Noether theorem and classical proof of electric charge conservation

Question

How to prove conservation of electric charge using Noether's theorem according to classical (non-quantum) mechanics? I know the proof based on using Klein–Gordon field, but that derivation use quantum mechanics particularly.

Answer 1

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

$$\tag{1} {\cal L} ~=~{\cal L}_{\rm Maxwell} + {\cal L}_{\rm matter} , $$

$$\tag{2} {\cal L}_{\rm Maxwell}~=~ -\frac{1}{4}F_{\mu\nu}F^{\mu\nu},$$

$$\tag{3} {\cal L}_{\rm matter}~=~{\cal L}_{\rm matter}^{\rm QED}+{\cal L}_{\rm matter}^{\rm scalar QED} + \ldots, $$

$$\tag{4} {\cal L}_{\rm matter}^{\rm QED} ~:=~ \overline{\Psi}( i\gamma^{\mu} D_{\mu}-m)\Psi , $$

$$\tag{5} {\cal L}_{\rm matter}^{\rm scalar QED}~:=~ -(D_{\mu}\phi)^{\dagger} D^{\mu}\phi -m^2\phi^{\dagger}\phi -\frac{\lambda}{4} (\phi^{\dagger}\phi)^2, $$

with covariant derivative

$$ \tag{6} D_{\mu}~=~d_{\mu}-ieA_{\mu}. $$

(Here we are too lazy to denote various matter masses $m$ and charges $e$ differently.) The matter equations of motion (eom) are

$$ \tag{7}( i\gamma^{\mu} D_{\mu}-m)\Psi ~\approx~0, \qquad D_{\mu}D^{\mu}\phi~\approx~m^2\phi+\frac{\lambda}{2} \phi^{\dagger}\phi^2, \qquad \ldots.$$

(The $\approx$ symbol means equality modulo eom, i.e. an on-shell equality.)

The infinitesimal global off-shell gauge transformation is

$$ \delta A_{\mu} ~=~0, \qquad \delta\Psi~=~-i\epsilon \Psi, \qquad \delta\overline{\Psi}~=~i\epsilon \overline{\Psi}, $$ $$ \tag{8} \delta\phi~=~-i\epsilon \phi,\qquad \delta\phi^{\dagger}~=~i\epsilon \phi^{\dagger}, \qquad \ldots, \qquad\delta {\cal L} ~=~0, $$

where the infinitesimal parameter $\epsilon$ does not depend on $x$.

The Noether current is the electric $4$-current$^1$

$$ \tag{9} j^{\mu}~=~e\overline{\Psi}\gamma^{\mu}\Psi - ie\{\phi^{\dagger} D^{\mu}\phi-(D^{\mu}\phi)^{\dagger}\phi\}+\ldots. $$

Noether's Theorem is a theorem about classical field theory. It yields an on-shell conservation law

$$ \tag{10} d_{\mu}j^{\mu}~\approx~0.$$

Hence the electric charge

$$\tag{11} Q~=~\int\! d^3x~ j^0$$

is conserved on-shell.

References:

1. M. Srednicki, QFT.

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$^1$ Interestingly, the electric $4$-current $j^{\mu}$ depends on the gauge potential $A_{\mu}$ in case of scalar QED matter.