New formula for ln(e^Ae^B) in terms of commutators of A and B

Abstract

We establish the formula

$$\ln (e^B e^A ) = \smallint _0^t \psi (e^{ - \tau ad_A } e^{ - \tau ad_B } ) e^{ - \tau ad_A } d\tau (A + B),$$

where Ψ(x)=(In x)/(x − 1); here A and B are elements of a. finite-dimensional Lie algebra which satisfy certain conditions. This formula enables us, in particular, to give a simple proof of the Campbell-Hausdorff theorem. We also give a generalization of the formula to the case of an arbitrary number of factors.