# Chiral Anomaly

In physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this (global) chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.

The non-conservation happens in a tunneling process from one vacuum to another. Such a process is called an instanton. In the case of a symmetry related to the conservation of a fermionic particle number, one may understand the creation of such particles as follows. The definition of a particle is different in the two vacuum states between which the tunneling occurs; therefore a state of no particles in one vacuum corresponds to a state with some particles in the other vacuum.

In particular, there is a Dirac sea of fermions and, when such a tunneling happens, it causes the energy levels of the sea fermions to gradually shift upwards for the particles and downwards for the anti-particles, or vice versa. This means particles which once belonged to the Dirac sea become real (positive energy) particles and particle creation happens.

Technically, an anomalous symmetry is a symmetry of the action, but not of the measure, and therefore not of the generating functional of the quantized theory ( is Planck's action-quantum divided by ). The measure consists of a part depending of the fermion field and a part depending on its complex conjugate . The transformations of both parts under a chiral symmetry do not cancel in general. Note that if is a Dirac fermion, then the chiral symmetry can be written as where is some matrix acting on .

From the formula for one also sees explicitly that in the classical limit, anomalies don't come into play, since in this limit only the extrema of are relevant.

The anomaly is in fact proportional to the instanton number of a gauge field to which the fermions are coupled (note that the gauge symmetry is always non-anomalous and is exactly respected, as is required by the consistency of the theory).