Anomalies and gravity

Anomalies can be viewed as a breaking of some Noether symmetry through the effects of the vacuum. In relativistic quantum field theory (QFT), such a (classical) symmetry is broken by field quantization, cf. Refs [20, 7, 51] for recent reviews. This has important implications on such physical processes as the decay of the neutral π-meson [6], induced instanton effects [43], or underlies the postulation of the axion in quantum chromodynamics (QCD), cf. Ref. [41]. In quantum electrodynamics (QED), Schwinger [46] demonstrated that the charge current j can be retained conserved, i.e. 〈d j〉 = 0, whereas the conservation of the axial current j5 is broken, 〈d j5〉 6 = 0. In the Fujikawa approach [14], the right-hand side can obtain from considering the point-splitted current j5(x;ε) := ψ(x)γ5γψ(x + ε), where ε is an infinitesimal fourvector in spacetime. Such an expression can be rendered invariant by dressing it with a path-ordered exponential