of the particle asymptotic momentum such that $$p_{\|}={\bm p}\cdot{\bm e}_z,\quad {\bm p}_\perp={\bm p}-p_{\|}{\bm e}_z.
\label{kg5}$$ To interpret the solutions of Eq. , we realize that in the remote past ($t\rightarrow -\infty$) this equation becomes $$\Bigl[\frac{{\mathrm{d}}^2}{{\mathrm{d}}t^2}+\omega_{\bm p}^2\Bigr]\Phi_{\bm p}(t)=0,
\label{kg6}$$ where $$\omega_{\bm p}=\sqrt{(mc^2)^2+c^2{\bm p}^2}.
\label{kg7}$$ Therefore, there exist two linearly independent solutions of Eq.  which we will label by the parameter $\beta$, $$\Phi_{\bm p}^{(\beta)}(t)\underset{t\rightarrow -\infty}{\sim}{\mathrm{e}}^{-{\mathrm{i}}\beta\omega_{\bm p}t}.
\label{kg8}$$ The one corresponding to a positive energy (with $\beta=+$) will be interpreted as a boson whereas the other one (with $\beta=-$) as an antiboson. In this way, we have determined two sets of solutions of the Klein-Gordon equation that appear in , $$\Phi^{(\beta)}_{\bm p}(x)={\mathrm{e}}^{{\mathrm{i}}{\bm p}\cdot {\bm x}}\Phi_{\bm p}^{(\beta)}(t),
\label{kg9}$$ where $\Phi_{\bm p}^{(\beta)}(t)$ solves Eq.  and asymptotically behaves according to . Note that we have chosen the same symbol for the space-time- and time-dependent solutions, discriminating them by the argument $x=(ct,{\bm x})$ and $t$.

It is crucial that $\Phi^{(\beta)}_{\bm p}(x)$ form an orthonormal and complete set of solutions of the Klein-Gordon equation. This is provided that the inner product of two such wave functions, $\Phi_{\bm p}^{(\beta)}(x)$ and $\Phi_{{\bm p}'}^{(\beta')}(x)$, is defined as [@Grib1; @Grib2; @Greiner1; @Greiner2] $$\begin{aligned}
\langle\Phi_{\bm p}^{(\beta)}|\Phi_{{\bm p}'}^{(\beta')}\rangle&={\mathrm{i}}\int{\mathrm{d}}^3{\bm x}\,[\Phi_{\bm p}^{(\beta)}(x)]^*\overleftrightarrow{\partial_0}\Phi_{{\bm p}'}^{(\beta')}(x)\nonumber\\
&={\mathrm{i}}\int{\mathrm{d}}^3{\bm x}\,\Bigl([\Phi_{\bm p}^{(\beta)}(x)]^*[\partial_0\Phi_{{\bm 