Noether's theorems/Glossary

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

Each entry names the page that defines the term. Where a term is later than 1918 and not Noether's own, the entry says so.

Terms from the 1918 paper

Action. The number a scoring rule assigns to a whole history. Noether calls it the integral and writes $I$. Definition, or in plain language.

Lagrange expressions. The left-hand sides of the equations of the variational problem, written ${\psi }_i$. Kept as expressions, never set to zero inside either proof. Definition, or in plain language.

Divergence. A sum of partial derivatives of a system of functions, one per independent variable. Integrates to a boundary term. Same page.

Finite continuous group. A group whose transformations depend on a fixed number of essential constant parameters. Definition, or in plain language.

Infinite continuous group. A group whose transformations depend on arbitrary functions of the independent variables. Not the same as a group with infinitely many parameters, which Noether treats as an intermediate case. Definition, or in plain language.

Mixed group. A group depending on both arbitrary functions and parameters. Both theorems apply and give independent results. Same page.

Generator. The infinitesimal transformation belonging to a one-parameter family, from which the finite transformations are recovered by integration. Definition, or in plain language.

Prolongation. Extension of a transformation to the derivatives, in the way the chain rule forces. Definition, or in plain language.

Noether current, Noether charge. The system of functions whose divergence appears in the first theorem's output, and the integral of one of its components over a cross-section. The current is not unique. Definition, or in plain language.

Proper, improper. A divergence relation is improper when its conserved quantity can be assembled from the Lagrange expressions and their derivatives. Noether's own terms, from her sixth section. Definition, or in plain language.

Energy relations, energy components. Her names for the divergence relations belonging to the translation group and for the quantities appearing in them. Same page.

Terms later than 1918

On-shell, off-shell. Holding for actual solutions, against holding regardless. Noether says instead that she passes to the variational problem. Discussion, or in plain language.

Noether identities. The relations the second theorem returns. She calls them dependencies, or identical relations. Discussion, or in plain language.

Divergence symmetry, quasi-symmetry. Invariance up to a divergence. Bessel-Hagen's, 1921, not Noether's. Discussion, or in plain language.

Jet. A position bundled with its derivatives to some order, treated as a single point of a larger space. Discussion, or in plain language.

Moment map. The rule assigning to each generator the quantity whose flow it is. Written $J$ here, to avoid Noether's $\mu$. Discussion, or in plain language.

Killing vector. A direction along which a metric can be slid without distortion. Definition, or in plain language.

Canonical stress-energy tensor. The two-index object the first theorem returns for translations. Neither unique nor generally symmetric. Definition, or in plain language.

Belinfante-Rosenfeld improvement. The addition of a divergence term that symmetrises the canonical tensor without changing the conserved charges. Same page.

Anomaly. A classical symmetry that does not survive quantisation. Not a failure of either theorem. Discussion, or in plain language.

Topological charge. A conserved quantity whose conservation does not use the field equations, and which therefore has no symmetry behind it. Discussion, or in plain language.