The _Glauber Formula_ is a formula for evaluating products of [matrix exponential](Matrix%20exponentials.md) that's stated as $e^Xe^Y = e^{X+Y+\frac{1}{2}[X,Y]}$ ^8576b3 where the following [commutation relation](Commutators.md) holds. $[X,[X,Y]] = [Y,[X,Y]]=0.$ If $[X,Y] = 0,$ the Glauber formula reduces to $e^{X}e^{Y} = e^{Y}e^{X} = e^{X+Y}.$ Thus we only need it when $X$ and $Y$ don't commute. We prove the Glauber formula [here.](Glauber%20formula.md#Proof%20of%20the%20Glauber%20formula) # Properties of the Glauber formula We note the following subtle property $e^{X+Y+\frac{1}{2}[X,Y]}=e^{Y+X-\frac{1}{2}[X,Y]}$ This comes about simply from switching the order of $X$ and $Y$ in the [commutator](Commutators.md). # Derivation of the Glauber formula from the BCH formula The _Glauber formula_ follows from a special case of the [Baker-Campbell-Hausdorff formula](Baker-Campbell-Hausdorff%20formula.md). Here the summation in the full BCH formula truncates after the first order term. Thus reducing to  with no additional terms if $[X,[X,Y]] = [Y,[X,Y]]=0.$ ^90e320 It then follows that $e^Xe^Y = e^{X+Y+\frac{1}{2}[X,Y]} =e^{X+Y}e^{\frac{1}{2}[X,Y]}.$ --- # Proofs and Examples ## Proof of the Glauber formula          #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators/Matrices