Noether theorem, gauge symmetry and conservation of charge

Question

I'm trying to understand Noether's theorem, and it's application to gauge symmetry. Below what I've done so far.

First, the global gauge symmetry. I'm starting with the Lagragian $$L_{1}=\partial^{\mu}\Psi\partial_{\mu}\Psi^{\ast}-m^{2}\left|\Psi\right|^{2}$$ with classical complex fields. This Lagragian is invariant with respect to the global gauge symmetry $\Psi\rightarrow\tilde{\Psi}=e^{\mathbf{i}\theta}\Psi$, ... such that I end up with $$\delta S=\int dv\left[\dfrac{\delta L_{1}}{\delta\Psi}\delta\Psi+\dfrac{\delta L_{1}}{\delta\Psi^{\ast}}\delta\Psi^{\ast}+\mathbf{i}\left(\Psi\partial^{\mu}\Psi^{\ast}-\Psi^{\ast}\partial^{\mu}\Psi\right)\partial_{\mu}\delta\theta\right]=\int dv\left[\partial_{\mu}j^{\mu}\right]\delta\theta$$ provided the equations of motion ($\delta L / \delta \Psi = 0$, ...) are valid. All along I'm using that $$\dfrac{\delta L}{\delta\phi}=\dfrac{\partial L}{\partial\phi}-\partial_{\mu}\dfrac{\partial L}{\partial\left[\partial_{\mu}\phi\right]}$$ and that $\int dv=\int d^{3}xdt$ for short. The conserved current is of course $$j_{1}^{\mu}=\mathbf{i}\left(\Psi^{\ast}\partial^{\mu}\Psi-\Psi\partial^{\mu}\Psi^{\ast}\right)$$ since $\delta S / \delta \theta =0 \Rightarrow\partial_{\mu}j_{1}^{\mu}=0$.

Here is my first question: Is this really the demonstration for conservation of charge ? Up to now, it seems to me that I only demonstrated that the particle number is conserved, there is no charge for the moment...

Then, I switch to the local gauge symmetry. I'm starting with the following Lagrangian $$L_{2}=\left(\partial^{\mu}+\mathbf{i}qA^{\mu}\right)\Psi\left(\partial_{\mu}-\mathbf{i}qA_{\mu}\right)\Psi^{\ast} -m^{2}\left|\Psi\right|^{2} -\dfrac{F_{\mu\nu}F^{\mu\nu}}{4}$$ with $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$. This Lagrangian is invariant with respect to the local gauge transformation $$L_{2}\left[\tilde{\Psi}=e^{\mathbf{i}q\varphi\left(x\right)}\Psi\left(x\right),\tilde{\Psi}^{\ast}=e^{-\mathbf{i}q\varphi\left(x\right)}\Psi^{\ast},\tilde{A}_{\mu}=A_{\mu}-\partial_{\mu}\varphi\right]=L_{2}\left[\Psi,\Psi^{\ast},A_{\mu}\right]$$

Then I have $$\delta S=\int dv\left[\dfrac{\delta L_{2}}{\delta\Psi}\delta\Psi+\dfrac{\delta L_{2}}{\delta\Psi^{\ast}}\delta\Psi^{\ast}+\dfrac{\delta L_{2}}{\delta A_{\mu}}\delta A_{\mu}\right]$$ with $\delta\Psi=\mathbf{i}q\Psi\delta\varphi$, $\delta A_{\mu}=-\partial_{\mu}\delta\varphi$, ... such that I end up with $$\dfrac{\delta S}{\delta\varphi}=\int dv\left[\mathbf{i}q\Psi\dfrac{\delta L_{2}}{\delta\Psi}+c.c.+\partial_{\mu}\left[j_{2}^{\mu}-\partial_{\nu}F^{\nu\mu}\right]\right]$$ with $j_{2}^{\mu}=\partial L_{2}/\partial A_{\mu}$ and $F^{\nu\mu}=\partial L_{2}/\partial\left[\partial_{\nu}A_{\mu}\right]$

Then, by application of the equations of motion, I have $$\partial_{\mu}\left[j_{2}^{\mu}-\partial_{\nu}F^{\nu\mu}\right]=0\Rightarrow\partial_{\mu}j_{2}^{\mu}=0$$ since $\partial_{\mu}\partial_{\nu}F^{\nu\mu}=0$ by construction. Of course the new current is $$j_{2}^{\mu}=\mathbf{i}q\left(\Psi^{\ast}\left(\partial^{\mu}+\mathbf{i}qA^{\mu}\right)\Psi-\Psi\left(\partial^{\mu}-\mathbf{i}qA^{\mu}\right)\Psi^{\ast}\right)$$ and is explicitly dependent on the charge. So it seems to me this one is a better candidate for the conservation of charge.

NB: As remarked in http://arxiv.org/abs/hep-th/0009058, Eq.(27) one can also suppose the Maxwell's equations to be valid ($j_{2}^{\mu}-\partial_{\nu}F^{\nu\mu} = 0$, since they are also part of the equation of motion after all, I'll come later to this point, which sounds weird to me), and we end up with the same current, once again conserved.

Nevertheless, I still have some troubles. Indeed, if I abruptly calculate the equations of motions from the Lagrangian, I end up with (for the $A_{\mu}$ equation of motion) $$j_{2}^{\mu}-\partial_{\nu}F^{\nu\mu}\Rightarrow\partial_{\mu}j_{2}^{\mu}=0$$ by definition of the $F^{\mu \nu}$ tensor.

