Publications of the Research Institute for Mathematical Sciences


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Volume 31, Issue 5, 1995, pp. 755–804
DOI: 10.2977/prims/1195163718

Published online: 1995-10-31

Quadratic Representations of the Canonical Commutation Relations

Martin Proksch[1], George Reents[2] and Stephen J. Summers[3]

(1) Universität Würzburg, Germany
(2) Universität Würzburg, Germany
(3) University of Florida, Gainesville, USA

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.

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Proksch Martin, Reents George, Summers Stephen: Quadratic Representations of the Canonical Commutation Relations. Publ. Res. Inst. Math. Sci. 31 (1995), 755-804. doi: 10.2977/prims/1195163718