Noether's theorems/Explainers/The theorem after quantum mechanics

Noether's theorems: map
Reading the volume How to read this volume Notation Glossary Index of symmetries Figure gallery
Reading by level Start here: the plain-language primer The first theorem, stated and proved The second theorem, stated and proved The two theorems side by side The mathematics: Lie groups and Lie algebras The applications: General relativity and the energy question The history: Göttingen and the energy problem
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Related Hamilton's principle On Physical Lines of Force

Mathematical counterpart: The Ward-Takahashi identity.

A note about this page. Its counterpart does not derive anything either. It states a result and marks it as unchecked, because deriving it needs quantum field theory, which this volume does not set up. So this page explains a shape rather than an argument, and you should hold it more loosely than the rest of the volume.

Why the question arises

Both of Noether's theorems are about classical physics. They are about a system that follows one history, chosen because a nudge to it changes nothing.

Quantum mechanics does not work like that. A quantum system does not follow one history. Every history contributes, including the ones classical physics rejects, and what you can ask about is the average behaviour over all of them.

So the classical statement cannot simply carry across. There is no single history for the conserved quantity to be conserved along.

The question is what survives.

What is said to survive

Unchecked, all of the following.

A symmetry of the classical scoring rule does constrain the quantum theory. What it constrains is not a quantity but a set of averages — the numbers you get when you ask what the theory predicts for various measurements.

The constraint says certain combinations of those averages are equal to each other. It carries the name of Ward and Takahashi, and it reduces to the classical conservation law when the quantum contributions are switched off.

Notice how the sentence changed shape. Classically you say "this quantity is conserved". Quantum mechanically you say "these predictions are related". The conserved current is still in there, but it is inside the statement rather than being its subject.

What it is used for

Mostly, for keeping theories honest.

Recall from the counting page that a theory with a local symmetry carries slack — parts of the description that mean nothing. The classical theory handles that cleanly.

The quantum theory has to be checked. Nothing guarantees in advance that the slack stays meaningless once you start summing over all histories, and if it does not, the theory predicts nonsense. These identities are what does the checking.

So the second theorem's structure reappears in the quantum theory as a consistency requirement, and it is the reason the identities matter rather than being a formal curiosity.

The claim to be careful about

You will read that this is Noether's theorem restated for quantum mechanics. Treat that as loose.

The two are connected by a limit, not by a translation. Noether's theorems hold identically, for every conceivable configuration, in a framework with no probabilities in it. The quantum identity is about expectation values in one particular quantised theory, and it depends on how the quantising was done.

That dependence is not a technicality. It is exactly what makes the next page possible: sometimes the quantising cannot be done in a way that respects the symmetry at all. See When a symmetry does not survive.

The honest summary

The classical theorems have a quantum descendant. It has a name, it does real work, and this volume has not derived it and does not depend on it anywhere.

What "average over all histories" actually means

The phrase does a lot of work above and deserves unpacking, since it is where the classical picture breaks.

Classically you ask: what does the system do? There is one answer.

Quantum mechanically you cannot ask that. What you can ask is: if I set things up this way and then measure this, what do I get? And the answer is a distribution, whose average is computable by adding up a contribution from every conceivable history — including the ones that violate the classical equations, each weighted in its own way.

Now put those two facts side by side with the identity page, and something interesting appears.

The classical conservation law was a statement about the one real history. It has no obvious quantum counterpart, because there is no one real history.

But the identity — the statement that held for every conceivable configuration, real or not — is in much better shape. Every conceivable configuration is exactly what the quantum theory sums over.

That is not a derivation and this page does not claim it is one. But it is the reason the off-shell statement, which looks like the weaker and less useful of the two, is the one with a future.

No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.