Symmetry and conservation: the first theorem
Mathematical counterpart: The first theorem stated and proved.
This is the primer. If you read one page of this volume, read this one.
The statement
Take a scoring rule for whole histories — an action. Suppose there is a smoothly adjustable change that leaves every score alone. Count its knobs.
Then for each knob there is a quantity that flows without accumulating anywhere. And once the system is actually obeying its equations of motion, that quantity is conserved: it has the same value at the end as at the beginning.
Turn a knob for time and you get energy. Turn one for place and you get momentum. Turn one for direction and you get angular momentum.
What is surprising about this
Nothing in the setup mentions energy. Nothing mentions conservation. You put in a statement about the shape of a scoring rule and you get out a quantity that nature keeps constant.
The three quantities above were known to be conserved for centuries before 1918. Noether did not discover them. What she supplied was the reason — and the reason turned out to be the same reason in all three cases, which nobody had known.
She also supplied the count. Three knobs for rotation, three conserved quantities, and no fourth to look for.
Two things the theorem does not say
It does not say every conserved quantity comes from a symmetry. Some do not, and this volume has a page on one.
It does not say conservation is the output. This is the point most compressed accounts lose, and it is worth being slow about.
What the theorem gives you first is a statement true of every possible history, including impossible ones. It says a certain combination flows without piling up. That statement holds for a ball on a path it would never take.
Conservation arrives only when you add the extra condition that the system is doing what physics says. That is a second step, Noether takes it separately and visibly, and there is a whole page on why keeping the two apart is worth the trouble.
It works backwards too
Noether proved the reverse as well: find such a flowing quantity, and there must be a symmetry behind it.
Most textbooks give one direction only. The reverse is the direction that does real work, because it turns an observation into evidence. Something is conserved and you do not know why? The theorem says look for the symmetry, and it is there. See Reading the theorem backwards.
And there is a second theorem
The page you are reading covers one of Noether's two theorems.
The other one takes a different kind of symmetry — one that can be dialled differently from place to place — and it does not give conservation laws at all. It gives something stranger and, for modern physics, more important.
It is the theorem the paper was written to prove, and it is the one that usually gets left out. See The second theorem, and why it is not the first.
One slogan to be careful with
You will often see the theorem compressed to four words: symmetry implies conservation.
It is a good slogan and it is not the theorem. It drops the identity that holds before physics is imposed, it drops the reverse direction, and it leaves no room at all for the second theorem. Use it to remember the shape. Do not use it to decide what is true.
One case, all the way through
A pendulum clock in a sealed room.
The scoring rule. At each instant the pendulum has a position and a speed. The rule combines them into a number, and adding that number up over the whole swing gives the score for that history.
The symmetry. Nothing in the rule mentions what time it is. Start the clock now or in an hour, and the rule is the same rule. One knob: how far you shift the start.
The count. One knob, so exactly one conserved quantity, and no second one to hunt for.
The quantity. It turns out to be the sum of the energy of motion and the energy of position — the total energy. Nobody put energy into the setup. It fell out of the fact that the rule does not mention the time.
The check. Now open a window and let the room warm through the afternoon, so that the air resistance changes as the day goes on. The rule now does mention the time. The symmetry is gone, and so is the conservation law — the pendulum really does lose energy, and the theorem correctly predicts that it may.
That last step is the one worth remembering. The theorem is not a guarantee that energy is conserved. It is a statement about exactly when it is.
No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.