﻿ Bakerâ€“Campbellâ€“Hausdorff formula

# Bakerâ€“Campbellâ€“Hausdorff formula

In mathematics, the Baker -Campbell - Hausdorff formula is a named after the mathematicians Henry Frederick Baker, John Edward Campbell and Felix Hausdorff equation that specifies a Vertauschungsgesetz for certain linear operators.

## Preliminary definitions

If X is a continuous linear operator of a Banach space into itself, then one can define the exponential of this operator as follows as a series:

The multiplication of a sequential execution and the addition is a point-wise addition of the operators involved. The commutator (also Lie bracket ) of two linear operators X and Y is defined as

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

With and.

## The formula

Case and apply the simple Baker -Campbell - Hausdorff formula

For arbitrary and the formula is very extensive and is

With

## 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 Saxon Akad Wiss Leipzig 58 (1906 ) 19-48.
• W. Magnus: Comm Pure Appl Math VII (1954 ) 649-673.
• W. Miller: Symmetry Groups and Their Applications, Academic Press, New York, 1972, p 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.
• M. W. 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.
• A. A. Sagle & R. E. 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 read here).
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