Baker–Campbell–Hausdorff formula

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Baker–Campbell–Hausdorff formula gives the value of 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 that solves the equation

e
          
            X
          
        
        
          e
          
            Y
          
        
        =
        
          e
          
            Z
          
        
      
    
    {\displaystyle e^{X}e^{Y}=e^{Z}}

for possibly <a href="/facts/Noncommutative/WaYxJLxd">noncommutative</a> X and Y in the <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> of a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a>. There are various ways of writing the formula, but all ultimately yield an expression for 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 and iterated commutators thereof. The first few terms of this series are:

Z
        =
        X
        +
        Y
        +
        
          
            1
            2
          
        
        [
        X
        ,
        Y
        ]
        +
        
          
            1
            12
          
        
        [
        X
        ,
        [
        X
        ,
        Y
        ]
        ]
        +
        
          
            1
            12
          
        
        [
        Y
        ,
        [
        Y
        ,
        X
        ]
        ]
        +
        ⋯
        
        ,
      
    
    {\displaystyle Z=X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}[X,[X,Y]]+{\frac {1}{12}}[Y,[Y,X]]+\cdots \,,}

where "
 
 
 
 ⋯
 
 
 {\displaystyle \cdots }
 
" indicates terms involving higher <a href="/facts/Commutator/eYxzFmFY">commutators</a> of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
. If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are sufficiently small elements of the Lie algebra 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
 of a Lie group 
 
 
 
 G
 
 
 {\displaystyle G}
 
, the series is convergent. Meanwhile, every element 
 
 
 
 g
 
 
 {\displaystyle g}
 
 sufficiently close to the identity in 
 
 
 
 G
 
 
 {\displaystyle G}
 
 can be expressed as 
 
 
 
 g
 =
 
 e
 
 X
 
 
 
 
 {\displaystyle g=e^{X}}
 
 for a small 
 
 
 
 X
 
 
 {\displaystyle X}
 
 in 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
. Thus, we can say that near the identity the group multiplication in 
 
 
 
 G
 
 
 {\displaystyle G}
 
—written as 
 
 
 
 
 e
 
 X
 
 
 
 e
 
 Y
 
 
 =
 
 e
 
 Z
 
 
 
 
 {\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 <a href="/facts/Lie_group%E2%80%93Lie_algebra_correspondence/zAnzPAV1">Lie group–Lie algebra correspondence</a>.
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are sufficiently small 
 
 
 
 n
 ×
 n
 
 
 {\displaystyle n\times n}
 
 matrices, then 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 can be computed as the logarithm of 
 
 
 
 
 e
 
 X
 
 
 
 e
 
 Y
 
 
 
 
 {\displaystyle e^{X}e^{Y}}
 
, where the exponentials and the logarithm can be computed as <a href="/facts/Power_series/vYh6sTps">power series</a>. The point of the Baker–Campbell–Hausdorff formula is then the highly nonobvious claim that 
 
 
 
 Z
 :=
 log
 ⁡
 
 (
 
 
 e
 
 X
 
 
 
 e
 
 Y
 
 
 
 )
 
 
 
 {\displaystyle Z:=\log \left(e^{X}e^{Y}\right)}
 
 can be expressed as a series in repeated commutators of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
.
Modern expositions of the formula can be found in, among other places, the books of Rossmann and Hall.

Baker–Campbell–Hausdorff formula open-in-new

Baker–Campbell–Hausdorff formula