generating a classically absent Schwinger term. In 1+1 dimensions, this gives the familiar central term directly, without any additional regulator, making it possible to calculate the anomaly and interpret its physical origin in a transparent way. In particular, no renormalisation beyond that required to define a consistent quantum theory in an external background field is needed to derive the anomaly, making it, in this sense, a free-theory effect.Finally, I discuss the relation between this current-algebra perspective and other regularisation schemes, including Fujikawa’s method, heat-kernel regularisation, and dimensional regularisation.