How to prove conservation of electric charge using Noether's first theorem according to classical (non-quantum) mechanics? I know the proof based on using Klein–Gordon field, but that derivation use quantum mechanics particularly.
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By the word classical we will mean $\hbar=0$, and we will use the conventions of Ref. 1.
The Lagrangian density for Maxwell theory with various matter content is$^1$
$${\cal L} ~=~{\cal L}_{\rm Maxwell} + {\cal L}_{\rm matter} ,\tag{1} $$
$${\cal L}_{\rm Maxwell}~=~ -\frac{1}{4}F_{\mu\nu}F^{\mu\nu},\tag{2}$$
$$ {\cal L}_{\rm matter}~=~{\cal L}_{\rm matter}^{\rm QED}+{\cal L}_{\rm matter}^{\rm scalar QED} + \ldots,\tag{3} $$
$$ {\cal L}_{\rm matter}^{\rm QED} ~:=~ \overline{\Psi}( i\gamma^{\mu} D_{\mu}-m)\Psi ,\tag{4} $$
$$ {\cal L}_{\rm matter}^{\rm scalar QED}~:=~ -(D_{\mu}\phi)^{\dagger} D^{\mu}\phi -m^2\phi^{\dagger}\phi -\frac{\lambda}{4} (\phi^{\dagger}\phi)^2,\tag{5} $$
with covariant derivative
$$ D_{\mu}~=~d_{\mu}-ieA_{\mu}, \tag{6} $$ and with Minkowski sign convention (-,+,+,+). (Here we are too lazy to denote various matter masses $m$ and charges $e$ differently.) The matter equations of motion (eom) are
$$ ( i\gamma^{\mu} D_{\mu}-m)\Psi ~\stackrel{m}{\approx}~0, \qquad D_{\mu}D^{\mu}\phi~\stackrel{m}{\approx}~m^2\phi+\frac{\lambda}{2} \phi^{\dagger}\phi^2, \qquad \ldots.\tag{7}$$
(The $\stackrel{m}{\approx}$ symbol means equality modulo matter eom, i.e. an on-shell equality.)
The infinitesimal global off-shell gauge transformation is
$$ \delta A_{\mu} ~=~0, \qquad \delta\Psi~=~-i\epsilon \Psi, \qquad \delta\overline{\Psi}~=~i\epsilon \overline{\Psi}, $$ $$ \delta\phi~=~-i\epsilon \phi,\qquad \delta\phi^{\dagger}~=~i\epsilon \phi^{\dagger}, \qquad \ldots, \qquad\delta {\cal L} ~=~0,\tag{8} $$
where the infinitesimal parameter $\epsilon$ does not depend on $x$.
The Noether current is the electric $4$-current$^2$
$$ j^{\mu}~=~e\overline{\Psi}\gamma^{\mu}\Psi - ie\{\phi^{\dagger} D^{\mu}\phi-(D^{\mu}\phi)^{\dagger}\phi\}+\ldots. \tag{9}$$
Noether's first Theorem is a theorem about classical field theory. It yields an on-shell continuity equation$^3$
$$ d_{\mu}j^{\mu}~\stackrel{m}{\approx}~0.\tag{10}$$
Hence the electric charge
$$ Q~=~\int\! d^3x~ j^0\tag{11}$$
is conserved on-shell.
References:
- M. Srednicki, QFT.
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$^1$ Note that the matter Lagrangian density ${\cal L}_{\rm matter}$ may depend on the gauge field $A_{\mu}$
$^2$ Interestingly, the electric $4$-current $j^{\mu}$ depends on the gauge potential $A_{\mu}$ in case of scalar QED matter.
$^3$ Note that the above proof of the continuity equation (10) via Noether's first theorem (as OP requested) never uses Maxwell's equations.
