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	''Mathematical counterpart: [[Noether's theorems/Mathematics/The Ward-Takahashi identity\|The Ward-Takahashi identity]].''		''Mathematical counterpart: [[Noether's theorems/Mathematics/The Ward-Takahashi identity\|The Ward-Takahashi identity]].''
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NT>NoetherVolumeBuild: Noether theorems volume, initial import (B-26)

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Noether theorems volume, initial import (B-26)

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{{DISPLAYTITLE:The theorem after quantum mechanics}}

''Mathematical counterpart: [[Noether's theorems/Mathematics/The Ward-Takahashi identity|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
[[Noether's theorems/Explainers/Counting the equations a gauge theory really has|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
[[Noether's theorems/Explainers/When a symmetry does not survive|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
[[Noether's theorems/Explainers/True everywhere against true for real motions|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.''

Noether's theorems/Explainers/The theorem after quantum mechanics - Revision history

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