foundation of scalar QED.

Scalar QED {#BB}
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We define the Klein-Gordon boson field operator $\hat{\Phi}(x)$ as $$\hat{\Phi}(x)=\int\frac{{\mathrm{d}}^3{\bm p}}{(2\pi)^3}\Bigl(\Phi_{\bm p}^{(+)}(x)\hat{b}_{\bm p}+\Phi_{\bm p}^{(-)}(x)\hat{d}_{-{\bm p}}^\dagger\Bigr),
\label{bf1}$$ where $\Phi_{\bm p}^{(\pm)}(x)$ are the one-particle solutions of the Klein-Gordon equation, whereas ${\hat{b}}_{\bm p}$ (${\hat{d}}_{\bm p}$) is the annihilation operator of a boson (antiboson) with a given momentum ${\bm p}$. These operators define the in-vacuum state through the conditions: ${\hat{b}}_{\bm p}{\vert{0_{-\infty}}\rangle}=0$ and ${\hat{d}}_{\bm p}{\vert{0_{-\infty}}\rangle}=0$. They also satisfy the commutation relations, $$[{\hat{b}}_{\bm p},{\hat{b}}_{{\bm p}'}^\dagger]=[{\hat{d}}_{\bm p},{\hat{d}}_{{\bm p}'}^\dagger]=\delta^{(3)}({\bm p}-{\bm p}'),
\label{bf2}$$ with the remaining commutators equal to zero. $\Phi_{\bm p}^{\pm}(x)$, on the other hand, will be constructed in the next section.

### One-particle solutions of the Klein-Gordon equation {#one-particle-bosons}

Our aim is to solve the scalar Klein-Gordon equation coupled to the external electromagnetic field, $$\Bigl[\bigl({\mathrm{i}}\partial-eA\bigr)^2-(mc)^2\Bigr]\Phi(x)=0.
\label{kg1}$$ Since the problem has translational symmetry, one can look for those solutions $\Phi(x)$ in the separable form, $$\Phi(x)={\mathrm{e}}^{{\mathrm{i}}{\bm p}\cdot {\bm x}}\Phi_{\bm p}(t),
\label{kg2}$$ where ${\bm p}$ is the particle asymptotic momentum. Substituting  into , we obtain that the function $\Phi_{\bm p}(t)$ satisfies the harmonic oscillator equation, $$\Bigl[\frac{{\mathrm{d}}^2}{{\mathrm{d}}t^2}+\omega_{\bm p}^2(t)\Bigr]\Phi_{\bm p}(t)=0,
\label{kg3}$$ with the time-dependent frequency $\omega_{\bm p}(t)$, $$\omega_{\bm p}(t)=\sqrt{(mc^2)^2+c^2{\bm p}_\perp^2+c^2(p_{\|}-eA(t))^2}.
\label{kg4}$$ Here, we have introduced the longitudinal $p_{\|}$ and the transverse ${\bm p}_\perp$ components 