A conserved quantity Noether does not explain
Mathematical counterpart: Topological charge, which is not a Noether charge.
The slogan says every continuous symmetry gives a conserved quantity. People usually hear it as saying every conserved quantity comes from a symmetry. This page is the counterexample.
A knot you cannot untie
Take a long strip of rubber, put a twist in it, and join the ends into a loop.
You can stretch the loop, deform it, wobble it about. What you cannot do, without cutting, is remove the twist. Count the twists and you have a number that stays the same no matter what you do to the rubber.
The number is conserved. Nothing about symmetry came into it. It is conserved because of the shape of the thing — because there is no continuous way to get from one twist to none.
Physics has fields that behave this way. A field configuration can wrap around in a way that cannot be smoothly undone, the wrapping count is a whole number, and it does not change as the field evolves. These are called topological charges.
Why it is not a Noether charge, in one question
Recall the two-step structure from the identity page.
A Noether current satisfies an identity that holds for every configuration, real or not. It becomes conserved only when you add the condition that the system obeys its equations of motion. That second step is where physics enters, and without it there is no conservation law at all.
A topological charge skips that step. Its conservation follows from how the quantity is built, and it holds for every configuration, including ones that violate every law of physics you care to name.
So the test is a single question.
Were the equations of motion used?
If yes, it may well be a Noether charge, and by the reverse direction there is definitely a symmetry to find. If no, it is not, and looking for a symmetry is looking for something that is not there.
Why the slogan misleads
The theorem runs one way. Continuous symmetry gives conserved quantity.
Noether also proved a reverse direction, and it is genuinely useful — but it is narrower than the slogan suggests. What she proved is that every conserved quantity of the kind her theorem produces comes from a symmetry. That qualification is not decoration. It is what excludes cases like this one.
Drop the qualification and you get a false statement that sounds like a theorem, and a physicist who believes it will hunt for a symmetry behind a topological charge indefinitely.
Why this matters more than it sounds
Because whole-number conserved quantities are unusually valuable in physics.
A Noether charge is generally a continuous quantity — an amount of something, which can be any value. A topological charge is a count, and a count cannot drift a little. It is either three or it is four.
That makes such quantities exceptionally robust, and physics uses them where robustness matters: for particle-like objects that cannot decay because their wrapping has nowhere to go, and for states of matter that keep their properties despite disorder and imperfection.
So the one place in this volume where Noether's theorems fall silent is also one of the more useful corners of modern physics. Knowing where a theorem stops is part of knowing what it says.
Why a count cannot drift
The most useful property of these quantities follows from their being whole numbers, and it is worth seeing why that matters so much.
An ordinary conserved quantity is an amount. In a real system, subject to imperfections and disturbances, an amount that is supposed to stay fixed will generally stay nearly fixed. Small effects the theory left out produce small drifts.
A count cannot do that. There is no such thing as three and a bit twists in the rubber loop. Whatever disturbs the system either has enough effect to move the count by a whole unit — which usually means cutting the loop — or it has no effect on the count at all.
Small perturbations do not produce small errors. They produce no errors.
Physics uses this wherever robustness matters. Certain particle-like objects cannot decay, because decaying would require their wrapping to unwind and there is nowhere for it to go — no gradual route from three to two. Certain states of matter keep their electrical properties to extraordinary precision despite being full of impurities and defects, because the property being measured is a count.
So the corner of the subject where Noether's theorems have nothing to say is also the corner where conservation is at its most reliable. The two facts are connected: what makes these quantities immune to the theorem — that no equation of motion was needed — is exactly what makes them immune to disturbance.
No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.