The Baker-Campbell-Hausdorff formula

The Baker-Campbell-Hausdorff formula

0.00 In the last video, we found a local inverse for the matrix exponential $\exp :𝔤𝔩(n,𝐑)\to GL(n,𝐑)$ . In other words, we found a neighbourhood $U$ of $0\in 𝔤𝔩(n,𝐑)$ and a neighbourhood $V$ of $I\in GL(n,𝐑)$ such that ${\exp |}_U:U\to V$ is a bijection whose inverse we called $\log :V\to U$ .

0.45 We also wrote down a power series expansion: $$\log (I+M)=M-\frac{1}{2}{M}^2+\frac{1}{3}{M}^3-\mathrm{\cdots }$$ valid for small $M$ (as $V$ is a neighbourhood of the identity, anything in $V$ can be written as $I+M$ for some small $M$ ).

1.10 If $XY=YX$ then we saw that $\exp (X)\exp (Y)=\exp (X+Y)$ . What happens if $XY\ne YX$ ? We can figure out a formula for $\log \exp (X)\exp (Y)$ using our power series for $\log$ .

2.25 Let's compute. $$\log (\exp (X)\exp (Y))=\log \left(\left(I+X+\frac{X^2}{2}+\mathrm{\cdots }\right)\left(I+Y+\frac{Y^2}{2}+\mathrm{\cdots }\right)\right).$$ Multiplying the power series together, we get $$\log \left(I+X+Y+XY+\frac{X^2}{2}+\frac{Y^2}{2}+\frac{X{Y}^2}{2}+\frac{X^{2}Y}{2}+\mathrm{\cdots }\right).$$ 3.25 Taking $M=X+Y+XY+\frac{X^2}{2}+\frac{Y^2}{2}+\mathrm{\cdots }$ and using our power series for $\log (I+M)$ we get $$M-\frac{1}{2}{M}^2+\frac{1}{3}{M}^3+\mathrm{\cdots }=X+Y+XY+\frac{1}{2}{X}^2+\frac{1}{2}{Y}^2+\mathrm{\cdots }-\frac{1}{2}(X^2+XY+YX+Y^2+\mathrm{\cdots })+\mathrm{\cdots }$$ where the dots stand for terms which are at least cubic in $X$ and $Y$ . For now we'll ignore all cubic and higher order terms.

5.20 This simplifies to give $$\log (\exp X\exp Y)=X+Y+\frac{1}{2}(XY-YX)+\mathrm{\cdots }$$

6.25 This quantity $XY-YX$ is going to turn out to be very important in this course. It's called the commutator bracket or Lie bracket of $X$ and $Y$ and is written $[X,Y]$ . This vanishes if $X$ and $Y$ commute.

7.00 We're now saying that $\log (\exp (X)\exp (Y))$ is:

• to first order just $X+Y$ ,

• has a second order correction term $[X,Y]$ , which vanishes if $X$ and $Y$ commute,

and remarkably, all the cubic and higher order correction terms can be expressed as iterated commutators of $X$ and $Y$ . For example, the cubic term is $$\frac{1}{12}[X,[X,Y]]-\frac{1}{12}[Y,[X,Y]].$$ This fact that the higher order terms can be written in this way is genuinely miraculous and not at all obvious: proving it is one of the possible projects you could do.

8.30 You can write a general formula (due to Dynkin) for the $n$ th order term, but this is not very illuminating. We will state the existence of this formula as a theorem:

10.25 The main point of the Baker-Campbell-Hausdorff formula is that the group multiplication of the invertible matrices $\exp (X)$ and $\exp (Y)$ in $GL(n,𝐑)$ is determined by the bracket operation (on $𝔤𝔩(n,𝐑)$ ).

Pre-class exercise