# Eikonal Blog

## 2010.02.15

### Magnus’ Bracket Operator

Filed under: mathematics — Tags: — sandokan65 @ 22:46

Definition: Define $z(x,y)$ via product of exponentials $e^z :\equiv e^x e^y$.

Properties:

• $z = x+y+\frac12 [x,y] + \frac1{12} (\{xy^2\} + \{yx^2\}) + \frac1{24}\{xy^2x\} + \cdots$
• $e^{-y} e^x e^y = e^{x+\frac1{1!}[x,y] + \frac1{24}\{xy^2\}+\cdots} = e^{\{x e^y\}_M}$

Definition: Here “the bracket operator” $\{\cdots\}_M$ is defined by following properties:

1. $\{G\}_M = G$ if $G$ is homogeneous of the first degree,
2. $\{r G\}_M = r \{G\}_M$ if $r$ is a rational number,
3. $\{G_1 + G_2\}_M = \{G_1\}_M + \{G_2\}_M$,
4. $\{G x\}_M = [\{G\}_M , x]$, $\{G y\}_M = [\{G\}_M , y]$

So:

• $\{x^2 G\}_M = \{y^2 G\}_M = 0$,
• $\{x y^n\}_M = x y^n - \binom{n}{1} y x y^{n-1} + \cdots + (-)^k \binom{n}{k}^k y^k x y^{n-k} + \cdots + (-)^n y^n x$.
• $\{\omega_n\}_M = n \omega_n$ for $\omega_n$ homogeneous in $x$ and $y$ of degree $n$, i.e. $\{\omega\}_M = \left(x D_x + y D_y\right) \omega$.

Source: T1272 = W. Magnus “A Connection between the Baker-Hausdorff formula and a Problem of Burnside”; Annals of Mathematics; Vol 52, No 1, July 1950.

Related here: Magnus on matrix exponentials – https://eikonal.wordpress.com/2010/01/26/magnus-on-matrix-exponentials/