View source for Noether's theorems/First theorem/The Noether current and the Noether charge

{{DISPLAYTITLE:The Noether current and the Noether charge}}

''Plain-language counterpart: [[Noether's theorems/Explainers/A flowing quantity and a standing total|A flowing quantity and a standing total]].''

[[File:NT-F-04.png|thumb|right|360px|alt=A tube drawn in perspective with arrows flowing along it, labelled the current. A dashed elliptical cross-section partway along is labelled the charge, with a note that the amount crossing is the same at every slice. A caption states that conservation means the books balance rather than that nothing moves.|The Noether current as a flux through a tube]]

[[Noether's theorems/First theorem/The first theorem stated and proved|The first
theorem]] produced <math>\varrho</math> identities of the form
<math>\sum\psi_{i}\bar{\delta}u_{i}^{(\lambda)}=\mathrm{Div}\,B^{(\lambda)}</math>.
This page says what the <math>B^{(\lambda)}</math> are.

== Where B comes from ==

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

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

<math>f\cdot\Delta x</math> 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
<math>x</math> obtains. Energy and momentum come from moving them, so the second
term is not optional.

== The current, and its non-uniqueness ==

Call <math>B^{(\lambda)}</math> the current belonging to the
<math>\lambda</math>th parameter. It is a system of <math>n</math> 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 <math>B^{(\lambda)}</math> 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
<math>\mathrm{Div}\,(B-\Gamma)=0</math> for a function system
<math>B-\Gamma</math> 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
[[Noether's theorems/Mathematics/The Belinfante-Rosenfeld improvement|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: <math>dB^{(\lambda)}/dx=0</math>, so
<math>B^{(\lambda)}</math> is itself constant. These are the first integrals of
mechanics.

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

== Linear independence is not functional independence ==

The <math>\varrho</math> 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 <math>f=\tfrac{1}{2}u'^{2}</math>. The linearly independent infinitesimal
transformations <math>\Delta u=\varepsilon_{1}</math> and
<math>\Delta x=\varepsilon_{2}</math> give the linearly independent relations

<math>u''=\frac{d}{dx}u';\qquad u''\cdot u'=\frac{1}{2}\frac{d}{dx}\left(u'\right)^{2}\qquad\qquad\mathrm{(1)}</math>

while between the first integrals

<math>u'=\mathrm{const.};\qquad u'^{2}=\mathrm{const.}\qquad\qquad\mathrm{(2)}</math>

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
<math>\psi_{i}=0</math>, or slightly more generally
<math>\psi_{i}=T_{i}</math> with <math>T_{i}</math> newly added functions, the
field equations. In the second case the identities (N13) become

<math>\mathrm{Div}\,B^{(\lambda)}=\sum T_{i}\bar{\delta}u_{i}^{(\lambda)}\qquad\qquad\mathrm{(3)}</math>

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
[[Noether's theorems/First theorem/On-shell and off-shell|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
[[Noether's theorems/Sources|Sources]].''