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I'm confused about Noether's theorem applied to gauge symmetry. Say we have

$$\mathcal L=-\frac14F_{ab}F^{ab}.$$

Then it's invariant under $A_a\rightarrow A_a+\partial_a\Lambda.$

But can I say that the conserved current here is

$$J^a=\frac{\partial\mathcal L}{\partial(\partial_aA_b)}\delta A_b=-\frac12 F^{ab}\partial_b\Lambda~?$$

Why do I never see such a current written? If Noether's theorem doesn't apply here, then is space-time translation symmetry the only candidate to produce Noether currents for this Lagrangian?

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For local symmetries, one uses Noether's second theorem, while Noether's first theorem is for global symmetries, cf. physics.stackexchange.com/a/13881/2451 , physics.stackexchange.com/q/66092/2451 and links therein. For global gauge symmetry, see physics.stackexchange.com/q/48305/2451 and links therein. –  Qmechanic May 12 at 15:08

2 Answers 2

up vote 11 down vote accepted

Indeed, nothing is wrong with Noether theorem here, $J^\mu = F^{\mu \nu} \partial_\nu \Lambda$ is a conserved current for every choice of the smooth scalar function $\Lambda$. It can be proved by direct inspection, since $$\partial_\mu J^\mu = \partial_\mu (F^{\mu \nu} \partial_\nu \Lambda)= (\partial_\mu F^{\mu \nu}) \partial_\nu \Lambda+ F^{\mu \nu} \partial_\mu\partial_\nu \Lambda = 0 + 0 =0\:.$$ Above, $\partial_\mu F^{\mu \nu}=0 $ due to field equations and $F^{\mu \nu} \partial_\mu\partial_\nu \Lambda=0$ because $F^{\mu \nu}=-F^{\nu \mu}$ whereas $\partial_\mu\partial_\nu \Lambda =\partial_\nu\partial_\mu \Lambda$.

ADDENDUM. I show here that $J^\mu$ arises from the standard Noether theorem. The relevant symmetry transformation, for every fixed $\Lambda$, is $$A_\mu \to A'_\mu = A_\mu + \epsilon \partial_\mu \Lambda\:.$$ One immediately sees that $$\int_\Omega {\cal L}(A', \partial A') d^4x = \int_\Omega {\cal L}(A, \partial A) d^4x\tag{0}$$ since even ${\cal L}$ is invariant. Hence, $$\frac{d}{d\epsilon}|_{\epsilon=0} \int_\Omega {\cal L}(A, \partial A) d^4x=0\:.\tag{1}$$ Swapping the symbol of derivative and that of integral (assuming $\Omega$ bounded) and exploiting Euler-Lagrange equations, (1) can be re-written as: $$\int_\Omega \partial_\nu \left(\frac{\partial {\cal L}}{\partial \partial_\nu A_\mu} \partial_\mu \Lambda\right) \: d^4 x =0\:.\tag{2}$$ Since the integrand is continuous and $\Omega$ arbitrary, (2) is equivalent to $$\partial_\nu \left(\frac{\partial {\cal L}}{\partial \partial_\nu A_\mu} \partial_\mu \Lambda\right) =0\:,$$ which is the identity discussed by the OP (I omit a constant factor): $$\partial_\mu (F^{\mu \nu} \partial_\nu \Lambda)=0\:.$$

ADDENDUM2. The charge associated to any of these currents is related with the electrical flux at spatial infinity. Indeed one has: $$Q = \int_{t=t_0} J^0 d^3x = \int_{t=t_0} \sum_{i=1}^3 F^{0i}\partial_i \Lambda d^3x = \int_{t=t_0} \partial_i\sum_{i=1}^3 F^{0i} \Lambda d^3x - \int_{t=t_0} ( \sum_{i=1}^3 \partial_i F^{0i}) \Lambda d^3x \:.$$ As $\sum_{i=1}^3 \partial_i F^{0i} = -\partial_\mu F^{\mu 0}=0$, the last integral does not give any contribution and we have $$Q = \int_{t=t_0} \partial_i\left(\Lambda \sum_{i=1}^3 F^{0i} \right) d^3x = \lim_{R\to +\infty}\oint_{t=t_0, |\vec{x}| =R} \Lambda \vec{E} \cdot \vec{n} \: dS\:.$$ If $\Lambda$ becomes constant in space outside a bounded region $\Omega_0$ and if, for instance, that constant does not vanish, $Q$ is just the flux of $\vec{E}$ at infinity up to a constant factor. In this case $Q$ is the electric charge up to a constant factor (as stressed by ramanujan_dirac in a comment below). In that case, however, $Q=0$ since we are dealing with the free EM field.

