
Appendix E
The Baker–Campbell–Hausdorff Formula
A problem which often occurs in applications and in the study of Lie groups and Lie
algebras (see Appendix D) is that of expressing the product of two exponential operators
in an equivalent form. Clearly, if the operators X and Y commute, that is, [X, Y ]=0,then
e
X
e
Y
= e
X+Y
,wheree
X
=exp(X) is defined by the usual power series. The generalization
of this formula to the case when X and Y do not commute is the content of the Baker–
Campbell–Hausdorff formula. To formulate the theorem, we introduce a concise notation
for nested commutators (or brackets). Thus, we abbreviate
[X, [X,...,[X
! "
r
1
, [Y,[Y,...,[Y
! ...