Baker–Campbell–Hausdorff formula

In mathematics, the Baker–Campbell–Hausdorff formula gives the value of ${\displaystyle Z}$ that solves the equation

for possibly noncommutative X and Y in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for ${\displaystyle Z}$ in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in ${\displaystyle X}$ and ${\displaystyle Y}$ and iterated commutators thereof. The first few terms of this series are:

where "${\displaystyle \cdots }$" indicates terms involving higher commutators of ${\displaystyle X}$ and ${\displaystyle Y}$. If ${\displaystyle X}$ and ${\displaystyle Y}$ are sufficiently small elements of the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of a Lie group ${\displaystyle G}$, the series is convergent. Meanwhile, every element ${\displaystyle g}$ sufficiently close to the identity in ${\displaystyle G}$ can be expressed as ${\displaystyle g=e^{X}}$ for a small ${\displaystyle X}$ in ${\displaystyle {\mathfrak {g}}}$. Thus, we can say that near the identity the group multiplication in ${\displaystyle G}$—written as ${\displaystyle e^{X}e^{Y}=e^{Z}}$—can be expressed in purely Lie algebraic terms. The Baker–Campbell–Hausdorff formula can be used to give comparatively simple proofs of deep results in the Lie group–Lie algebra correspondence.

If ${\displaystyle X}$ and ${\displaystyle Y}$ are sufficiently small ${\displaystyle n\times n}$ matrices, then ${\displaystyle Z}$ can be computed as the logarithm of ${\displaystyle e^{X}e^{Y}}$, where the exponentials and the logarithm can be computed as power series. The point of the Baker–Campbell–Hausdorff formula is then the highly nonobvious claim that ${\displaystyle Z:=\log \left(e^{X}e^{Y}\right)}$ can be expressed as a series in repeated commutators of ${\displaystyle X}$ and ${\displaystyle Y}$.

Modern expositions of the formula can be found in, among other places, the books of Rossmann and Hall.