Noether's theorems/Explainers/Reading the second theorem backwards

Noether's theorems: map
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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 converse of the second theorem.

Noether proved both her theorems in both directions. This page is the reverse of the second one, and it is the direction working physicists actually use.

The reverse

Forward: the theory has a local symmetry, therefore its equations have relations among them.

Reverse: the equations have relations among them, therefore the theory has a local symmetry.

Why the reverse is the working direction

You do not usually start by knowing a theory's symmetries. You start with equations somebody wrote down, often for reasons that had nothing to do with symmetry, and you want to know what you are holding.

So you count. How many equations, how many variables, and do the equations determine everything? If you find that some of them follow from the others, the reverse direction tells you what that means: there is a local symmetry in there, and it is not an accident of how the equations were written.

This is a diagnosis, not a curiosity. Finding a redundancy tells you that some of your variables are description rather than physics, that your equations will not determine everything, and that you will need to make a choice — fix a convention — before you can compute anything. All of that is forced, and the reverse direction is what tells you it is forced rather than a mistake.

A restriction that turns out not to be one

Before the reverse direction proper, Noether clears away an assumption she had been making, and the argument for it is unusually pretty.

She had assumed the symmetry depends on your free shape in the simplest possible way: draw twice the shape and you get twice the effect.

Suppose it did not. Suppose the effect involved the shape squared, or the shape multiplied by how fast it varies.

Now do two of these transformations one after the other, and look at what comes out. Terms appear that cannot be produced by any single free shape — they need more shapes than you started with. But the family was supposed to be built from a fixed number of free shapes, and now composing two of its members has produced something outside it.

So either the family was never closed under composition, in which case it was not the sort of family we are discussing, or the complicated dependence was not there after all.

The simple dependence is not an assumption. It is forced by the requirement that doing two symmetries in a row is a symmetry.

Getting the whole symmetry back

The reverse direction proper runs the forward proof backwards, and it recovers the smallest steps of the symmetry — the infinitesimal version from the earlier page.

Getting from those back to the full-sized transformations is where Noether leans on Lie, and it is real work rather than a formality: a result of his guarantees that a family of smallest steps of this kind always belongs to a genuine family of full-sized transformations, and that every one of those is built up out of the small ones.

She notes one wrinkle. Sometimes you have to let your free shape change as you build the transformation up, rather than holding it fixed and repeating. When you can hold it fixed, everything is simpler.

And, as with the first theorem's reverse, there is a case involving symmetries that depend on speeds where extra care is needed before the same conclusions can be drawn.

One more assumption removed

Throughout, the free shapes were allowed to vary with position but not with the quantities being described.

Noether closes the section by showing that this too costs nothing. Allow the shapes to depend on the quantities as well and you get relations of the same kind, just more of them — and running the reverse direction on those brings you back to the case already handled.

The same argument settles the mixed case, where a theory has global knobs and local shapes at once, and with that the classification is complete.

How a symmetry gets discovered

The reverse direction is not only a diagnosis. It is sometimes how the symmetry is found in the first place.

Somebody writes down equations for a new situation, often by analogy with something already understood. They count: so many equations, so many quantities to solve for. The count says the problem should be determined.

It is not. Solutions come in families, differing in ways no measurement distinguishes.

At that point the reverse direction says something definite. This is not a mistake in the equations and not a shortage of data. There is a local symmetry in the theory, it is the reason the solutions come in families, and it can be found by asking which changes to the description leave every prediction alone.

Several of the symmetries at the centre of modern physics were identified along exactly this route, from a redundancy noticed first and explained afterwards.

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