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Noether theorems volume, initial import (M-08)

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{{DISPLAYTITLE:Anomalies: when a classical symmetry does not survive quantisation}}

''Plain-language counterpart: [[Noether's theorems/Explainers/When a symmetry does not survive|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
[[Noether's theorems/Second theorem/Fewer independent field equations than fields|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 [[Noether's theorems/Sources|Sources]].''

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