Anomaly cancellation

In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physics, a classical anomaly is the failure of a symmetry to be restored in the limit in which the symmetry-breaking parameter goes to zero. Perhaps the first known anomaly was the dissipative anomaly in turbulence: time-reversibility remains broken (and energy dissipation rate finite) at the limit of vanishing viscosity.

Technically, an anomalous symmetry in a quantum theory is a symmetry of the action, but not of the measure, and so not of the partition function as a whole.

A global anomaly is anomaly associated with global symmetry.

The most prevalent global anomaly in physics is associated with the violation of scale invariance by quantum corrections, quantified in renormalization. Since regulators generally introduce a distance scale, the classically scale-invariant theories are subject to renormalization group flow, i.e., changing behavior with energy scale. For example, the large strength of the strong nuclear force results from a theory that is weakly coupled at short distances flowing to a strongly coupled theory at long distances, due to this scale anomaly.

Anomalies in abelian global symmetries pose no problems in a quantum field theory, and are often encountered (see the example of the chiral anomaly). In particular the corresponding symmetries can be fixed by fixing the boundary conditions of the path integral.

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