The Noether current and the Noether charge

Plain-language counterpart: A flowing quantity and a standing total.

The Noether current as a flux through a tube

The first theorem produced $\varrho$ identities of the form $\sum {\psi }_i\overline{\delta }{u}_i^{(\lambda )}=\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(\lambda )}$. This page says what the $B^{(\lambda )}$ are.

Where B comes from

From (N12), $B=A-f\cdot \Delta x$, and the two terms have different origins.

$A$ is the boundary term of the integration by parts, defined by (N3) and written out by (N4) to (N6). It is linear in $\overline{\delta }u$ and its derivatives. It records what happens because the dependent variables changed.

$f\cdot \Delta x$ is the term produced by (N8) when the region of integration was moved onto itself. It records what happens because the independent variables changed.

A current with only the first term is what a treatment that never moves the $x$ obtains. Energy and momentum come from moving them, so the second term is not optional.

The current, and its non-uniqueness

Call $B^{(\lambda )}$ the current belonging to the $\lambda$th parameter. It is a system of $n$ components, one for each independent variable, and the identity says its divergence equals a combination of Lagrange expressions.

The current is not unique. Add to $B^{(\lambda )}$ any system of functions whose divergence vanishes identically and the identity is unchanged. Noether meets exactly such a system in her second section, where she obtains $\mathrm{D}\mathrm{i}\mathrm{v}(B-\Gamma )=0$ for a function system $B-\Gamma$ that does not itself vanish identically.

The non-uniqueness is a property of the theorem, not a defect of a derivation, and it is the room in which the Belinfante-Rosenfeld improvement operates.

The charge

Where the divergence of the current vanishes, the integral of one component over a cross-section is independent of where the cross-section is taken. That integral is the charge.

In one independent variable the divergence degenerates to a total differential and the statement is simpler: $d{B}^{(\lambda )}/dx=0$, so $B^{(\lambda )}$ is itself constant. These are the first integrals of mechanics.

Noether counts them. By (N6) the $B$ contain at most $(2\varkappa -1)$th derivatives of the $u$, provided $\Delta u$ and $\Delta x$ contain no derivatives higher than the $\varkappa$th appearing in $f$. Since $\psi$ in general contains $2\varkappa$th derivatives, one has the existence of $\varrho$ first integrals.

Linear independence is not functional independence

The $\varrho$ relations are linearly independent. The first integrals they yield need not be independent in any stronger sense, and Noether gives the counterexample rather than leaving it to the reader.

Take $f=\frac{1}{2}u{\text{'}}^2$. The linearly independent infinitesimal transformations $\Delta u={\epsilon }_1$ and $\Delta x={\epsilon }_2$ give the linearly independent relations

$u^{″}=\frac{d}{dx}{u}^{\prime };u^{″}\cdot u^{\prime }=\frac{1}{2}\frac{d}{dx}{\left(u^{\prime }\right)}^2\mathrm{(}\mathrm{1}\mathrm{)}$

while between the first integrals

$u^{\prime }=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.};u{\text{'}}^2=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.}\mathrm{(}\mathrm{2}\mathrm{)}$

there is a nonlinear dependence. Two relations, two first integrals, one piece of information.

The theorem promises a count of relations. It does not promise a count of independent physical facts.

The sourced case

Noether notes in a footnote that physics calls ${\psi }_i=0$, or slightly more generally ${\psi }_i=T_i$ with $T_i$ newly added functions, the field equations. In the second case the identities (N13) become

$\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(\lambda )}=\sum T_i\overline{\delta }{u}_i^{(\lambda )}\mathrm{(}\mathrm{3}\mathrm{)}$

which physics also calls conservation laws. The current is then not conserved but sourced, and the source is the coupling. She writes this in 1918, in a footnote, without comment.

What has been obtained here is still an identity. The passage to a genuine conservation statement is the next page.

Provenance: the counting argument, the example (1) and (2), and the sourced case (3) are Noether's, given in her third section and its footnotes as running text; this volume has numbered the displays. Reproduced in her notation from the public-domain German original. Her prose is paraphrased. See Sources.