Noether's theorems/Mathematics/Anomalies

Plain-language counterpart: When a symmetry does not survive.

This page derives nothing. It is a summary, and it says so here. It exists to mark the ceiling of the subject and to prevent one specific misreading.

What an anomaly is

UNVERIFIED, all of the following.

A classical theory has a symmetry. The theory is quantised. The symmetry is not a symmetry of the quantised theory.

The failure is not a mistake in the quantisation and cannot be repaired by quantising more carefully. For certain symmetries it is unavoidable: the procedure that makes the quantum theory well defined is itself incompatible with the symmetry, and something has to give.

The standard case is the chiral anomaly, where a current conserved in the classical theory acquires a non-zero divergence in the quantum one. The effect is measurable and was confirmed experimentally before it was fully understood.

The misreading this page exists to prevent

An anomaly is not a failure of Noether's theorem.

Both theorems are conditional statements. Given an invariant integral, certain identities follow. Nothing in either theorem asserts that any particular integral is invariant, and nothing asserts that a classical symmetry must persist through a change of framework.

When a symmetry fails to survive quantisation, the hypothesis of the theorem is not satisfied by the quantum theory. A conditional whose hypothesis fails is not a conditional that has been refuted. The classical theorem remains exactly as true as it was, about the classical theory it was always about.

The correct sentence is: the symmetry did not survive. Not: the theorem broke.

Why it matters that some anomalies are fatal

One consequence is worth recording, because it is the reason anomalies are a constraint on theory-building rather than a curiosity.

If a global symmetry is anomalous, a classically conserved quantity is not conserved. That is a physical prediction and can be checked.

If a local symmetry is anomalous, the theory is inconsistent. The slack identified on the counting page was never physical, and the quantum theory needs it to stay unphysical. An anomaly in a local symmetry means it does not, and the resulting theory predicts nonsense.

So candidate theories are tested for local anomalies before anything else, and the requirement that they cancel is a strong constraint on what content a theory may have. That is the second theorem's structure reappearing as a design rule, and it is as far downstream of 1918 as this volume goes.

Where the volume stops

This is the ceiling. Beyond it the subject becomes quantum field theory proper, and the connection to Noether's paper becomes a matter of ancestry rather than of derivation.

Provenance: no source was reached for this page and nothing on it is derived. Every substantive statement is tagged UNVERIFIED above. The argument that an anomaly is not a counterexample to the theorem is this volume's own and is a point of logic rather than of physics. See Sources.