Is gauge symmetry necessary for charge conservation?

Question

The common view is that gauge symmetry is necessary for conservation of charge(s) in Yang-Mills theory. But one thing I have never been able to get out of my head is, if there isn't any other possible mechanism for implying conservation of something like a charge.

Suppose (only as a toy model, of course) that the electromagnetic Lagrange density was given by $${\cal L}=F^{\mu\nu}F_{\nu\mu}-\Omega(A_\mu A^\mu)$$ with an ordinary function $\Omega(x)$ of a scalar variable $x$.

Then the electromagnetic field equations would be something like (sorry if I messed up the constants...) $$\partial_\mu F^{\mu\nu}=\Omega^\prime(A_\mu A^\mu)A^\nu$$ or in other words, the current density would be an algebraic function of the four potential: $$j^\nu = \Omega^\prime(A_\mu A^\mu)A^\nu$$ While there are probably dozens of other reasons why this cannot represent valid physics, I think the current would still be conserved because of antisymmetry of the field tensor $$0=\partial_\mu\partial_\nu F^{\mu\nu}=\partial_\mu j^\mu$$ although the explicit dependency of the $\Omega$-term from the four potential breaks gauge symmetry.

Is there something wrong about this reasoning, and therefore, is gauge invariance really a necessary condition for charge conservation, or is it only a sufficient (and convenient, because everything works out so well in the standard model and the Higgs mechanism) condition?

Answer 1

Yes, OP is right, even without gauge symmetry, if the Lagrangian density is of the form

$${\cal L}(A,\partial A)~=~-\frac{1}{4}F_{\mu\nu} F^{\mu\nu}- \Omega(A),\tag{1}$$ then EL equations read $$ d_{\nu}F^{\mu\nu}~=~d_{\nu}\frac{\partial {\cal L}}{\partial A_{\mu,\nu}}~\approx~\frac{\partial {\cal L}}{\partial A_{\mu}}~=~-\frac{\partial \Omega}{\partial A_{\mu}}~=:~ j^{\mu}, \tag{2}$$ which leads to an on-shell continuity equation
$$d_{\mu}j^{\mu}~\approx~0, \tag{3}$$ and a conserved charge $$Q~:=~\int_V \! d^3x ~j^0.\tag{4}$$ And as far as the inverse Noether's first theorem goes, there should be a corresponding global symmetry.

The question remains of whether this notion of charge (4) is related to the usual notion of electric matter charge or not. This is less clear, especially because OP didn't discuss any matter content.

Answer 2

I'll address a comment under @Qmechanics's answer, since that's apparently where we get the real need fleshed out:

given that gauge symmetry is put at the outset of the standard model, I find it pretty interesting to note that the assumption of the necessity of gauge symmetry severely limits the range of possible theories

That's not a bug, it's a feature. The job of physical theories is to explain past observations and predict future ones. We can't do that with zero extra-mathematical assumptions, and occasionally we realize the need for a new one.

Einstein started a trend of seeing symmetries not as things we discover models otherwise motivated obtain, but as motivations for the most general otherwise-OK model we'll fit to data. ("Otherwise OK" includes things like "don't put in second- or higher-order derivatives, because of Ostrogradski instability.") His first move of this kind was arguing contra other contemporaneous physicists that, if Maxwell's equations contradict what we would now describe as the invariance of physical laws under Galilean transformations, because they are instead invariant under Lorentz transformations, Galileo is wrong; Maxwell is right; therefore, Lorentz is right. That last part is crucial because, whereas the first two are an unpredictive ad hoc rescue move, the three together determine what a Newton-recovering new physics looks like.

That was his reaction to $A_\mu\to \Lambda_\mu{}^\nu A_\nu$, with $\Lambda$ a Lorentz matrix. But Maxwell's equations are also invariant under the gauge transformation $A_\mu\to A_\mu+\partial_\mu\chi$. Einstein's insight is to never let that be an accident. How big a family of theories does it allow for? And more importantly, what does that family predict? If we introduce new fieds, which we can do ad infinitum, we can keep expanding the family, but that's neither theoretically satisfying nor empirically constraining, so it's clearly that's not what "most general" intends. As long as we only have $A^\mu$, we can't give it mass, and indeed the photon is massless and moves at speed $c$. If it didn't, 19th century experiments wouldn't have forced Maxwell upon us, the "long 19th century" (1789-1914) wouldn't have forced special relativity on us, and we wouldn't be in this mess in the first place. And until we need massive counterparts to the photon, that's as far as our theory should go.

But since the weak interaction decays with distance too fast to have a massless carrier, we need a bigger family, so we need to add another field, say a scalar $h$. But something funny happens when you expand a theory: the newcomer could, with the right history of science, have added its pieces in a different order. So imagine we believed in $h$ first: why would we posit $A^\mu$?

You already know the answer, but don't like it. If you'd instead posted a question here asking, "why should gauge invariance be so important we expand $\partial_\mu h$ to $D_\mu h$, thereby dreaming up $A_\mu$?", I wouldn't be surprised because it comes from the same concern as your current question. But look what such a commitment gets us: making $A_\mu$ dynamical introduces Maxwell (or more generally Yang-Mills for multiplet-valued $h$), and it turns out $h$ gives this vector a mass, which our old one couldn't. And yet, the gauge invariance has survived. Even nicer - since old theoretical successes must be maintained - this theory lets the photon stay massless (through $\mathsf{SU}(2)_L\times\mathsf{U}_Y(1)$, Goldstone's theorem, blah blah blah).

We have this model because, every time we worked out what solves one data-driven problem, we asked how it can be adapted for follow-up problems. Yang-Mills isn't just multiplet Maxwell; it's multiple Maxwell with mass installed, because the data says we need such mass. Yang-Mills changes $\partial_\mu F^{\mu\nu}$ to $D_\mu F^{\mu\nu}$, the difference being proportional to a Lie algebra's structure constants. Your toy model, by contrast, never gets round to this upgrade. Mathematics doesn't forbid it; the desire to fit the data does, in this case adding mass.

Back to the question:

gauge symmetry is necessary for conservation of charge(s)

Noether's theorem relates conservation laws to symmetries, but you either posit the list of them or you don't. You actually can't posit it all up front because of emergent results, e.g. from Lax pairs. But gauge theory rests on a surprisingly helpful result: the most general option compatible with our initial choice of symmetries isn't actually very general at all (as measured by how many parameters you have to fix by experiment), provided we use the minimum number of fields. Luckily, "one field per known particle species, plus just enough for the maths to check out" is a pragmatic definition of "minimum number", especially since it leads us to predict new particles before we find them. (Anyone who doubted this strategy ate humble pie when we found the W, Z and Higgs bosons, and measured their masses.) As an added bonus, the aforementioned adapt-old-theories approach gives us a good idea what to conserve. Why do we conserve electric charge? Because experiments have asked us to since the 1800s.