Noether's theorems/Sources

Noether's theorems: map
Reading the volume How to read this volume Notation Glossary Index of symmetries Figure gallery
Reading by level Start here: the plain-language primer The first theorem, stated and proved The second theorem, stated and proved The two theorems side by side The mathematics: Lie groups and Lie algebras The applications: General relativity and the energy question The history: Göttingen and the energy problem
Sources Sources Bibliography Chronology Further reading
Related Hamilton's principle On Physical Lines of Force

This page states what the rest of the volume is worth. It is not a bibliography; that is separate. It is an accounting of which text was read, which was not, what follows from each, and what would close the gaps.

The short version, before the detail. The equations are in the best condition this subject allows. The history is not. One of the nine primary sources was read in full. Five were never reached at all.

The three classes

Every passage in this volume is one of three things.

Verbatim. The source was opened and the text read from it. In this volume that means one source: the German original of Invariante Variationsprobleme, public domain, read in full, with its mathematics intact.

Editorial. The volume's own prose, argument or arrangement. Most of the volume is this, and the substantive judgements are listed below rather than left for a reader to find.

Reconstructed. A passage or equation restated at second hand, from secondary accounts rather than from the source itself.

This volume contains no reconstructed equations, and the claim is meant precisely rather than as a boast. Every mathematical expression in it is either reproduced from the public-domain German original in Noether's own notation, or derived in full on the page where it appears. There is no third category. Where a result could be neither read nor derived at this depth — the general moment map, the Ward-Takahashi identity, the Belinfante construction, the Bianchi identities themselves — it is stated and tagged as unverified, and no equation is given for it.

Seventeen of her twenty numbered equations appear, (N1) through (N17). Every expression in the volume was passed through MediaWiki's own TeX checker before delivery: 710 expressions, none rejected.

The quotation rule, and why

Noether's German is quotable. Nothing else in this volume is.

The 1918 original is public domain. Every full English translation is under copyright — Tavel's of 1971 is the standard one, and the copies circulating are reproductions rather than re-licensed texts. Klein's papers were read only in a modern English translation whose terms of reuse were not established, and his German was not reached.

The rule that follows, applied on every page:

• her equations are reproduced verbatim, in her own symbols;
• her prose is paraphrased throughout and marked as paraphrased;
• no English wording of hers is transcribed anywhere in this volume;
• no other primary source is quoted at all, in any language.

German technical terms appear where she coined them and no English equivalent is standard — eigentlich and uneigentlich, Lagrangesche Ausdrücke, Erhaltungssätze. Those are her words, given as terms rather than as quotation.

Reachability

Matching the bibliography row for row. A disagreement between the two pages would be a defect in one of them.

source	state	what it cost
Noether 1918, German	read in full, text-bearing, mathematics intact	nothing. This is the volume's foundation
Tavel translation 1971	located, under copyright	no English wording of hers appears
Klein 1918, second note	read in English translation only	content verified, wording not quotable
Klein 1917, first note	an open translation located, not read	the Hilbert correspondence is summarised at second hand
Klein 1918, third note	not reached	not used
Hilbert 1915, first communication	not reached; only an OCR-damaged copy of the second was found	his Theorem I and his remark on prediction are paraphrased from a corrupt text and never quoted
Bessel-Hagen 1921	not reached	the account of divergence invariance is entirely secondary, on the one point where this volume corrects the standard attribution
Weyl 1918	not reached	appears only as a precursor Noether names
Lie 1891	not reached	her three citations to it, including Theorem VII at page 391, are reported as hers

Three transcription discrepancies, and the collation they require

The German original was read in a transcription proofread against the source, not from page images. Three places disagree with what the surrounding argument requires. All three are written here as the argument requires and flagged on their pages.

1. Her equation (5): the electronic text carries one index where the pattern of the surrounding terms requires another. Flagged there.
2. Her equation (14): the electronic text carries the unbarred variation where its own derivation requires the barred one. Flagged there.
3. Her equation (14) again: the sign factor is punctuated with the exponent inside the parenthesis, where the same factor in her (16) puts it outside. Same page.

Whether each slip belongs to the 1918 printing or to the transcription cannot be decided without the page images.

This is scheduled work, not a note. The transcription is backed by a page-by-page apparatus over a scan, so the collation is a defined task with a defined endpoint: open the three pages, compare, and either correct the volume or record the printing's own error. Until it is done, those three equations carry the volume's only unresolved textual questions.

What the five unreached sources cost, page by page

Not a list of absences. Each one changes what a particular page can say.

Hilbert. The page on Hilbert and Klein describes his Theorem I and his statement about prediction from an OCR-damaged copy, at word level. It quotes nothing and cannot. Reaching the 1915 Nachrichten would let that page quote the assertion Noether's final section was written to prove.

Bessel-Hagen. This is the sharpest cost. The volume corrects a standard misattribution — divergence invariance is his, not hers — on four pages, and does so entirely from secondary literature. His page says so. The correction is very likely right and it is not verified.

Weyl and Lie. Both are named by Noether as precursors, and the precursor page reports whom she names, with her loci, and makes no priority claim whatever. That restraint is forced by not having read them.

Klein's German. His content is verified; his words are not available. Every statement attributed to him in this volume is a paraphrase of a paraphrase.

What is editorial

The volume makes judgements a reader should be able to identify and reject. These are the substantive ones.

• That Theorem II is the theorem the paper was written to prove, and that quoting only Theorem I gives the corollary without the argument. Stated on the side-by-side page.
• That the moment map is not Noether's theorem in another language, and the same refusal for the Ward-Takahashi identity. Both rest on the off-shell content having no counterpart in a phase space or in a set of expectation values.
• That an anomaly does not refute either theorem, because a conditional whose hypothesis fails has not been contradicted. This is a point of logic, not of physics.
• That the internal-versus-spacetime distinction is one physics adds afterwards. The theorem cannot tell the two apart.
• That absent symmetry and empty conservation law are different failures, and that general relativity exhibits both.
• That Noether's third variation symbol exists because it commutes with differentiation. She does not say so; the property is derived and the inference is the volume's.
• That the gauge identity forces charge conservation rather than permitting it. The argument is exhibited; the framing is the volume's.
• That "symmetry implies conservation" is not her statement of Theorem I.
• The account of what the name now means to physicists and to mathematicians, which is assessment rather than sourced history.

What would close the remaining gaps

For the equations. Collate the three flagged places against the page images. That is the whole of what is outstanding on the mathematics, and it is a short task.

For the history. Reach the Göttingen Nachrichten for 1915 and 1918, and Mathematische Annalen volume 84. Those three volumes carry Hilbert's first communication, Klein's German, and Bessel-Hagen. Together they would move six pages from summary to quotation and verify the volume's one substantive correction to the standard account.

For the reception. A citation trace of the 1918 paper from 1918 to 1960, separated by discipline. Nothing of the kind was attempted, and the page on the neglect reports the usual explanations as claims rather than as findings.

For the precursors. Read the eight papers Noether names, against her two theorems, and decide for each what it establishes and under what restriction. No priority verdict should be made before that reading, and none is made here.