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	[[File:NT-F-07.png\|thumb\|right\|360px\|alt=Five symmetries in a left column, each joined by an arrow to the quantity it conserves in a right column: clock shift to energy, spatial motion to momentum, rotation to angular momentum, field rotation to electric charge, and steady-speed change to uniform centre-of-mass motion. The last arrow is dashed to mark that it needs the weaker invariance condition.\|The symmetry-to-conservation map]]		[[File:NT-F-07.png\|thumb\|right\|360px\|alt=Five symmetries in a left column, each joined by an arrow to the quantity it conserves in a right column: clock shift to energy, spatial motion to momentum, rotation to angular momentum, field rotation to electric charge, and steady-speed change to uniform centre-of-mass motion. The last arrow is dashed to mark that it needs the weaker invariance condition.\|The symmetry-to-conservation map]]

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	this volume; her equations are reproduced in her own notation, from the German		this volume; her equations are reproduced in her own notation, from the German
	original. See [[Noether's theorems/Sources\|Sources]].''		original. See [[Noether's theorems/Sources\|Sources]].''

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Noether theorems volume, initial import (L-01)

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[[File:NT-F-07.png|thumb|right|360px|alt=Five symmetries in a left column, each joined by an arrow to the quantity it conserves in a right column: clock shift to energy, spatial motion to momentum, rotation to angular momentum, field rotation to electric charge, and steady-speed change to uniform centre-of-mass motion. The last arrow is dashed to mark that it needs the weaker invariance condition.|The symmetry-to-conservation map]]

Emmy Noether read her results to the Göttingen Mathematical Society on 23 July
1918. Three days later Felix Klein presented the paper, ''Invariante
Variationsprobleme'', to the Göttingen Scientific Society, and it appeared in
that Society's ''Nachrichten'' for the year. Noether held no post at Göttingen
when she wrote it.

The paper contains two theorems. They are not two statements of one result, and
they do not do the same work. Most accounts of "Noether's theorem" give the
first and pass over the second. The paper was written to prove the second.

Both theorems start from the same object. Physics writes down an integral,
called the action, built from a function of some variables and their
derivatives. Both theorems ask what follows when a continuous group of
transformations leaves that integral unchanged. They differ in one thing only:
what kind of group is permitted. The difference in what comes back is large
enough that treating the two as one result loses the paper.

== The first theorem ==

'''What it takes.''' A continuous group of transformations that leaves the
action unchanged and depends on a fixed, finite number of constant parameters.
Rotation of space is such a group, with three parameters. Shift in time is
another, with one.

'''What it returns.''' One relation for each parameter. Each relation says that
a particular combination of the expressions on the left of the equations of
motion is a divergence — the mathematical form of something flowing without
accumulating anywhere.

'''Where the result holds.''' The relations themselves are identities. They are
true of the action whether or not anything obeys the equations of motion. Impose
those equations and each relation becomes a conservation law: a quantity that
does not change as the system evolves. The identity is unconditional; the
conservation law is not.

The theorem runs backwards as well. Noether proved the converse: the relations
imply the invariance they came from. That direction is what makes an observed
conservation law evidence of a symmetry, rather than a coincidence that happens
to be consistent with one.

== The second theorem ==

'''What it takes.''' A continuous group that depends on arbitrary functions of
the independent variables rather than on constants. The transformation is free
to differ from place to place, and its derivatives up to some order may enter.
The coordinate freedom of general relativity is such a group. So is the
freedom in the potentials of electromagnetism.

'''What it returns.''' One identity for each arbitrary function, holding among
the expressions on the left of the field equations and their derivatives.

'''Where the result holds.''' Everywhere, and unconditionally. Nothing has to
obey a field equation for these to be true. They are not conservation laws and
they do not become conservation laws. They are relations the field equations
satisfy by construction, which is a stronger and stranger thing to have.

This theorem also has a converse, and Noether proved it.

== The distinction ==

Set the two sets of slots beside each other and the difference reads off them.

