Noether's theorems/Applications/Electromagnetism and charge conservation

Charge conservation as a continuity equation on a spacetime slab

This page works its case. It is also the first place where both theorems apply to the same theory and give different things.

Charge from the global symmetry

The first theorem's charge computation rotated a pair of dependent variables into each other with the independent variables held fixed, and produced a current whose divergence vanishes on the field equations.

That is charge conservation, obtained from the first theorem, and it requires the matter fields only. The electromagnetic field plays no part in it.

The gauge identity, worked

Now the second theorem. The electromagnetic potential is determined only up to the addition of the gradient of an arbitrary function — one free shape, so by the counting page exactly one identity among the field equations.

Write the field strength as antisymmetric,

$F^{ab}=-F^{ba}\mathrm{(}\mathrm{1}\mathrm{)}$

The field equations couple it to the current,

${\partial }_a{F}^{ab}=j^b\mathrm{(}\mathrm{2}\mathrm{)}$

Take the divergence of the left side:

${\partial }_b{\partial }_a{F}^{ab}=0\mathrm{(}\mathrm{3}\mathrm{)}$

because ${\partial }_b{\partial }_a$ is symmetric under interchange of $a$ and $b$ while $F^{ab}$ is antisymmetric by (1). This is an identity: it holds whatever $F$ is, solution or not. It is the single Noether identity the gauge freedom promised.

Combining (3) with (2),

${\partial }_b{j}^b=0\mathrm{(}\mathrm{4}\mathrm{)}$

Charge is conserved twice over

Equation (4) is charge conservation again, and it did not come from the first theorem. It came from the identity forced by the local symmetry, together with the field equations.

The two routes differ in what they need.

The first theorem route needs the matter fields to have the rotational symmetry, and gives a conserved current whether or not electromagnetism exists.

The second theorem route needs the gauge freedom and the coupling (2), and gives conservation of whatever is on the right of the field equations, whatever that turns out to be.

The consequence is not symmetric between them. Because of (3), a current that is not conserved cannot be coupled to electromagnetism at all: equation (2) would be inconsistent. So the gauge symmetry does not merely permit charge conservation. It requires it, on pain of the theory having no solutions.

That is a much stronger statement than the first theorem's, and it is the sort of thing the second theorem produces routinely.

The same argument twice

Step (3) is the argument of the improvement page, character for character: a symmetric derivative pair contracted with an antisymmetric object. It appears again on the topological charge page, where it does something different and more surprising.

Provenance: (1) to (4) are standard and are derived above. The observation that the gauge identity forces charge conservation rather than merely permitting it is this volume's own arrangement. No source was consulted. See Sources.