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Noether theorems volume, initial import (A-04)

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{{DISPLAYTITLE:Electromagnetism and charge conservation}}

[[File:NT-F-23.png|thumb|right|360px|alt=A rectangular slab of spacetime with time running upward and space across. Arrows run from the lower slice to the upper one. Side arrows mark flow in and out. A caption states that the total is the same on every slice provided nothing crosses the sides.|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 ==

[[Noether's theorems/First theorem/Centre-of-mass motion and electric charge|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
[[Noether's theorems/Second theorem/Fewer independent field equations than fields|the
counting page]] exactly one identity among the field equations.

Write the field strength as antisymmetric,

<math>F^{ab}=-F^{ba}\qquad\qquad\mathrm{(1)}</math>

The field equations couple it to the current,

<math>\partial_{a}F^{ab}=j^{b}\qquad\qquad\mathrm{(2)}</math>

Take the divergence of the left side:

<math>\partial_{b}\partial_{a}F^{ab}=0\qquad\qquad\mathrm{(3)}</math>

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

Combining (3) with (2),

<math>\partial_{b}j^{b}=0\qquad\qquad\mathrm{(4)}</math>

== 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
[[Noether's theorems/Mathematics/The Belinfante-Rosenfeld improvement|the
improvement page]], character for character: a symmetric derivative pair
contracted with an antisymmetric object. It appears again on
[[Noether's theorems/Applications/Topological charge, which is not a Noether charge|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
[[Noether's theorems/Sources|Sources]].''

Noether's theorems/Applications/Electromagnetism and charge conservation - Revision history

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