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Published from the Principia Machina source

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NT>NoetherVolumeBuild: Noether theorems volume, initial import (R-07)

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Noether theorems volume, initial import (R-07)

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{{DISPLAYTITLE:Further reading}}

This page names what the volume declined, and why, so that a reader knows where
its edges are rather than discovering them.

== Read the paper ==

Before anything else. The German original is public domain, is freely available
online in a transcription proofread against the source, and carries its
mathematics intact — which was not true for most of the period during which the
textbook version of "Noether's theorem" formed.

It is twenty-three pages in six sections.
[[Noether's theorems/History/The 1918 paper read in its own order|A guide to
reading it in its own order]] is in this volume.

The English translations are all under copyright, and this volume quotes none of
them. If you read English only, the paper is still worth having beside a
translation you obtain elsewhere, because the equations are legible in any
language.

== What this volume does not derive ==

Each item below is discussed somewhere in the volume, marked as underived, and
would need a text of its own.

'''Lie theory proper.''' The classification of continuous groups. Neither theorem
needs it;
[[Noether's theorems/Mathematics/Lie groups and Lie algebras|the Lie page]]
carries only the commutator and the closure condition.

'''Jet bundles and the variational bicomplex.''' The modern framework in which
both theorems are corollaries of a statement about a differential complex.
Noether proves both without it.
[[Noether's theorems/Mathematics/Jet space and prolongation|Discussion]], or [[Noether's theorems/Explainers/Why Noether needed a third symbol|in plain language]].

'''The general moment map.''' Existence, equivariance, and the relation to the
Lie algebra's structure. Two cases are checked in this volume and the general
construction is not.
[[Noether's theorems/Mathematics/The symplectic form of the theorem and the moment map|Discussion]], or [[Noether's theorems/Explainers/Why the moment map resists plain language|in plain language]].

'''Quantum field theory.''' The Ward-Takahashi identity and the anomaly are
named, their shape is described, and neither is derived. This is the ceiling of
the volume.
[[Noether's theorems/Mathematics/Anomalies|Discussion]], or [[Noether's theorems/Explainers/When a symmetry does not survive|in plain language]].

'''The Belinfante construction.''' Why the improvement is permitted is derived
here; that a symmetrising choice always exists is not.
[[Noether's theorems/Mathematics/The Belinfante-Rosenfeld improvement|Discussion]], or [[Noether's theorems/Explainers/Moving the bookkeeping without changing the total|in plain language]].

'''The symmetry behind the Runge-Lenz vector.''' The conserved quantity is
exhibited and checked; the symmetry is not.
[[Noether's theorems/Applications/The Runge-Lenz vector|Discussion]].

== Open questions this volume leaves ==

'''Which precursor anticipated what.''' Noether names eight people. Measuring
what each of them established, and under what restriction, requires reading eight
papers against her two theorems. No priority verdict is made or implied.
[[Noether's theorems/History/The precursors|Discussion]].

'''How the paper was actually received between 1918 and 1960.''' No citation
trace was performed. The usual explanations are recorded as claims.
[[Noether's theorems/History/The long neglect|Discussion]].

'''Which theorem Kelvin's circulation theorem belongs to.''' The symmetry is a
free shape, which puts it on the second theorem's side, and the derivation that
would settle it was not done.
[[Noether's theorems/Applications/Kelvin's circulation theorem|Discussion]].

== For the mathematics without the history ==

Any graduate text on classical field theory carries the first theorem in its
field-theory special case. Be aware, reading one, of the four things that version
usually drops: the second theorem, both converses, the general derivative order,
and the correct attribution of divergence invariance.
[[Noether's theorems/History/The recovery in field theory|The page on the
recovery]] explains how each was lost.

Noether's theorems/Further reading - Revision history

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