The two theorems compared
Mathematical counterpart: The two theorems side by side.
Two theorems, one paper, one page to keep them apart.
Side by side
| First theorem | Second theorem | |
|---|---|---|
| What you feed it | a symmetry with a fixed number of knobs, each one number for everywhere | a symmetry you can dial differently at every point |
| What comes back | one flowing quantity per knob | one relation among the equations of motion, per free shape |
| Is there something conserved? | yes, once the laws of physics are imposed | no. Nothing flows and there is nothing to measure |
| When is it true? | always, of every conceivable configuration | always, of every conceivable configuration |
| What happens when you impose the laws of physics? | the flow balances: a conservation law | some of the laws turn out to follow from the others |
| Does it work backwards? | yes, with one case its author left undecided | yes |
Nine tenths shared, then one step
Both proofs are the same proof until near the end. The setup, the nudge, the care about which comparison you are making, the loss of the integral sign — all common.
Then one step, and it is forced by the input.
A fixed number of knobs can be separated from each other, because each one is independent. The relation falls apart into one piece per knob, and each piece says something flows.
A free shape cannot be separated, because the shape and how fast it varies from place to place are not independent of each other. Instead, the varying is traded away, and then the freedom to draw a shape that fades out near the edges kills everything that is left except a plain zero.
One theorem ends with something flowing. The other ends with zero. Every other difference in the table follows from that.
Three things people say that are not right
"The second theorem is the local version of the first." It is not a version of the first. Different input, different output, and a different thing happens when you impose the laws of physics. Rows two, three and five of the table are the check.
"The second theorem gives conservation laws for local symmetries." It gives relations among equations. When a theory with a local symmetry does produce something shaped like a conservation law, that law is empty — its conserved quantity is built out of the equations of motion and reports nothing they did not already say.
"General relativity is why energy conservation goes strange." The rule is about symmetries, and it works in both directions. General relativity meets its condition. So does a symmetry Noether builds out of a logarithm with no physics in it at all, which is exactly why she built it.
Which one is the paper about
Both, and not equally.
The first theorem is stated first, and it is the one that reached physics textbooks. It explains why energy, momentum and angular momentum are conserved, and it is genuinely famous.
The second theorem is the one the paper was written to prove. It occupies the final section, it answers the question that Göttingen had been arguing about since 1915, and it turned out fifty years later to describe the structure of every theory of every fundamental force.
This volume's own view, offered as a view: quoting only the first theorem gives you the corollary without the argument. You are free to disagree, and the history track explains how the other reading came to be the standard one.
How to tell which one you are reading
A short test, for anybody else's account of this subject.
Ask what the symmetry is. One choice for everywhere, or a choice at every point? That alone settles it.
Ask what came out. Something that flows and can be totalled up, or a statement that a combination equals zero?
Ask what happens when the laws of physics are imposed. Does something become conserved, or do some of the laws turn out to be consequences of the others?
If an account never poses any of these questions, it is almost certainly giving the first theorem and calling it "Noether's theorem", which is the ordinary usage and is not wrong so much as incomplete.
Where the second theorem shows up under other names
Part of why the second theorem seems obscure is that physics met it repeatedly without recognising it, and gave it a new name each time.
Constraints, in the study of how to put general relativity into a form suitable for computation. Identities among the field equations, in general relativity itself. Gauge redundancy, in the theories of the fundamental forces. Ghosts and gauge fixing, in the quantum versions of those theories.
All of these are the same theorem, met from different directions by people solving different problems. The connection back to a 1918 paper was often not made at all.
No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.