Topological charge, which is not a Noether charge

Plain-language counterpart: A conserved quantity Noether does not explain.

Three dashed circles containing configurations with winding number zero, one and two. A note says you cannot pass from one to another without cutting. A red panel records that the conservation does not use the field equations, so no symmetry is responsible and neither theorem explains it. — Topological charge: winding, not Noether

This page works the argument that matters. The argument is short and it is the sharpest available statement of the theorem's boundary.

A current that is conserved for free

Suppose a current can be written as the divergence of an object antisymmetric in its two indices:

$J^{a} = \partial_{b} K^{a b}, K^{a b} = - K^{b a} (1)$

Take its divergence:

$\partial_{a} J^{a} = \partial_{a} \partial_{b} K^{a b} = 0 (2)$

because $\partial_{a} \partial_{b}$ is symmetric under interchange while $K^{a b}$ is antisymmetric. The current is conserved.

Nothing was imposed. No field equation, no symmetry, no action. Equation (2) holds for any antisymmetric $K$ whatever, including ones describing no physical situation at all.

Why that disqualifies it

Compare what the on-shell page established. A Noether current satisfies an identity off-shell and is conserved only once the field equations are imposed. The imposition is the step where physics enters, and without it there is no conservation law.

A current of the form (1) skips that step. It is conserved before the physics and regardless of it.

So a quantity conserved this way is not conserved because of a symmetry, and Noether's theorem does not explain it. It is conserved because of how it was constructed.

Charges of this kind occur in real physics, where the antisymmetric object is built from the fields and the conserved integer counts something about the shape of the field configuration — how many times it wraps, in the standard cases. They are called topological charges, and they are as conserved as any Noether charge and for an entirely different reason.

The test

Given a conserved current, ask a single question: did the field equations get used?

If yes, it may be a Noether charge and there is a symmetry to look for — and by the converse there certainly is one.

If no, it is not, and looking for a symmetry is looking for something that is not there.

Why the volume needs this page

Because the slogan on the history page runs in one direction only.

Every continuous symmetry gives a conserved quantity — true, and it is the theorem. Every conserved quantity comes from a continuous symmetry — false, and this page is the counterexample.

The converse Noether proved is narrower and correct: every conserved quantity of the kind her theorem produces comes from a symmetry. That qualification is what the slogan drops, and this is where dropping it costs something.

Provenance: (1) and (2) are standard and are derived above. The reading of the off-shell distinction as the disqualifying criterion is this volume's own argument. No source was consulted. See Sources.