**Appendix A**

Let *<sup>ψ</sup>*<sup>→</sup>*r* , *t*be a real function satisfying the homogeneous scalar wave equation in free space. The energy transport equation for such a wavefunction is given by

$$\nabla \cdot \stackrel{\rightarrow}{S} + \frac{\partial}{\partial t} \mathcal{W} = 0,\tag{A1}$$

where

$$\mathcal{W} = \frac{1}{2} \left[ \frac{1}{c^2} \left( \frac{\partial}{\partial t} \psi \right)^2 + \nabla \psi \cdot \nabla \psi \right], \stackrel{\rightarrow}{\mathcal{S}} = -\frac{\partial}{\partial t} \psi \,\nabla \psi \tag{A2}$$

are, respectively, the real energy density in units J/m<sup>3</sup> and the real energy flow vector (W/m2).

Let the real field *<sup>ψ</sup>*<sup>→</sup>*r* , *t*be expressed in terms of its spatial Fourier spectrum as follows:

$$
\psi\left(\stackrel{\rightarrow}{r},t\right) = \frac{1}{\left(2\pi\right)^{\overline{3}}} \int d\stackrel{\rightarrow}{k} \Psi\left(\stackrel{\rightarrow}{k},t\right) e^{i\stackrel{\rightarrow}{k}\cdot\stackrel{\rightarrow}{r}}.\tag{A3}
$$

Then, the total energy density can be written as follows:

$$\mathcal{W}\_{\text{total}} = \int d\vec{r} \,\, \mathcal{W}\left(\stackrel{\rightarrow}{r}, t\right) = \frac{1}{2} \frac{1}{\left(2\pi\right)^{3}} \int d\vec{k} \left[\frac{1}{c^{2}} \dot{\Psi}\left(\stackrel{\rightarrow}{k}, t\right) \dot{\Psi}^{\*}\left(\stackrel{\rightarrow}{k}, t\right) + k^{2} \Psi\left(\stackrel{\rightarrow}{k}, t\right) \Psi^{\*}\left(\stackrel{\rightarrow}{k}, t\right)\right],\tag{A4}$$

where the dot indicates differentiation with respect of time. Bearing in mind the dispersion relationship −*k*<sup>2</sup> + *ω*2/*c*<sup>2</sup> = 0, we assume, next, the form

$$
\Psi\left(\stackrel{\rightarrow}{k},t\right) = \Psi\_-\left(\stackrel{\rightarrow}{k}\right)e^{ikct} + \Psi\_+\left(\stackrel{\rightarrow}{k}\right)e^{-ikct}.\tag{A5}
$$

Then, we have

$$\mathcal{W}\_{\text{total}} = \frac{1}{2} \frac{1}{\left(2\pi\right)^{3}} \int d\vec{k} \, k^{2} \left[ \left| \Psi\_{-} \left( \stackrel{\rightarrow}{k} \right) \right|^{2} + \left| \Psi\_{+} \left( \stackrel{\rightarrow}{k} \right) \right|^{2} \right]. \tag{A6}$$

Next, let the wavefunction *<sup>ψ</sup>*<sup>→</sup>*r* , *t* be defined as *<sup>ψ</sup>*<sup>→</sup>*r* , *t* = *∂φ*<sup>→</sup>*r* , *t*/*∂t* and consider the integral

$$\begin{split} S\_{\Psi} \left( \stackrel{\rightarrow}{r} \right) &= \int\_{-\infty}^{\infty} \Psi \left( \stackrel{\rightarrow}{r}, t \right) dt = \operatorname{Lim}|\_{T \to \infty} \int\_{-T}^{T} \Psi \left( \stackrel{\rightarrow}{r}, t \right) dt \\ &= \operatorname{Lim}|\_{T \to \infty} \left[ \Phi \left( \stackrel{\rightarrow}{r}, T \right) - \Phi \left( \stackrel{\rightarrow}{r}, -T \right) \right] \end{split} \tag{A7}$$

The wavefunction *<sup>φ</sup>*<sup>→</sup>*r* , *T*is expressed as

$$\Phi\left(\stackrel{\rightarrow}{r},T\right) = \int d\stackrel{\rightarrow}{k} \epsilon^{\stackrel{\rightarrow}{i}\stackrel{\rightarrow}{k}\stackrel{\rightarrow}{r}} \left[\Phi\_{-}\left(\stackrel{\rightarrow}{k}\right)\epsilon^{ik\stackrel{\rightarrow}{i}T} + \Phi\_{+}\left(\stackrel{\rightarrow}{k}\right)\epsilon^{-ik\stackrel{\rightarrow}{i}T}\right].\tag{A8}$$

According to the Riemann–Lebesgue theorem, *<sup>S</sup>ψ*<sup>→</sup>*<sup>r</sup>* vanishes provided that

$$\int\_0^\infty dk \, k^2 \left| \int d\widehat{\Omega} \, \Phi\_\pm \left( \stackrel{\rightarrow}{k} \right) e^{i \frac{\gamma}{k} \cdot \frac{\rightarrow}{r}} \right| \tag{A9}$$

converges. However, <sup>Φ</sup>±<sup>→</sup>*<sup>k</sup>* = <sup>∓</sup>Φ±<sup>→</sup>*<sup>k</sup>* /*ck*. Therefore, *<sup>S</sup>ψ*<sup>→</sup>*<sup>r</sup>* vanishes and the field is usual provided that the integral

$$\int\_0^\infty dk \, k \left| \int d\widehat{\Omega} \, \Psi\_\pm \left( \stackrel{\rightarrow}{k} \right) e^{i \frac{\rightarrow}{k} \cdot \frac{\rightarrow}{r}} \right| \tag{A10}$$

converges.
