グリーン–久保公式
Tr  ( ρ ′ ( t ) B ) = i ħ Tr  ( ∫ − ∞ t exp  ( − i H 0 ( t − t ′ ) ħ ) [ A , ρ 0 ] exp  ( i H 0 ( t − t ′ ) ħ ) F ( t ′ ) d t ′ B ) = i ħ ∫ − ∞ t Tr  ( B exp  ( − i H 0 ( t − t ′ ) ħ ) [ A , ρ 0 ] exp  ( i H 0 ( t − t ′ ) ħ ) F ( t ′ ) ) d t ′ = i ħ ∫ − ∞ t Tr  ( exp  ( i H 0 ( t − t ′ ) ħ ) B exp  ( − i H 0 ( t − t ′ ) ħ ) [ A , ρ 0 ] F ( t ′ ) ) d t ′ {\displaystyle {\begin{aligned}\operatorname {Tr} (\rho '(t)B)&={i \over {\hbar }}\operatorname {Tr} \left(\int _{-\infty }^{t}\exp \left({-{iH_{0}(t-t') \over \hbar }}\right)[A,\rho _{0}]\exp \left({iH_{0}(t-t') \over \hbar }\right)F(t')dt'B\right)\\&={i \over {\hbar }}\int _{-\infty }^{t}\operatorname {Tr} \left(B\exp \left({-{iH_{0}(t-t') \over \hbar }}\right)[A,\rho _{0}]\exp \left({iH_{0}(t-t') \over \hbar }\right)F(t')\right)dt'\\&={i \over {\hbar }}\int _{-\infty }^{t}\operatorname {Tr} \left(\exp \left({iH_{0}(t-t') \over \hbar }\right)B\exp \left({-{iH_{0}(t-t') \over \hbar }}\right)[A,\rho _{0}]F(t')\right)dt'\end{aligned}}}