- #1

lbrits

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**[SOLVED] Baker, Campbell, Hausdorff and all that**

I'm posting this here because, although it is a mathematics problem, it is related to perturbation theory and is the kind of problem physicists might be more skilled at answering.

Does anyone know an elegant proof of

[tex]e^{A+B} = \int_0^1 d\alpha_1 \,\delta(1-\alpha_1) e^{\alpha_1 A} + \int_0^1 d\alpha_1 d\alpha_2 \,\delta(1-\alpha_1 - \alpha_2) e^{\alpha_1 A} B e^{\alpha_2 A} + \frac{1}{2!}\int_0^1 d\alpha_1 d\alpha_2 d\alpha_3 \,\delta(1-\alpha_1 - \alpha_2 - \alpha_3) e^{\alpha_1 A} B e^{\alpha_2 A} B e^{\alpha_3 A} + \dots[/tex]

where of course [tex]A[/tex] and [tex]B[/tex] are matrices? I can prove it starting from the easy to prove identity

[tex]\frac{d}{ds} e^{A + s B} = \left( \int_0^1\!dt\, e^{t(A + s B)} B e^{-t(A + s B)} \right) e^{A + s B}[/tex]

but the proof gets a bit messy. I was hoping maybe someone recognizes the formula or knows a good references.