Baker-Campbell-Hausdorff formula question

  • Thread starter Karliski
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In summary, the Hadamard lemma is easy to show by constructing a Taylor series for the function f(s) = e^{sA} B e^{-sA} and evaluating it at s=1. This allows for the relation to be shown when computing s=1.
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The Hadamard formula is easy to show. The full BCH formula is a ***** (I spent several hours yesterday trying to do it, but I didn't understand enough about Lie groups to get there). Anyway, start with this function:

[tex]f(s) = e^{sA} B e^{-sA}[/tex]

Then differentiate it a few times with respect to s:

[tex]f'(s) = e^{sA} A B e^{-sA} - e^{sA} B A e^{-sA} = e^{sA} [A,B] e^{-sA}[/tex]

[tex]f''(s) = e^{sA} A [A,B] e^{-sA} - e^{sA} [A,B] A e^{-sA} = e^{sA} [A, [A,B]] e^{-sA}[/tex]

[tex]f'''(s) = e^{sA} [A, [A, [A,B]]] e^{-sA}[/tex]

etc.

Now construct the Taylor series for f(s):

[tex]f(s) = f(0) + s f'(0) + \frac12 s^2 f''(0) + \frac1{3!} s^3 f'''(0) + ...[/tex]

[tex]e^{sA} B e^{-sA} = B + [A,B] s + \frac12 [A, [A, B]] s^2 + \frac1{3!} [A, [A, [A, B]]] s^3 + ...[/tex]

Finally, evaluate the above at s=1 to get the result.
 
  • #3
Hi,

Thanks, yes that is what I also did. The "parametric induction" term threw me off.
 

1. What is the Baker-Campbell-Hausdorff formula?

The Baker-Campbell-Hausdorff formula, also known as the BCH formula, is an important mathematical tool used in the field of Lie algebras to calculate the composition of two exponentials. It is named after the mathematicians Henry Baker, John Herschel Campell, and Felix Hausdorff who independently derived the formula in the late 19th and early 20th centuries.

2. What is the significance of the Baker-Campbell-Hausdorff formula?

The Baker-Campbell-Hausdorff formula is significant because it allows for the simplification of complex exponential expressions involving elements of a Lie algebra. This simplification is useful in many areas of mathematics and physics, including quantum mechanics, differential geometry, and group theory.

3. How is the Baker-Campbell-Hausdorff formula derived?

The Baker-Campbell-Hausdorff formula is derived using the Taylor series expansion of exponential functions and the Jacobi identity for Lie algebras. The formula involves nested commutators and can be written in several different forms depending on the specific Lie algebra being studied.

4. Can the Baker-Campbell-Hausdorff formula be generalized to higher dimensions?

Yes, the Baker-Campbell-Hausdorff formula can be generalized to higher dimensions by using the concept of multilinear maps. In this case, the formula becomes more complex and involves multiple nested commutators.

5. What are the applications of the Baker-Campbell-Hausdorff formula?

The Baker-Campbell-Hausdorff formula has numerous applications in mathematics and physics, including in the study of Lie groups, quantum mechanics, differential geometry, and dynamical systems. It is also used in the development of numerical methods for solving differential equations and in the theory of stochastic processes.

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