# Why does charge conservation due to gauge symmetry only hold on-shell?

While deriving Noether's theorem or the generator(and hence conserved current) for a continuous symmetry, we work modulo the assumption that the field equations hold. Considering the case of gauge symmetry: to my understanding, it's a redundancy in the "formulation" of a theory itself. So, shouldn't it lead to quantities which are conserved irrespective of whether the field equations hold?

• Yes. In fact, that's precisely what happens if you try and go through applying Noether's theorem to a gauge symmetry. You obtain that the conservation is off-shell too. – genneth Oct 20 '11 at 8:09
• In Classical Electrodynamics a charge $q$ is defined to be constant whatever happens to the particle. Of course, the motion equations are compatible and should be compatible with this definition. There is no physical symmetry behind it. Of course, one can invent equations incompatible with the charge conservation, for example, a diffusion equation with a sink. If no sinks/sources are implemented, the charge will be conserved. – Vladimir Kalitvianski Oct 20 '11 at 10:19
• One can express $\frac{dq}{dt}$ from the equations. If you use the true solutions in such expression, the charge will not depend on time. If you put arbitrary functions of time in this expression, there is no guarantee that $\frac{dq}{dt}=0$. Off-shell "solutions" may have such a drawback. – Vladimir Kalitvianski Oct 20 '11 at 10:31
• – Qmechanic Jan 17 '13 at 23:07
• @genneth Your statement is incorrect; see arxiv.org/abs/hep-th/0009058 for a detailed discussion. Conservation of charge charge follows from either the gauge field's or the matter field's equation of motion, but you do need to assume one. There are certainly "completely off-shell" field configurations in which charge is not conserved. – tparker Jul 21 '17 at 3:09

Whether your current $j^\mu$ is conserved off-shell depends on your definition of $j^\mu$. If you define it via the Dirac and other charged fields, it will only be conserved assuming the equations of motion.

However, if you define $j^\mu$ via $$j^\mu = \partial^\nu F_{\mu\nu},$$ i.e. as a function of the electromagnetic field and its derivatives, then $\partial_\mu j^\mu=0$ holds tautologically because it is $$\partial_\mu j^\mu= \partial_\mu\partial_\nu F^{\mu\nu} =0$$ which vanishes because the $\mu\nu$-symmetric second derivatives are applied to a $\mu\nu$-antisymmetric field strength tensor. The possibility to make the local conservation law tautological is indeed linked to the existence of a gauge symmetry. Why? Because it's the equation of motion one may derive from variations of the fields that are equivalent to gauge transformations: the vanishing of the variation of the action under such variations is guaranteed even without the equations of motion, by the gauge symmetry, so the corresponding combination of the currents, $\partial_\mu j^\mu$, has to vanish identically.

This logic also guarantees that the Dirac and other charged field coupled to electromagnetism will have equations of motion that guarantee the local charge conservation.

An analogous statement exists in the case of the diffeomorphism symmetry: $$\nabla_\mu G^{\mu\nu} = 0$$ also holds tautologically for the Einstein tensor $G$ defined in terms of the metric tensor and its derivatives.

• There is at least a stretch in your "proof" (tautology=gauge invariance). The equation $j=\partial F$ is not a definition of a charge but that of fields. So one can easily write down a $j$ incompatible with charge conservation. ;-) – Vladimir Kalitvianski Oct 20 '11 at 11:04
• Ah, okay. So charge conservation just reproduces (Bianchi) identities and then we can use the field equations to get it's implications for the source fields. Thanks @Lubos! – Siva Oct 21 '11 at 4:55
• @VladimirKalitvianski , If $j$, in the equation $j=\partial F$, is incompatible with charge conservation, then the Gauge invariance is broken. The variation of the action with respect to a gauge transformation is given by $\partial (\partial F-j) - eom´s$, so if the equations of motion are satisfied the charge must be conserved in order to preserve the gauge invariance – Nogueira Jun 1 '17 at 21:27