distribution $$\begin{aligned}
n_\text{P}({\vec p})=N_\text{BE}({\vec p})={1 \over e^\frac{p^0}{k_\text{B} T}-1}\end{aligned}$$ for photon and by Fermi-Dirac distribution $$\begin{aligned}
n_e({\vec p})=N_\text{FD}({\vec p})={1 \over e^{{p^0-\mu} \over k_\text{B} T}+1}\end{aligned}$$ for electron, where $\mu$ is the chemical potential, $k_\text{B}$ is Boltzmann constant, and $T$ is the temperature that is determined from the average energy. A many-body wave function that satisfies these conditions mentioned above is the coherent state of the Boson field and the state of the Fermi field constructed in the same manner. An explicit form is given in Appendix C.

Thus, we assume that the amplitude for one photon and one electron is given by $$\begin{aligned}
T(1)= N_\text{BE}(p_2)^{1/2}N_\text{FD}(k_2)^{1/2}\tilde T(1),\end{aligned}$$ where $\tilde T(1)$ does not depend on ${\vec p_2}$ and ${\vec k_2}$. The whole amplitude is written as $$\begin{aligned}
T({\vec k}_1,{\vec p}_1)&=&  \int d{\vec k}_2 d{\vec p}_2 T({\vec p}_1,{\vec k}_1;{\vec p}_2,{\vec k}_2)N_\text{F}(p_2)^{1/2}N_\text{B}(k_2)^{1/2}
\delta({\vec p}_2+{\vec k}_2-{\vec p}_1-{\vec k}_1)\nonumber \\
& &\times \tilde T(1),\end{aligned}$$ where $T({\vec p}_1,{\vec k}_1;{\vec p}_2,{\vec k}_2)$ is the amplitude of Thomson scattering and is constant in low energy.

The momentum correlation function $C({\vec p}_1,{\vec p}_2)$ is a product of the amplitude of the photon momentum ${\vec p}_1$ and its complex conjugate of the photon momentum ${\vec p}_2$ and is given by $$\begin{aligned}
& &C({\vec 