Baker-Campbell-Hausdorff formula(e)

Given a linear operatorMathworldPlanetmath A, we define:

expA:=k=01k!Ak. (1)

It follows that

τeτA=AeτA=eτAA. (2)

Consider another linear operator B. Let B(τ)=eτABe-τA. Then one can prove the following series representation for B(τ):

B(τ)=m=0τmm!Bm, (3)

where Bm=[A,B]m:=[A,[A,B]m-1] and B0:=B. A very important special case of eq. (3) is known as the Baker-Campbell-Hausdorff (BCH) formula. Namely, for τ=1 we get:

eABe-A=m=01m!Bm. (4)

Alternatively, this expression may be rewritten as

[B,e-A]=e-A([A,B]+12[A,[A,B]]+), (5)


[eA,B]=([A,B]+12[A,[A,B]]+)eA. (6)

There is a descendent of the BCH formula, which often is also referred to as BCH formula. It provides us with the multiplication law for two exponentialsPlanetmathPlanetmath of linear operators: Suppose [A,[A,B]]=[B,[B,A]]=0. Then,

eAeB=eA+Be12[A,B]. (7)

Thus, if we want to commute two exponentials, we get an extra factor

eAeB=eBeAe[A,B]. (8)
Title Baker-Campbell-Hausdorff formula(e)
Canonical name BakerCampbellHausdorffFormulae
Date of creation 2013-03-22 13:39:51
Last modified on 2013-03-22 13:39:51
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 13
Author Mathprof (13753)
Entry type Definition
Classification msc 47A05
Synonym BCH formula
Synonym Baker-Campbell-Hausdorff formula
Synonym Baker-Campbell-Hausdorff formulae