So, my other questions: Is there a better way to show the conservation of EM charge ? Is there something wrong with what I did so far ? Why the Noether theorem does not seem to give me something which are not in the equations of motions ? said differently: Why should I use the Noether machinery for something which is intrinsically implemented in the Lagrangian, and thus in the equations of motion for the independent fields ? (Is it because my Lagrangian is too simple ? Is it due to the multiple boundary terms I cancel ?)

Thanks in advance.

PS: I've the feeling that part of the answer would be in the difference between what high-energy physicists call "on-shell" and "off-shell" structure. So far, I never understood the difference. That's should be my last question today :-)

Answer 1

Comments to the question (v1):

1. Last thing first. On-shell means (in this context) that equations of motion (eom) are satisfied. Equations of motion means Euler-Lagrange equations. Off-shell means strictly speaking not on-shell, but in practice it is always used in the sense not necessarily on-shell. [Let us stress that every infinitesimal transformation is an on-shell symmetry of an action, so an on-shell symmetry is a vacuous notion. Therefore in physics, when we claim that an action has a symmetry, it is always implicitly understood that the symmetry is an off-shell symmetry.]

2. OP wrote: Here is my first question: Is this really the demonstration for conservation of (electric) charge? For that particular action: Yes. More generally for QED: No, because the $4$-gauge-potential $A_{\mu}$, the Maxwell term $F_{\mu\nu}F^{\mu\nu}$, and the minimal coupling are missing in OP's action. It is in principle not enough to only look at the matter sector. On the other hand, global gauge symmetry for the full action $S[A,\Psi]$ leads to electric charge conservation, cf. Noether's first Theorem. [Two comments to drive home the point that it is necessary to also consider the gauge sector: (i) If we were doing scalar QED (rather than ordinary QED), it is known that the Noether current $j^{\mu}$ actually depends on the $4$-gauge-potential $A_{\mu}$, so the gauge sector is important, cf. this Phys.SE post. (ii) Another issue is that if we follow OP's method and are supposed to treat the $4$-gauge potential $A_{\mu}$ as a classical background (which OP puts to zero), then presumably we should also assume Maxwell's equations $d_{\mu}F^{\mu\nu}=-j^{\nu}$. Maxwell's equations imply by themselves the continuity equation $d_{\mu}J^{\mu}=0$ even before we apply Noether's Theorems.]

3. There is no conserved quantity associated with local gauge symmetry per se, cf. Noether's second Theorem. (Its off-shell Noether identity is a triviality. See also this Phys.SE question.)

4. Perhaps a helpful comparison. It is possible to consider an EM model of the form $$S[A]~=~\int\! d^4x~ \left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+J^{\mu}A_{\mu}\right),$$ where $J^{\mu}$ are treated as passive non-dynamical classical background matter sources. In other words, only the gauge fields $A_{\mu}$ are dynamical variables in this model. Before we even get started, we have to ensure local (off-shell) gauge symmetry of the action $S[A]$ up to boundary terms. This implies that the classical background sources $J^{\mu}$ must satisfy the continuity equation $d_{\mu}J^{\mu}=0$ off-shell. Thus a conservation law is forced upon us even before we apply Noether's Theorems. Note that global gauge symmetry is an empty statement in this model.

Answer 2

Is this really the demonstration for conservation of charge ?

Yes. The charge is defined as $Q = \int d^3x~j^0$, so $\partial_\mu j^\mu = 0$ shows that it is conserved.

Up to now, it seems to me that I only demonstrated that the probability flow is conserved, there is no charge for the moment...

What you demonstrated is that the current is conserved. I don't think you should call this a "probability flow"; it sounds like you're confusing $\Psi$ with a wavefunction, when in fact it's a quantum field.

Answer 3

The OP asked me to give an answer to this question. Well, all the questions seem to be about the "necessity" of the Noether's theorem.

So the answer is that Noether's procedure is the way to derive the current from a known symmetry. This is very useful because we usually know very well how a symmetry acts – because we know how field transform under it or how things rotate or shift under spacetime operations etc. On the other hand, the precise form of the conserved current becomes much less obvious, especially once we start to add various interactions. There's "pretty much" just one solution what the current may be to be conserved and Noether's procedure is a way to get this right form. Well, yes, the form of the current is "contained" in the Lagrangian or the equations of motion but it is not obvious how to "extract it" – and that's why we cherish Noether's procedure. If you have a different algorithm how to extract it, tell us, but I know that there can't be any different procedure that wouldn't be equivalent to Noether's one at all.

Now, back to the first example in the question.

For the non-interacting fields, the number of particles – their quanta – is conserved completely. In fact, every free field of species $s$ in every state given by a momentum $k$ and polarization $\lambda$ etc. is conserved, $N_{s,\lambda,\dots}(k,\dots)={\rm const}$. But this is clearly just a special situation when the interactions don't exist and this case isn't physically interesting.

The interesting theories only begin once we have some interactions. They destroy almost all of these "conservation laws". In particular, it is not true that the number of particles is conserved in quantum field theory. We may create electron-positron pairs out of pure energy, and so on. Only some quantities such as charges, energy/momentum, angular momentum are conserved, they are in one-to-one correspondence with the symmetries, and the corresponding currents (which includes the stress-energy tensor) may be derived via the Noether's procedure.