As the title says, I can not understand whether the assumptions are reasonable. If an interaction $1+2\leftrightarrow 3+4$ is taken into account, the number variance in time is proportional to
$$|\mathcal{M}_{12\rightarrow 34}|^2f_3f_4(1\pm f_1)(1\pm f_2)-|\mathcal{M}_{34\rightarrow 12}|^2f_1f_2(1\pm f_3)(1\pm f_4)$$
In many textbooks, the following assumptions are taken as
(1) $|\mathcal{M}_{12\rightarrow 34}|^2$=$|\mathcal{M}_{34\rightarrow 12}|^2$
and
(2) $E-\mu>>T$,$\;\;$ satisfying$\;\;$ $\dfrac{1}{e^{(E-\mu)/T}\mp 1}\approx e^{-(E-\mu)/T}$.
The first assumption, as I understand it, is because the probabilities in both directions are equal at high energy (due to the dominance of the tree-level process resulting in the hermitian $T$ matrix involved in $S=1+iT$), or the T invariance, i.e. CP invariance is violated slightly only when we consider the Baryogenesis (Kolb and Turner). Is the first argument correct?
However, it is too difficult to accept the second assumption. Definitely, the number variance is integrated over the energy $E$ of each particle ranging from $m$ to $\infty$ due to the phase space factor. In addition, the assumption enables us to neglect the statistical effects
$1\mp f\approx 1$.
It is really weird for me... I recognized why the stimulated emission and blocking factor appear in the Boltzmann equation by tracking the method to obtain the Einstein coefficient. Even in that tracking, I had to assume that only one of $f$ follows the exact distribution form.
My guess is that the transition amplitude at the low collision energy is much smaller than that at the high one, resulting that the distribution function with low energy can be neglected. However, I can not specify the energy behavior of an arbitrary amplitude.
Can anyone give me help...??
