Baker-Campbell-Hausdorff formula(e)

Given a linear operator $A$, we define:

$$\exp A:=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}{A}^k.$$(1)

It follows that

$$\frac{\partial }{\partial \tau }{e}^{\tau A}=A{e}^{\tau A}=e^{\tau A}A.$$(2)

Consider another linear operator $B$. Let $B(\tau )=e^{\tau A}B{e}^{-\tau A}$. Then one can prove the following series representation for $B(\tau )$:

$$B(\tau )=\sum _{m=0}^{\mathrm{\infty }}\frac{{\tau }^m}{m!}{B}_m,$$(3)

where $B_m={[A,B]}_m:=[A,{[A,B]}_{m-1}]$ and $B_0:=B$.A very important special case of eq. (3) is known as theBaker-Campbell-Hausdorff (BCH) formula. Namely, for $\tau =1$ we get:

$$e^{A}B{e}^{-A}=\sum _{m=0}^{\mathrm{\infty }}\frac{1}{m!}{B}_m.$$(4)

Alternatively, this expression may be rewritten as

$$[B,e^{-A}]=e^{-A}\left([A,B]+\frac{1}{2}[A,[A,B]]+\mathrm{\cdots }\right),$$(5)

or

$$[e^A,B]=\left([A,B]+\frac{1}{2}[A,[A,B]]+\mathrm{\cdots }\right)e^A.$$(6)

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

$$e^A{e}^B=e^{A+B}{e}^{\frac{1}{2}[A,B]}.$$(7)

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

$$e^A{e}^B=e^B{e}^A{e}^{[A,B]}.$$(8)