Noether's theorems/Explainers/Drifting centres and electric charge

Mathematical counterpart: Centre-of-mass motion and electric charge.

Two more cases. The first needs a condition Noether did not use. The second needs nothing new and is quietly the same calculation as angular momentum.

Two skaters push apart

Two skaters stand still on ice, facing each other, and push. They slide apart.

Watch the point midway between them, weighted for their masses. Before the push it was still. After the push it is still still — it has not moved, and it will not.

Now do the same on a moving train. From the platform, that midpoint was already travelling at the train's speed before the push, and afterwards it carries on at exactly that speed. Not faster, not slower, not wobbling.

The centre of mass of an isolated system moves at a constant rate. That is a conservation law, and it is less famous than energy or momentum because it sounds like a restatement of them. It is not.

Where it comes from, and the catch

The symmetry is: run the experiment from a laboratory sliding at a steady speed instead of a parked one. One knob — how fast you slide.

Here is the catch. Under Noether's own condition, this symmetry gives nothing at all. The scoring rule does not survive the change unaltered.

What it does is change by a term that only ever contributes at the beginning and the end. That is the weaker condition from Two ways a thing can stay the same, and it is Bessel-Hagen's, from 1921.

Under his condition, the calculation runs and delivers the drifting centre of mass.

The conserved quantity is real, the derivation is standard, and the condition that licenses it is not in the 1918 paper. This volume says so wherever the case appears, because saying "by Noether's theorem" here is the single most common misattribution in the subject.

Charge, and a coincidence that is not one

Now something that looks entirely different and turns out to be the same arithmetic.

Some quantities in physics are described by a pair of numbers at every point, and the scoring rule only ever cares about the pair through a combination that stays fixed when you rotate one number into the other — the way a compass needle's length does not change when it swings.

Rotating that pair is a symmetry. One knob: the angle.

One knob, one conserved quantity. That quantity is electric charge.

Why that is worth pausing on

Compare this with angular momentum, from the previous page.

Angular momentum came from rotating space — turning the apparatus so it faces a different way.

Charge comes from rotating something that is not space at all. There is no direction you could point to. The two numbers are properties the field carries at each point, and the rotation happens among those properties.

The theorem cannot tell the difference. The calculation is character for character the same. Nothing in it knows whether the things being rotated are directions in a room or numbers attached to a field.

Physics draws the distinction afterwards and calls the second kind an internal symmetry. Charge is its first and simplest example, and every force in the Standard Model was found by taking the idea seriously.

That an eighteenth-century bookkeeping trick about rotating a pendulum and a twentieth-century account of electric charge are the same line of algebra is not a coincidence. It is what a theorem is for.

What happened when people took the idea seriously

The rotation described above is the simplest possible internal symmetry: one knob, one conserved quantity, electric charge.

In the 1950s physicists asked an obvious follow-up question. What if the internal thing being rotated has more than two components, so that the rotations can be done in several independent ways and do not commute with one another?

The answer turned out to describe the strong and weak nuclear forces. Every force in the Standard Model is now understood as an internal symmetry of exactly this kind, with a different number of knobs in each case, and the conserved quantities are the various charges those forces respond to.

None of that was visible in 1918. What was visible was that the calculation does not care whether you are rotating a room or rotating a pair of numbers, and that turned out to be the door.

And then the same question again, locally

There is one more step, and it is where the second theorem comes back.

Ask what happens if the internal rotation may be dialled differently at every point rather than once for everywhere. That is the local case from the local symmetry page, and it is how the forces are actually built. See The symmetry behind the strong and weak forces.

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