The first theorem is fed constants and returns quantities that are conserved
once the physics holds. The second is fed functions and returns relations that
hold before any physics is imposed. The first tells you what a system keeps. The
second tells you that some of its equations were never independent.

The consequence is blunt. A theory with a symmetry that varies from place to
place has fewer independent field equations than it has fields. You cannot
specify all of them freely, because the theory itself has already fixed some in
terms of the rest.

== What the theorems are about ==

Both theorems are statements about the action. Neither is a statement about the
equations of motion.

This distinction is the most common error in accounts of the subject. A
transformation that carries solutions to solutions need not leave the action
unchanged, and if it does not, no conservation law follows from it. Symmetry of
the equations and symmetry of the action are different properties, and only the
second feeds either theorem.

Noether also worked at a generality that later treatments usually drop. Her
statement allows several independent variables, derivatives of any order, and
transformations that mix the dependent and independent variables together. Most
textbook versions state a special case and leave the reader to suppose it is the
theorem.

== Why it matters ==

The first theorem's results are famous, and citing them proves little: energy,
momentum and angular momentum were known to be conserved long before 1918. What
the theorem added was the reason.

The second theorem's results were not known, and without it they remain
unexplained.

The final section of the paper settles a claim of Hilbert's. Energy conservation
in general relativity does not take the form it takes in every other field
theory. Attempts to write a local energy density for the gravitational field
produce quantities that depend on the choice of coordinates, and can be made to
vanish at any chosen point. This was read for years as a defect of the
formalism, something a better construction might repair.

Noether showed it is a prediction. A theory invariant under a group of
transformations that vary from place to place, and containing the shifts among
that group, has energy relations of exactly this degenerate kind — and has them
if and only if that condition is met. The property is group-theoretic. Gravity
is one instance of it, not the cause of it. No better construction is coming,
because there is nothing to repair.

The same theorem accounts for the constraint structure of every gauge theory
built since, and for the identity that Einstein's equations satisfy
automatically. These are not applications discovered later and traced back. They
are the theorem.

Knowing the boundary matters too. Not every conserved quantity comes from a
symmetry of an action, and this volume marks the cases that do not.

== How to read this volume ==

The volume has eight tracks. Every page carrying mathematics names a
plain-language counterpart at the top, and each counterpart links back, so a
reader can enter at either level and cross over at any point.

* [[Noether's theorems/Explainers/Symmetry and conservation, the first theorem|Start with the plain-language primer]] if you want the argument with no mathematics at all.
* [[Noether's theorems/History/Gottingen and the energy problem|Göttingen and the energy problem]] gives the question the paper was written to settle, and the history track follows it to [[Noether's theorems/History/What Noether's theorem now means|what the name means today]].
* [[Noether's theorems/First theorem/The first theorem stated and proved|The first theorem]] is proved in full, with worked derivations of the standard conserved quantities.
* [[Noether's theorems/Second theorem/The second theorem stated and proved|The second theorem]] is proved separately and at the same length, and [[Noether's theorems/Second theorem/The two theorems side by side|the two are set side by side]] on one page.
* [[Noether's theorems/Mathematics/Lie groups and Lie algebras|The mathematics track]] carries the machinery the theorems need, from Lie groups to the quantum statement and its failure.
* [[Noether's theorems/Applications/General relativity and the energy question|The applications track]] works the physics case by case, including [[Noether's theorems/Applications/Topological charge, which is not a Noether charge|one conserved quantity the theorems do not explain]].
* [[Noether's theorems/Notation|Notation]], [[Noether's theorems/Glossary|glossary]], [[Noether's theorems/Chronology|chronology]], [[Noether's theorems/Index of symmetries|an index of symmetries against their conserved quantities]] and a [[Noether's theorems/Bibliography|bibliography]] weighted toward primary sources make up the reference apparatus.

''Provenance: this page carries no equations and no quotation. Noether's prose is
paraphrased throughout and no English wording of hers is transcribed anywhere in
this volume; her equations are reproduced in her own notation, from the German
original. See [[Noether's theorems/Sources|Sources]].''

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