Divergence Anomaly and Schwinger Terms: Towards a Consistent Theory of Anomalous Classical Fluid

Anomaly, a generic feature of relativistic quantum field theory, is shown to be present in non-relativistic classical ideal fluid. A new result is the presence of anomalous terms in current algebra, an obvious analogue of Schwinger terms present in quantum field theory. We work in Hamiltonian framework where Eulerian dynamical variables obey an anomalous algebra (with Schwinger terms) that is inherited from modified Poisson brackets, with Berry curvature corrections, among Lagrangian discrete coordinates. The divergence anomaly appears in the Hamiltonian equations of motion. A generalized form of fluid velocity field can be identified with the "anomalous velocity" of Bloch band electrons appearing in quantum Hall effect in condensed matter physics. We finally show that the divergence anomaly and Schwinger terms satisfy well known Adler consistency condition. Lastly we mention possible scenarios where this new anomalous fluid theory can impact.