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Noether theorems volume, initial import (H-09)

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{{DISPLAYTITLE:What "Noether's theorem" now means}}

[[File:NT-F-12.png|thumb|right|360px|alt=A horizontal timeline from 1891 to 2018 with nine marked events, Noether 1918 highlighted. A bracket beneath spans roughly 1921 to 1951, labelled as three decades in which the paper was little used.|A chronology from Lie to the anomaly]]

The name is used by two communities and does not denote the same thing in both.
The divergence is not a matter of emphasis. It is a difference in which theorem
is meant, at what generality, and in which direction.

== What a physicist means ==

Singular: '''Noether's theorem'''. It means the first theorem, in the
field-theory special case, and it is usually stated in one line — every
continuous symmetry of the action yields a conserved current.

Around that line, the standard presentation:

* proves the forward direction only;
* states it for first derivatives and for transformations that do not mix dependent and independent variables;
* includes divergence symmetries, which Bessel-Hagen added;
* often states it on-shell from the start, so the identity that holds before the field equations are imposed never appears;
* treats the second theorem, where it appears at all, under other names and usually without attribution to the same paper.

The theorem in this sense is a tool. It is applied several times per course, it
works, and the presentation is adequate to what it is used for.

== What a mathematician means ==

Plural: '''Noether's theorems'''. Two results about variational problems
admitting continuous groups, at the generality of the 1918 paper — any number of
independent variables, derivatives of arbitrary order, transformations mixing
dependent and independent variables — with both converses proved.

In this sense the second theorem is the more interesting of the two. It is the
origin of the identities among the field equations of a gauge system, of the
constraint structure of such systems, and of the modern treatment of variational
problems on jet bundles. The first theorem, from here, is the easier corollary.

== Why they differ ==

The two readings descend from different routes.

The physicist's descends from
[[Noether's theorems/History/The recovery in field theory|the recovery of the
1950s and 1960s]], which took from the paper what field theory needed at the
time and left the rest. Once that version was in the textbooks it reproduced
itself, because the next textbook was written from the last one and not from the
paper.

The mathematician's descends through the calculus of variations and Lie theory,
where the paper had always been read as what it is, and where the generality was
never the obstacle it was for physics.

Neither community was wrong about its own subject. They stopped reading the same
document.

== One slogan to distrust ==

'''"Symmetry implies conservation" is not her statement of the first theorem.'''
Her statement is that invariance of the integral makes a certain number of
linearly independent combinations of the Lagrange expressions into divergences —
an identity, holding whether or not anything satisfies the field equations.
Conservation appears only on passing to the variational problem, which she does
explicitly and separately. The slogan compresses an identity and its
specialisation into one clause, drops the converse, and leaves no room at all for
the second theorem.

It is a good slogan. It is not the theorem.

== What has changed recently ==

The historical record has been repaired, and the repair has not yet reached the
teaching.

Kosmann-Schwarzbach's book-length study of the theorems, their precursors and
their reception appeared in French in 2004 and in English in 2011. The
centenary in 2018 produced a body of work returning to the paper itself. The
German original is now freely readable online, in a transcription proofread
against the source, with its mathematics intact — which was not true for most of
the period during which the textbook version formed.

A reader today can therefore do what almost no physicist between 1930 and 2000
could conveniently do: read the paper. This volume exists to make the difference
between the paper and the slogan visible, and the shortest route to it is
[[Noether's theorems/History/The 1918 paper read in its own order|the paper read
in its own order]] followed by
[[Noether's theorems/Second theorem/The two theorems side by side|the two
theorems set side by side]].

''Provenance: the characterisation of Noether's own statement is read from the
German original, which is public domain; her prose is paraphrased and no English
wording of hers is transcribed. The account of current usage in the two
communities is this volume's own assessment and is editorial, not sourced. See
[[Noether's theorems/Sources|Sources]].''

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