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/

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