Time dependent Baker-Hausdorf formula

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SUMMARY

The discussion centers on the time-dependent Baker-Hausdorff formula for the product of exponentials in the context of operators. Participants clarify that the formula applies universally to all operators, provided certain assumptions, such as commutators commuting with all operators, are met. A specific interest in a version that includes only double commutators and time-ordered exponentials is highlighted. A relevant paper was identified, providing the necessary formula for computation.

PREREQUISITES
  • Understanding of the Baker-Hausdorff formula
  • Familiarity with operator theory in quantum mechanics
  • Knowledge of commutators and their properties
  • Experience with time-ordered exponentials
NEXT STEPS
  • Study the full Baker-Hausdorff theorem, including double and triple commutators
  • Research time-ordered exponentials in quantum mechanics
  • Read the paper referenced in the discussion for detailed applications
  • Explore advanced operator algebra techniques
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Physicists, mathematicians, and researchers working in quantum mechanics or operator theory, particularly those interested in the applications of the Baker-Hausdorff formula.

paweld
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Could anyone show me the Baker-Hausdorf formula for product of exponentials in case of
operators which are time dependent. I know that there is a time-dependent version of this
formula which works under some assumptions are imposed on the operators which appear
in exponentials, like e.g. commutator commutes with all operators.
Thanks.
 
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Forgive me, but I don't see why any time-dependence would matter. It's an identity which applies to any and all operators. (I assume you're talking about the full theorem that includes double commutators, triple commutators, etc, not one that has been truncated.)
 
Thanks for answer. I looked for the identity which involves time ordered exponentials
and includes only double commutators. Fortunantely I've just found paper
(http://www.sciencedirect.com/science/article/pii/S0167691101001943)
where a formula which enabled me to compute what I wanted was given.
 

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