= Fredholm determinant of an exponential commutator
{title2=$\det(e^Ae^Be^{-A}e^{-B})=e^{\operatorname{Tr}[A,B]}$}
For bounded operators with trace-class commutator, the multiplicative commutator differs from identity by a <trace-class operator>. Along $F(t)=e^{tA}e^Be^{-tA}e^{-B}$, the trace of its logarithmic derivative is the constant $\operatorname{Tr}[A,B]$. This follows by writing $e^BAe^{-B}-A$ as the integral of conjugates of $[B,A]$ and using trace invariance under bounded similarities. Integrating the <Fredholm determinant> derivative from $F(0)=I$ proves the formula. Individual traces of $A$ and $B$ need not exist.
Back to article page
