JournalsprimsVol. 31 , No. 5DOI 10.2977/prims/1195163718

Quadratic Representations of the Canonical Commutation Relations

  • Martin Proksch

    Universität Würzburg, Germany
  • George Reents

    Universität Würzburg, Germany
  • Stephen J. Summers

    University of Florida, Gainesville, USA
Quadratic Representations of the Canonical Commutation Relations cover


This paper studies a class of representations (called quadratic) of the canonical commutation relations over symplectic spaces of arbitrary dimension, which naturally generalizes coherent and symplectic (i.e. quasifree) representations and which has previously been heuristically employed in the special case of finite degrees of freedom in the physics literature. An explicit characterization of canonical quadratic transformations in terms of a 'standard form' is given, and it is shown that they can be exponentiated to give representations of the Weyl algebra. Necessary and sufficient conditions are presented for the unitary equivalence of these representations with the Fock representation. Possible applications to quantum optics and quantum field theory are briefly indicated.