Lets have the scalar Klein-Gordon field interacting with EM field:
$$ L = \partial_{\mu}\varphi \partial^{\mu}\varphi - m^2\varphi \varphi^{*} - j_{\mu}A^{\mu} + q^{2} A_{\mu}A^{\mu}\varphi \varphi^{*} - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}. \qquad (1) $$ I heard that the normalization of Klein-Gordon field in a theory $(1)$ is invariant under gauge transformations. What normalization is meaned? Does it refer to the factor $\frac{1}{\sqrt{2(2 \pi)^{3} E_{\mathbf p}}}$? How to prove it?
An edit.
It was the invariance of condition $\int j^{0}d^{3}\mathbf r = q$ under $U(1)$ local gauge transformations. $j^{0} = \frac{q}{2m}(\psi^{*}\partial^{0}\psi - \psi \partial^{0}\psi^{*}) - \frac{q^2}{m}A^{0}|\Psi |^{2}$.