How can I prove that the Noether charge represents actually conservation of electric charge?

Question

I have a question about Noether's theorem for global gauge invariance of a complex scalar field. Starting from \begin{equation} \mathscr{L} = \partial_{\mu}\Phi \partial^{\mu}\Phi^{*} + \frac{m^2c^2}{2\hbar^2}\Phi\Phi^{*}, \end{equation} since the field is invariant globally I have a conserved quantity which express conservation of charge. The conserved current for the above field is \begin{equation} J^{\mu} = i\lambda(\Phi \partial^{\mu}\Phi^{*} - \Phi^{*}\partial^{\mu}\Phi), \end{equation} which means my conserved quantity at a certain time is \begin{equation} Q = \int{J^0}d^3x = i\lambda\int{\bigg(\Phi \frac{\partial}{\partial t}\Phi^{*} - \Phi^{*}\frac{\partial}{\partial t}\Phi}\bigg)d^3x = constant. \end{equation}

I considered the surface part of the integral removed by surface extension to infinity, in which I consider it zero. My question is: how can I prove that the integrand above represents actually conservation of electric charge?

I supposed I could explain it by not removing the spacial integral and recall Gauss theorem or redefining the $\lambda$ parameter, but this could be done also for other conserved quantities which do not represent conservation of electric charge. So what can be a way I can prove it?

Answer 1

1. To identify the Noether 4-current with the electric 4-current, one would in principle have to show that the Noether 4-current indeed appears as the source term in Maxwell's equations.

2. The Maxwell equations with sources (Gauss's + Ampere's laws) are derived by adding the Maxwell Lagrangian $-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ to a minimally coupled, gauge-invariant matter Lagrangian, and vary wrt. the 4-gauge potential $A_{\mu}$.

3. See also this related Phys.SE post.