Quantum anomalies and consistency

A Ward identity is derived by changing variables in a path integral. Classically invariant action terms are only part of the calculation: the functional measure and ultraviolet definition must also be transformed. For a chiral rotation of fermions, the regulated Jacobian can be nontrivial. The axial anomaly is therefore not the claim that classical algebra was performed incorrectly; it says that the classical symmetry cannot be maintained together with the other required quantum properties in the specified theory.

A global current may have an anomalous divergence in background gauge fields while the theory remains perfectly consistent. The familiar axial-current example is compatible with preserving the electromagnetic gauge current. By contrast, an uncanceled anomaly in a dynamical gauge redundancy obstructs the Ward or BRST identities needed to remove unphysical polarizations. A four-dimensional chiral gauge theory must cancel its relevant gauge anomalies, or supply an explicitly specified additional mechanism that restores consistency. Ignoring a gauge anomaly is not an acceptable approximation.

A right-handed Weyl field of charge $q$ contributes with the opposite chirality sign from a left-handed field of the same charge. A convenient bookkeeping method rewrites it as a left-handed charge-conjugate field of charge $-q$. A vectorlike Dirac pair then appears as left-handed charges $q$ and $-q$, whose cubic and linear sums vanish. Forgetting this conversion produces a fictitious anomaly even in ordinary vectorlike electrodynamics.

Cancellation of the linear sum does not imply cancellation of the cubic sum. Nor do these two Abelian checks exhaust a general model: non-Abelian cubic anomalies, mixed gauge factors, and possible global gauge anomalies must be considered for the actual group and representation. For example, a mixed non-Abelian-squared–U(1) coefficient weights the Abelian charge by the representation's Dynkin index and multiplicities. State the complete particle representations and chirality convention before declaring a proposed model anomaly free.

An 't Hooft anomaly obstructs coupling a global symmetry consistently to arbitrary background gauge fields while preserving the desired symmetry properties. The original theory with those nondynamical sources switched off can still be consistent. Integrating out short-distance modes cannot erase the obstruction, so an infrared description must reproduce it. Gapless particles, symmetry breaking with suitable effective terms, or more subtle long-distance structures can carry the required information.

Anomaly matching is a constraint rather than a unique solution algorithm. Several infrared phases may satisfy the same matching conditions, and one must track the same symmetry, normalization, and allowed local counterterms on both sides. A global anomaly also differs from a local anomaly detected by a triangle diagram: topology can reveal obstructions invisible to the simple cubic sums above. The practical habit is to state exactly which anomaly has been checked and which consistency questions remain open.