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Another Noether's theorem question, this time about electrical charge.

According to Noether's theorem, all conservation laws originate from invariance of a system to shifts in a certain space. For example conservation of energy stems from invariance to time translation.

What kind of symmetry creates the conservation of electrical charge?

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4 Answers 4

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Remember that voltage is always expressed as a "potential difference." You can't measure the absolute value of voltage because everything is invariant when you add a constant voltage everywhere. That expresses a symmetry just like time translation invariance.

When you bring in the magnetic field this invariance or symmetry can be generalised to a bigger gauge invariance transforming the electromagnetic potential as a vector field. Charge particles are also described by fields such as Dirac spinors, which are multiplied by a phase factor under the action of this symmetry, making it a U(1) invariance. Electric charge is the conserved quantity that Noether's theorem gives for this symmetry.

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Global gauge invariance, cf. Wikipedia.

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  • $\begingroup$ To elaborate on Qmechanic's answer: In the case of the Dirac field, a global change in the phase gives rise to a conserved current $\bar{\psi} \gamma^\mu \psi$, which has a locally conserved (electric) charge $\int \mathrm{d}^3 x \, \, \psi^{\dagger}\psi$. $\endgroup$
    – JamalS
    Commented Apr 4, 2014 at 11:50
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    $\begingroup$ For details, see e.g. this Phys.SE post. $\endgroup$
    – Qmechanic ♦
    Commented Mar 8, 2015 at 21:16
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    $\begingroup$ This is a confusing terminology "Global gauge invariance" is a contradiction in terms. I assume you mean the Global U(1) symmetry (implied by the existence of a Local U(1) Gauge Symmetry). $\endgroup$
    – Kvothe
    Commented May 29, 2020 at 8:38
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In CED written in terms of field strengths there is no a notion of gauge invariance. The charge value is a constant in time parameter by definition. There is also a continuity equation that governs charge flows. So it is a sequence of definitions and physical equations. Charge of a system is not a dynamical variable, nor a function of dynamical variables. The Noether theorem has nothing to do with its conservation.

The masses, despite being constant, do not have a continuity equation in CED so they are not obliged to conserve ;-).

Edit 1: I see this question is not so easy for many. OK, the charge value of one particle is constant by definition (like mass) so its conservation is a sequence of definition. Another matter - whether the system charge is additive in particles? Does it evolve with time? Does it depend on interactions? To answer these questions, we have to employ the equations of motion. The charge continuity equation $\partial \rho /\partial t = div(\rho v)$ is valid for any v, so the additivity is an exact sequence of this equation: $\rho$ is additive in particles and a single charge is constant.

For the masses we can write such a continuity equations too but the system mass is generally not a sum of particle masses. The system mass is defined differently as it depends also on interactions.

Edit 2: The number of particles, charged or not, is also conserved in many theories. Do you really think it is a consequence of ambiguity in the potential definition?

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  • $\begingroup$ Downvoters, explain your motivation, please. $\endgroup$
    – Vladimir Kalitvianski
    Commented Jan 22, 2011 at 15:40
  • $\begingroup$ I am not downvoting, I am 126 only. But note that the Wikipedia description "The full statement of gauge invariance is that the physics of an electromagnetic field are unchanged when the scalar and vector potential are shifted by the gradient of an arbitrary scalar field" applies to Classical ElectroDynamics. $\endgroup$
    – arivero
    Commented Jan 22, 2011 at 16:21
  • $\begingroup$ The EMF strengths do not depend on gauge transformations of potentials, nobody argues with it. But, since under gauge transformations the Lagrangian expressed via field tensions $F_{\mu \nu}$ (no potentials) does not vary at all, there is no conservation because of this. $\endgroup$
    – Vladimir Kalitvianski
    Commented Jan 22, 2011 at 16:29
  • $\begingroup$ Vlad, your writting is confuse. Mass is not the topic of this question, it was the topic of another question. At least, from your last comment, you agree that there is a notion of global gauge transformations and global gauge invariance in classical electrodymamics, do you? A simple yes or not will be enough as answer. $\endgroup$
    – arivero
    Commented Jan 22, 2011 at 19:00
  • $\begingroup$ Yes, there is a gauge invariance in CED, of course. But it is an ambiguity of new variables (potentials) rather than some sort of physical symmetry. And tell me, why should we "derive" the charge conservation law from some symmetry if we define charge as independent from time constant? $\endgroup$
    – Vladimir Kalitvianski
    Commented Jan 22, 2011 at 19:24
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Charge conservation is related to the invariance of the Lagrangian under rotation in the complex plane, or, equivalently, under a complex phase shift, such as $$\phi \rightarrow \phi + i\delta \phi ~.$$

Often it is considered a consequence of electromagnetic gauge invariance. However, charge is also conserved in the absence of electromagnetism. It is therefore better to say that gauge invariant electromagnetism can only describe conserved charge.

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