In the quantum path integral formulation of a field theory model an anomaly arises when the functional measure is not invariant under a symmetry transformation of the Lagrangian. In this paper, generalizing previous work done on the point particle, we show that even at the classical level we can give a path integral formulation for any field theory model. Since classical mechanics cannot be affected by anomalies, the measure of the classical path integral of a field theory must be invariant under the symmetry. The classical path integral measure contains the fields of the quantum one plus some extra auxiliary ones. So, at the classical level, there must be a sort of "cancellation" of the quantum anomaly between the original fields and the auxiliary ones. In this paper we prove in detail how this occurs for the chiral anomaly.
The chiral anomaly originally referred to the anomalous decay rate of the neutral pion, as computed in the current algebra of the chiral model. These
In theoretical physics, a chiral anomaly is the anomalous nonconservation of a chiral current. Such events are expected to be prohibited according to classical conservation laws, but it is known there must be ways they can be broken, because we have evidence of charge–parity non-conservation ("CP violation"). It is possible that other imbalances have been caused by breaking of a chiral law of this
The chiral anomaly originally referred to the anomalous decay rate of the neutral pion, as computed in the current algebra of the chiral model. These calculations suggested that the decay of the pion was suppressed, clearly contradicting experimental results. The nature of the anomalous calculations was first explained in 1969 by Stephen L. Adler and John Stewart Bell and Roman Jackiw. This is called the Adler–Bell–Jackiw anomaly of quantum electrodynamics. This is a symmetry of classical electrodynamics that is violated by quantum corrections.
The Adler–Bell–Jackiw anomaly arises in the following way. If one considers the classical (non-quantized) theory of electromagnetism coupled to massless fermions (electrically charged Dirac spinors solving the Dirac equation), one expects to have not just one but two conserved currents: the ordinary electrical current (the vector current), described by the Dirac field
j
μ
=
ψ
¯
γ
μ
ψ
{\displaystyle j^{\mu }={\overline {\psi }}\gamma ^{\mu }\psi }
as well as an axial current
j
5
μ
=
ψ
¯
γ
5
γ
μ
ψ
.
{\displaystyle j_{5}^{\mu }={\overline {\psi }}\gamma ^{5}\gamma ^{\mu }\psi ~.}
When moving from the classical theory to the quantum theory, one may compute the quantum corrections to these currents; to first order, these are the one-loop Feynman diagrams. These are famously divergent, and require a regularization to be applied, to obtain the renormalized amplitudes. In order for the renormalization to be meaningful, coherent and consistent, the regularized diagrams must obey the same symmetries as the zero-loop (classical) amplitudes. This is…
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