Central-commutator exponential identity
= Central-commutator exponential identity
{title2=$e^{-B}e^{-A}e^Be^A=e^{-[A,B]}$}
For finite-dimensional complex <linear operators> with $C=[A,B]$ commuting with both $A$ and $B$, differentiating $e^{-vB}Ae^{vB}$ gives the constant derivative $C$. Hence $e^{-B}Ae^B=A+C$, and conjugating the <matrix exponential> gives $e^{-B}e^{-A}e^B=e^{-A-C}$. Since $[A,C]=0$, multiplication by $e^A$ proves the identity. The commutator order determines the minus sign.