The _Baker-Campbell Hausdorff formula_ (BCH formula) is a formula for evaluating [matrix logarithms](Matrix%20logarithms.md) expressed as $Z = \log({e^Xe^Y}) = X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}[X,[X,Y]]-\frac{1}{12}[Y,[X,Y]]+...$ A related expression known as the [[Baker-Campbell-Hausdorff lemma]] may be used for the [proof of the BCH formula.](Baker-Campbell-Hausdorff%20formula.md#Proof%20of%20the%20Baker-Campbell%20Hausdorff%20formula) # Role of Lie Groups, Algebras, and Representations Consider [matrix Lie group](Matrix%20Lie%20group.md)s, $\mbox{G}$ and $\mbox{H}$, which form a [Lie group homomorphism](Lie%20group%20homomorphisms.md), $\Phi: \mbox{G} \rightarrow \mbox{H}$. The corresponding [[Lie algebras]] $\mathfrak{g}$ and $\mathfrak{h}$ also form a [Lie algebra homomorphism](Lie%20algebra%20homomorphisms.md), $\phi: \mathfrak{g} \rightarrow \mathfrak{h}$. The [Lie group homomorphism](Lie%20group%20homomorphisms.md) is expressed as $\Phi(e^X) = e^{\phi(X)}$ where $e^X$, $e^Y$ as well as $e^Xe^Y$ are in the domain of $\Phi$. Therefore we must also be able to write $\Phi(e^Xe^Y) = e^{\phi(Z)}$. For a given [Lie group homomorphism](Lie%20group%20homomorphisms.md) As it turns out Z is evaluated by _Baker-Campbell-Hausdorff formula_ (BCH formula), which is valid for _sufficiently small_ $X$ and $Y$. # Glauber formula An important special case of the [BCH formula](Baker-Campbell-Hausdorff%20formula.md) is where $[X,[X,Y]] = [Y,[X,Y]]=0,$ leading to the truncated sum, $Z=\log{(e^Xe^Y)} = X + Y + \frac{1}{2}[X,Y]$ ^a0f4e0 This leads to the [_Glauber formula_,](Glauber%20formula.md)  Thus, its [proof](Proof%20of%20the%20Glauber%20formula.md) provides a partial proof to the [BCH formula](Baker-Campbell-Hausdorff%20formula.md). ([... see more](Glauber%20formula.md)) --- # Proofs and Examples ## Proof of the Glauber Formula The [Glauber formula](Glauber%20formula.md) provides a partial proof for the Baker-Campbell-Hausdorff lemma where [$[X,[X,Y]] = [Y,[X,Y]]=0$,](Glauber%20formula#^90e320) thus it is instructive to review the [proof of the Glauber formula](Proof%20of%20the%20Glauber%20formula.md) ([... see more](Proof%20of%20the%20Glauber%20formula.md)) ## Proof of the Baker-Campbell Hausdorff formula #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators/Matrices