k}_1,{\vec k'}_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\\
& &\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)|\tilde T|^2.\end{aligned}$$ This correlation $C({\vec k}_1,{\vec k'}_1)$ shows the wave packet nature of photons.

We compute the above function $C({\vec k}_1,{\vec k'}_1)$ numerically. The result is given in Fig. \[3500K\].

\[3500K\]

As seen from Fig. \[3500K\], the correlation shows that of a Gaussian wave packet of the width of $3.5k_\text{B}T$. The photon is regarded as a wave packet whose energy distribution is a Planck distribution but the momentum width is $3.5k_\text{B}T$. This wave packet size should be understood as a maximal possible value. The effect of the wave packet is show in Ref. .

Measurements of particle’s trajectory
-------------------------------------

When a particle is measured using an apparatus, its position and momentum are measured within certain uncertainties. Many physical processes are involved in measurements, but regardless of these processes when the position and momentum are determined within uncertainties, the final state after the measurement is expressed as the state with these uncertainties. Thus, a wave packet of suitable values of momentum width and coordinate width is used for this state. The 