Baker-Campbell-Hausdorff formula

In mathematics , the Baker-Campbell-Hausdorff formula is an equation named after the mathematicians Henry Frederick Baker , John Edward Campbell and Felix Hausdorff , which specifies an exchange law for certain linear operators .

Preparatory Definitions

If X is a continuous linear operator of a Banach space in itself, then the exponential of this operator can be defined as a series as follows :

${\displaystyle e^{X}=\sum _{k=0}^{\infty }{\frac {1}{k!}}X^{k}}$

The multiplication means that the operators are executed one after the other and the addition means that the operators involved are added point by point. The commutator (also known as Lie bracket ) of two linear operators X and Y is defined as

${\displaystyle [X,Y]:=XY-YX\,}$

It is a bilinear operator. From the definition, the so-called Hadamard lemma, also called Lie's development formula, follows:

${\displaystyle e^{X}Ye^{-X}=\sum _{m=0}^{\infty }{\frac {1}{m!}}[X,Y]_{m}}$

with and . ${\displaystyle [X,Y]_{m}=[X,[X,Y]_{m-1}]\,}$ ${\displaystyle [X,Y]_{0}=Y\,}$

The formula

If and , the simple Baker-Campbell-Hausdorff formulas apply ${\displaystyle [X,[X,Y]]=0\,}$ ${\displaystyle [Y,[Y,X]]=0\,}$

${\displaystyle e^{X}e^{Y}=e^{Y}e^{X}e^{[X,Y]}\,}$

${\displaystyle e^{X+Y}=e^{X}e^{Y}e^{-[X,Y]/2}\,}$.

For any and the formula is very extensive and only for in an environment of converging. It then reads ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X,Y}$ ${\displaystyle 0}$

${\displaystyle e^{X}e^{Y}=e^{Z(X,Y)}\,}$

With

${\displaystyle {\begin{aligned}Z(X,Y)&{}=X+Y\\&{}+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}[X,[X,Y]]-{\frac {1}{12}}[Y,[X,Y]]\\&{}\quad -{\frac {1}{24}}[Y,[X,[X,Y]]]\\&{}\quad -{\frac {1}{720}}([[[[X,Y],Y],Y],Y]+[[[[Y,X],X],X],X])\\&{}\quad +{\frac {1}{360}}([[[[X,Y],Y],Y],X]+[[[[Y,X],X],X],Y])\\&{}\quad +{\frac {1}{120}}([[[[Y,X],Y],X],Y]+[[[[X,Y],X],Y],X])+\cdots \end{aligned}}}$

credentials

• H. Baker: Proc Lond Math Soc (1) 34 (1902) 347-360; ibid (1) 35 (1903) 333-374; ibid (Ser 2) 3 (1905) 24-47.
• J. Campbell: Proc Lond Math Soc 28 (1897) 381-390; ibid 29 (1898) 14-32.
• L. Corwin & FP Greenleaf: Representation of nilpotent Lie groups and their applications, Part 1: Basic theory and examples , Cambridge University Press , New York, 1990, ISBN 0-521-36034-X .
• EB Dynkin: Calculation of the coefficients in the Campbell-Hausdorff formula , Doklady Akad Nauk USSR, 57 (1947) 323-326.
• Brian C. Hall: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction , Springer, 2003. ISBN 0-387-40122-9 .
• F. Hausdorff: Berl Verh Saechs Akad Wiss Leipzig 58 (1906) 19-48.
• W. Magnus : Comm Pur Appl Math VII (1954) 649-673.
• W. Miller: Symmetry Groups and their Applications , Academic Press , New York, 1972, pp. 159-161. ISBN 0-124-97460-0 .
• H. Poincaré: Compt Rend Acad Sci Paris 128 (1899) 1065-1069; Camb Philos Trans 18 (1899) 220-255.
• MW Reinsch: A simple expression for the terms in the Baker-Campbell-Hausdorff series . Jou Math Phys , 41 (4): 2434-2442, (2000). doi : 10.1063 / 1.533250 ( arXiv preprint )
• W. Rossmann: Lie Groups: An Introduction through Linear Groups . Oxford University Press, 2002.
• AA Sagle & RE Walde: Introduction to Lie Groups and Lie Algebras , Academic Press, New York, 1973. ISBN 0-12-614550-4 .
• J.-P. Serre: Lie algebras and Lie groups , Benjamin, 1965.
• H. Kleinert : Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets , 4th edition, World Scientific (Singapore, 2006) (also readable here ).

Web links

• An English representation of various Baker-Campbell-Hausdorff formulas
• Identities (with evidence) of the Baker-Campbell-Hausdorff formula