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Thank you very much. Do you mean then that Noether's (first) theorem is applicable here, although it's not a global symmetry? –  user46348 May 12 at 18:30
    
I provided a proof, what should I do further? ;) –  Valter Moretti May 12 at 18:32
    
I know... It's just that it conflicts with what others said here. Then can we say that as long as the symmetry is continuous, Noether's theorem applies? Also, why don't I see this current written anywhere? –  user46348 May 12 at 18:36
    
@user46348 I believe, but please correct me if I'm wrong, that Noether's first theorem in general does not apply to gauge transformations, it only applies to global transformations. However, apparently Noether's first theorem happens to work for $\mathrm{U(1)}$ electromagnetic theory. –  Hunter May 12 at 19:02
    
@V.Moretti: So is the Noether's charge, the usual electric charge, and what you are stating is the conservation of electric charge? –  user7757 May 13 at 9:42

I asked myself the same question a while back, and this is what I came up with:

I assumed that the Lagrangian of the Maxwell action is of the form: $$\mathcal{L} = \mathcal{L}(A_\mu,\,\partial_\nu A_\mu)$$ Then I assumed that under variations of the type $A_\mu \rightarrow A_\mu - \frac{1}{e}\partial_\mu\alpha$ where $\alpha\equiv \alpha(x)$ a local gauge parameter, the Lagrangian remains invariant: \begin{align} \delta\mathcal{L} &= \frac{\partial \mathcal{L}}{\partial A_\mu}\delta A_\mu + \frac{\partial \mathcal{L}}{\partial(\partial_\nu A_\mu)}\delta(\partial_\nu A_\mu)\\ &= \frac{\partial \mathcal{L}}{\partial A_\mu}({\textstyle \frac{-1}{e}\partial_\mu \alpha})+ \frac{\partial \mathcal{L}}{\partial (\partial_\nu A_\mu)}({\textstyle \frac{-1}{e}\partial_\mu \partial_\mu \alpha})=0\,. \end{align} The in the first term, I use the equation of motion to replace $\frac{\partial\mathcal{L}}{\partial A_\mu}=\partial_\nu (\frac{\partial\mathcal{L}}{\partial (\partial_\nu A_\mu)})$ to get \begin{equation} \phantom{wwwwww}\delta\mathcal{L}=\partial_\nu\big(\frac{\partial\mathcal{L}}{\partial (\partial_\nu A_\mu)}\big)({\textstyle \frac{-1}{e}\partial_\mu \alpha})+\frac{\partial \mathcal{L}}{\partial (\partial_\nu A_\mu)}({\textstyle \frac{-1}{e}\partial_\mu \partial_\mu \alpha})=0\,. \end{equation} Now, since $\alpha(x)$ is an arbitrary smooth function, first derivative is independent of second derivative. So the two terms vanish separately.

  • Vanishing second term implies the tensor $\frac{\partial \mathcal{L}}{\partial (\partial_\nu A_\mu)}$ is antisymmetric (as it contracts with symmetric $\partial_\mu\partial_\nu \alpha$). So just define it $\frac{\partial \mathcal{L}}{\partial (\partial_\nu A_\mu)}:=F^{\nu\mu}$

  • Vanishing first term implies $\partial_\nu \big(\frac{\partial \mathcal{L}}{\partial (\partial_\nu A_\mu)}\big) = \partial_\nu F^{\nu\mu} = 0.$

Therefore, I conclude that Noether's theorem for gauge transformations is not a conservation law, but an equation of motion. And, the requirement of gauge symmetry imposes that $F^{\mu\nu}$ is antisymmetric.

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