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Application of commutator theorems to the integration of representations of Lie algebras and commutation relations

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Abstract

Sufficient conditions on unbounded, symmetric operatorsA andB which imply that

$$\exp (itA)\exp (isB)\exp ( - itA)$$

satisfies the well known “multiple commutator” formula are derived. This formula is then applied to prove new necessary and sufficient conditions for the integrability of representations of Lie algebras and canonical commutation relations and the commutativity of the spectral projections of two commuting, unbounded, self-adjoint operators. A classic theorem of Nelson's is obtained as a corollary. Our results are useful in relativistic quantum field theory.

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, 08540, Princeton, NJ, USA

    J. Fröhlich

Authors
  1. J. Fröhlich
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Additional information

Communicated by H. Araki

Research supported in part by the US National Science Foundation under Grant MPS 75-11864

A Sloan Foundation Fellow

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Fröhlich, J. Application of commutator theorems to the integration of representations of Lie algebras and commutation relations. Commun.Math. Phys. 54, 135–150 (1977). https://doi.org/10.1007/BF01614134

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  • DOI: https://doi.org/10.1007/BF01614134

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