View source for Noether's theorems/Applications/Topological charge, which is not a Noether charge

{{DISPLAYTITLE:Topological charge, which is not a Noether charge}}

''Plain-language counterpart: [[Noether's theorems/Explainers/A conserved quantity Noether does not explain|A conserved quantity Noether does not explain]].''

[[File:NT-F-24.png|thumb|right|360px|alt=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:

<math>J^{a}=\partial_{b}K^{ab},\qquad K^{ab}=-K^{ba}\qquad\qquad\mathrm{(1)}</math>

Take its divergence:

<math>\partial_{a}J^{a}=\partial_{a}\partial_{b}K^{ab}=0\qquad\qquad\mathrm{(2)}</math>

because <math>\partial_{a}\partial_{b}</math> is symmetric under interchange while
<math>K^{ab}</math> is antisymmetric. The current is conserved.

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

== Why that disqualifies it ==

Compare what
[[Noether's theorems/First theorem/On-shell and off-shell|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
[[Noether's theorems/First theorem/The converse of the first theorem|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
[[Noether's theorems/History/What Noether's theorem now means|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 [[Noether's theorems/Sources|Sources]].''