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New formula for ln(eAeB) in terms of commutators of A and B

  • M. V. Mosolova
Article
  • 43 Downloads

Abstract

We establish the formula
$$\ln (e^B e^A ) = \smallint _0^t \psi (e^{ - \tau ad_A } e^{ - \tau ad_B } ) e^{ - \tau ad_A } d\tau (A + B),$$
where Ψ(x)=(In x)/(x − 1); here A and B are elements of a. finite-dimensional Lie algebra which satisfy certain conditions. This formula enables us, in particular, to give a simple proof of the Campbell-Hausdorff theorem. We also give a generalization of the formula to the case of an arbitrary number of factors.

Keywords

Arbitrary Number Simple Proof 
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Literature cited

  1. 1.
    R. Feynman, “An operator calculus having applications in quantum electrodynamics,” Phys. Rev.,84, No. 2, 108–128 (1951).Google Scholar
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    V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973).Google Scholar
  3. 3.
    E. B. Dynkin, “Representation of the series ln(exey) in noncommuting x and y in terms of commutators,” Mat. Sb.,25, No. 1, 155–162 (1949).Google Scholar
  4. 4.
    M. V. Karasev, “On ordered quantization,” Moscow Institute of Electrical Engineering (MIÉM) (1974), VINITI Deposit No. 304–75.Google Scholar

Copyright information

© Plenum Publishing Corporation 1978

Authors and Affiliations

  • M. V. Mosolova
    • 1
  1. 1.Moscow Institute of Electrical EngineeringUSSR

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