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	''Mathematical counterpart: [[Noether's theorems/Second theorem/The two theorems side by side\|The two theorems side by side]].''		''Mathematical counterpart: [[Noether's theorems/Second theorem/The two theorems side by side\|The two theorems side by side]].''
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Noether theorems volume, initial import (B-19)

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{{DISPLAYTITLE:The two theorems compared}}

''Mathematical counterpart: [[Noether's theorems/Second theorem/The two theorems side by side|The two theorems side by side]].''

[[File:NT-F-06.png|thumb|right|360px|alt=A three-column comparison table. Rows: what each theorem takes, what it returns, where it holds, what happens when the field equations are imposed, whether a conserved quantity results, and whether the converse is proved. Theorem II's conserved-quantity row reads none.|The first and second theorems side by side]]

Two theorems, one paper, one page to keep them apart.

== Side by side ==

{| class="wikitable"
! !! 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
[[Noether's theorems/Explainers/A conservation law that is only wearing the costume|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
[[Noether's theorems/History/What Noether's theorem now means|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.''

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