Centre-of-mass motion and electric charge

Plain-language counterpart: Drifting centres and electric charge.

Two more cases. The first needs a condition Noether did not use, and is included partly to show where her statement stops. The second needs nothing new at all and is the same computation as the angular-momentum case with the rotation moved out of space.

The Galilean boost, and why (N1) is not enough

Take a free particle, one independent variable $x$, one dependent $u$, and

$f=\frac{1}{2}mu{\text{'}}^2\mathrm{(}\mathrm{1}\mathrm{)}$

and the boost $\Delta x=0$, $\Delta u=\epsilon x$, so that $\overline{\delta }u=\epsilon x$.

Test the invariance condition first:

$\overline{\delta }f=\frac{\partial f}{\partial u^{\prime }}\overline{\delta }{u}^{\prime }=m{u}^{\prime }\epsilon =\frac{d}{dx}\left(mu\epsilon \right)\mathrm{(}\mathrm{2}\mathrm{)}$

This is not zero, so (N11) fails and Theorem I as Noether states it does not apply. The integrand changes by a total derivative. Under strict invariance the boost yields nothing.

Under Bessel-Hagen's 1921 weakening it does. The condition becomes $\overline{\delta }f+\mathrm{D}\mathrm{i}\mathrm{v}(f\cdot \Delta x)=\mathrm{D}\mathrm{i}\mathrm{v}C$ for some $C$, here $C=mu\epsilon$, and (N12) is replaced by $\sum {\psi }_i\overline{\delta }{u}_i=\mathrm{D}\mathrm{i}\mathrm{v}(B-C)$.

Carrying it through. By (1) of the previous page,

$A=B=-m{u}^{\prime }\cdot \epsilon x\mathrm{(}\mathrm{3}\mathrm{)}$

so

$B-C=-\epsilon (m{u}^{\prime }x-mu)\mathrm{(}\mathrm{4}\mathrm{)}$

and the modified relation gives the identity

$-m{u}^{″}\epsilon x=-\epsilon \frac{d}{dx}(m{u}^{\prime }x-mu)\mathrm{(}\mathrm{5}\mathrm{)}$

which is correct as an identity, since $\frac{d}{dx}(m{u}^{\prime }x-mu)=m{u}^{″}x$. On $\psi =-m{u}^{″}=0$,

$m{u}^{\prime }x-mu=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{(}\mathrm{6}\mathrm{)}$

Read with $x$ as time, this is momentum times time minus mass times position: the statement that the centre of mass moves uniformly.

The attribution matters here and this volume states it on every page where the case arises. The conserved quantity is real, the derivation is standard, and the condition that licenses it is Bessel-Hagen's, not Noether's.

Electric charge, as a rotation in field space

Now several independent variables $x_1,\dots ,x_n$, and two dependent variables $u_1,u_2$, with $f$ containing first derivatives, so that (N5) applies and

$A^{\lambda }=-\sum _i\frac{\partial f}{\partial \frac{\partial u_i}{\partial x_{\lambda }}}\overline{\delta }{u}_i\mathrm{(}\mathrm{7}\mathrm{)}$

Take the transformation that rotates the two dependent variables into each other and leaves the independent variables alone:

$\Delta x_{\lambda }=0;\Delta u_1=-u_2\epsilon ;\Delta u_2=u_1\epsilon \mathrm{(}\mathrm{8}\mathrm{)}$

By (N9), $\overline{\delta }{u}_i=\Delta u_i$, and since $\Delta x=0$ we have $B^{\lambda }=A^{\lambda }$:

$B^{\lambda }=\epsilon (\frac{\partial f}{\partial \frac{\partial u_1}{\partial x_{\lambda }}}{u}_2-\frac{\partial f}{\partial \frac{\partial u_2}{\partial x_{\lambda }}}{u}_1)\mathrm{(}\mathrm{9}\mathrm{)}$

and (N12) gives the identity

$-{\psi }_1{u}_2+{\psi }_2{u}_1=\mathrm{D}\mathrm{i}\mathrm{v}B\mathrm{(}\mathrm{10}\mathrm{)}$

On $\psi =0$, $\mathrm{D}\mathrm{i}\mathrm{v}B=0$, and the integral of the appropriate component of $B^{\lambda }$ over a cross-section is constant. That constant is the electric charge.

The point of putting these two together

Compare (9) with equation (13) of the previous page. They are the same expression. The angular-momentum case rotates two components of position; the charge case rotates two components of a field. Nothing in (N12) knows the difference, because nothing in (N12) knows what the $u_i$ mean.

A symmetry that acts only on the dependent variables and not on the independent ones is what physics calls an internal symmetry, and charge is its first example. The theorem does not distinguish internal from spacetime symmetries; the distinction is one physics imposes afterwards, on the basis of what the variables stand for.

The full electromagnetic case, with the gauge freedom that makes charge conservation an instance of the second theorem as well as the first, is on its own page.

Provenance: both derivations are this volume's own, carried out in Noether's notation with her (N9), (N12), (N4) and (N5). The weakened invariance condition used in the first is Bessel-Hagen's and is attributed to him wherever it is used. No equation above is attributed to Noether. See Sources.