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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?

Qmechanic
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Uri
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4 Answers4

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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.

Philip Gibbs - inactive
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Global gauge invariance, cf. Wikipedia.

Qmechanic
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Greg P
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  • 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 $\int \mathrm{d}^3 x , , \psi^{\dagger}\psi$. – JamalS Apr 04 '14 at 11:50
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    For details, see e.g. this Phys.SE post. – Qmechanic Mar 08 '15 at 21:16
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    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). – Marten May 29 '20 at 08:38
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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?

Vladimir Kalitvianski
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  • Downvoters, explain your motivation, please. – Vladimir Kalitvianski Jan 22 '11 at 15:40
  • 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. – arivero Jan 22 '11 at 16:21
  • 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. – Vladimir Kalitvianski Jan 22 '11 at 16:29
  • 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. – arivero Jan 22 '11 at 19:00
  • 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? – Vladimir Kalitvianski Jan 22 '11 at 19:24
  • I think we're getting confused over terminology here. Vladimir is completely correct that the local gauge symmetry does not generate a conserved current ala Noether's theorem. However, even after gauge fixing, there is a (physical) global U(1) symmetry. It is that symmetry which generates the conserved quantity. – genneth Jan 28 '11 at 15:14
  • To genneth: what conserved quantity is it and how it is expressed via dynamical variables, please? – Vladimir Kalitvianski Feb 11 '11 at 10:41
  • Are you saying that because the potential isn't physical, Noether's theorem doesn't apply to variation of the potential? – John McAndrew Apr 23 '16 at 22:24
  • @JohnMcVirgo: the charge conservation of an interacting system of charges follows from the equations. And one can obtain the total charge as a sum of constituents from the equations. Consider the total field of the system at a long distance which gets into the equations of motion of a distant probe charge. The leading term is determined with the total charge. Note, the sum of masses is also constant - by definition of masses, but we do not obtain it from equations because we must deal with the total energy involving the interaction energy too. – Vladimir Kalitvianski Apr 24 '16 at 19:29
  • If something is obtained from Noether's theorem, it does not mean it cannot be obtained differently. In case of the total charge, it is necessary for equations to have physical solutions and it is implied so while integrating the charge density. – Vladimir Kalitvianski Apr 24 '16 at 19:29
  • Likewise, you can derive conservation laws from Maxwell's equations that are interpreted as energy, momentum, angular momentum without using a Lagrangian and Noether's theorem. Maybe you doubt whether Noether's theorem is a valid way of generating conserved quantities in general for a physical system? – John McAndrew Apr 24 '16 at 20:07
  • @JohnMcVirgo: For a physical system, you say? We speak of equations and sometimes these equations have non physical solutions. Their Lagrangian exists and the Noether theorem gives formal formulas for conserving quantities as if the solutions were physical. What would you say in case of non physical solutions? What is conserved? – Vladimir Kalitvianski Apr 26 '16 at 16:06
  • I'm under the impression that one doesn't use a Lagrangian that gives unphysical solutions. This would make the whole point of using a Lagrangian in the first place pointless otherwise. – John McAndrew Apr 28 '16 at 02:51
  • @JohnMcVirgo: How about Feynman lectures, chapter 28 (http://www.feynmanlectures.caltech.edu/II_28.html)? How about Landau-Lifshitz textbook about the same matter? – Vladimir Kalitvianski Apr 28 '16 at 11:52
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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.

my2cts
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