Looking for a proof regarding Baker-Campell-Hausdorff

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In summary, the conversation discusses the proof that if [A,B] belongs to the same vector space as A and B, then C is also in the same vector space to all orders when e^Ae^B = e^C. It is clear up to the second order but at higher orders, terms may cancel and the proof becomes more difficult. It is important to note that C must not only be in the same vector space, but also in a Lie algebra formed by A, B, and higher commutators. A reference for a detailed proof is not available, but Friedrichs' theorem is mentioned as a possible outline.
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Hello!
Is there any nice proof that if [A,B] belongs to the same vectorspace as A and B, then C is in the same vectorspace to all orders, given that
[tex]e^Ae^B = e^C[/tex]
?
It is obvious to the second order but at higher orders it seems as if terms will cancel but I can't prove it.

Thanks!
 
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Yes, it's actually not enough for ##C## to be in the same vector space. The quantities ##A,B,\[A,B\]## and higher commutators actually close as a Lie algebra. This is still a vector space, but might be much smaller than the original vector space. ##C## will be in this Lie algebra. I don't have a reference for a detailed proof at hand, but the wiki discussion has an outline based on what is called Friedrichs' theorem.
 

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

The Baker-Campbell-Hausdorff formula is a mathematical expression that calculates the product of two exponentials in a Lie algebra. It is often used in physics and mathematics to simplify calculations involving non-commutative operators.

2. Why is the Baker-Campbell-Hausdorff formula important?

The Baker-Campbell-Hausdorff formula allows for the simplification of complex calculations involving non-commutative operators, making it a valuable tool in fields such as quantum mechanics, differential geometry, and group theory.

3. Is the Baker-Campbell-Hausdorff formula exact?

No, the Baker-Campbell-Hausdorff formula is an approximation and becomes more accurate as the number of terms in the formula increases. However, it is often a useful approximation for practical calculations.

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

The Baker-Campbell-Hausdorff formula is derived using a combination of mathematical techniques, including the Campbell-Baker-Hausdorff theorem and the Zassenhaus formula. It involves expanding the exponential of a Lie bracket into a power series and then rearranging the terms to achieve a simpler expression.

5. Can the Baker-Campbell-Hausdorff formula be applied to all Lie algebras?

No, the Baker-Campbell-Hausdorff formula can only be applied to certain types of Lie algebras, specifically those that are nilpotent or solvable. In addition, the formula may not always converge for all values of the Lie algebra elements, making its application limited in some cases.

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