Editing Chiral anomaly (section)

==General discussion==
In some theories of [[fermion]]s with [[chiral symmetry]], the [[Quantization (physics)|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 process of [[Quantum tunneling|tunneling]] from one [[Vacuum#Quantum mechanics|vacuum]] to another. Such a process is called an [[instanton]].

In the case of a symmetry related to the conservation of a [[particle number operator|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 level]]s 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, in the [[path integral formulation]], an '''anomalous symmetry''' is a symmetry of the [[action (physics)|action]] <math>\mathcal A</math>, but not of the [[measure (physics)|measure]] {{mvar| μ}} and  therefore ''not'' of the [[partition function (quantum field theory)|generating functional]] 
:<math>\mathcal Z=\int\! {\exp (i \mathcal A/\hbar) ~ \mathrm{d} \mu}</math>

of the quantized theory ({{mvar|ℏ}} is Planck's action-quantum divided by 2{{mvar|π}}). The measure <math>d\mu</math> consists of a part depending on the fermion field <math>[\mathrm{d}\psi]</math> and a part depending on its complex conjugate <math>[\mathrm{d}\bar{\psi}]</math>. The transformations of both parts under a chiral symmetry do not cancel in general. Note that if <math>\psi</math> is a [[Dirac fermion]], then the chiral symmetry can be written as <math>\psi \rightarrow e^{i \alpha \gamma^5}\psi</math> where <math>\gamma^5</math> is the chiral [[gamma matrix]] acting on <math>\psi</math>. From the formula for <math>\mathcal Z</math> one also sees explicitly that in the [[classical limit]], {{nowrap|{{mvar|ℏ}} → 0,}} anomalies don't come into play, since in this limit only the extrema of <math>\mathcal A</math> remain relevant.

The anomaly is 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 for the theory to be consistent.)