A.

For the coherence of photon and electron, surrounding protons are taken into account, and the Feynman diagram of these processes has many lines of particles, as in Fig. \[fig:proton-electron\].

![Feynman diagram in which an electron interacts with surrounding protons.](fig1.eps)

\[fig:proton-electron\]

The average number of Rutherford scatterings in the distance $l$, $N$, is given by $$\begin{aligned}
N={l \over l_\text{Th}}.\end{aligned}$$ Thus, the electron’s spread obtains contributions from the proton’s momentum spread $N$ times in the distance $l$. A photon obtains contributions from the electron’s momentum spread. As shown in Appendix B, the momentum spreads of the electron and photon, after the $N$ step of interactions, are given by $$\begin{aligned}
(\Delta p_\text{total})^2=\sum_i^N (\Delta p_i)^2,\end{aligned}$$ and its magnitude becomes large for large $N$. Particularly if $(\Delta p_i)^2=(\Delta p)^2$ is finite, $(\Delta p_\text{total})^2$ is given by $$\begin{aligned}
(\Delta p_\text{total})^2=N (\Delta p)^2\end{aligned}$$ and reaches the absolute value of the momentum, $|p|$, at a sufficiently large $N$. This is realized at a macroscopic distance $l$. Thus, the total spread becomes $$\begin{aligned}
\Delta p_\text{total} \rightarrow \infty,\end{aligned}$$ and the wave has a large momentum spread and the momentum conservation becomes effectively negligible.

Next, we focus the final Thomson scattering of one photon and one electron shown in Fig. 