Eikonal Blog

2010.01.22

Guenin on commutators

Filed under: Uncategorized — Tags: , — sandokan65 @ 15:46

A B^n = \sum_{j=0}^n \binom{n}{j} (-)^j B^{n-j} \ ad(B)^j A.

B^n A = \sum_{j=0}^n \binom{n}{j} (-)^j  \ (ad(B)^j A) B^{n-j}.

\partial_s f(F(s)) = \sum_{j=0}^\infty \frac{(-)^j}{(j+1)!} f^{(j+1)}(F(s)) \ ad(F(s)^j \partial_s F(s) = \sum_{j=0}^\infty \frac{1}{(j+1)!}  \ (ad(F(s))^j \partial_s F(s) ) f^{(j+1)}(F(s)).

[H, f(A)] = \sum_{j=0}^\infty \frac{(-)^j}{j!} f^{(j)}(A) \ ad(A)^j H = -  \sum_{j=0}^\infty \frac1{j!}\ (ad(A)^j H) f^{(j)}(A).
—-
Sources:

  • T1266: M. Guenin: “On the Derivation and Commutation of Operator Functionals”; Helv. Phys. Act. 41, 75-76, 1968.
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