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Publications of the Research Institute for Mathematical Sciences

Polynomial Weyl Representations

Paul Lee Robinson

Published Apr 30, 1999• 2 citations
Publisher

Abstract

For the canonical commutation relations in infinite dimensions, we offer an explicit direct construction of Weyl representations generated from the Fock representation by polynomial transformations of arbitrary degree, solving a problem posed by Proksch, Reents and Summers. Our solution employs new approaches to Hilbert–Schmidt polynomials and their Wick ordering.

DOI
10.2977/prims/1195143949

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For the canonical commutation relations in infinite dimensions, we offer an explicit direct construction of Weyl representations generated from the Fock representation by polynomial transformations of arbitrary degree, solving a problem posed by Proksch, Reents and Summers. Our solution employs new approaches to Hilbert–Schmidt polynomials and their Wick ordering.
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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.
We construct a new class of representations of the canonical commutation relations, which generalizes previously known classes. We perturb the infinitesimal generator of the initial Fock representation (in other words, the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize perturbations which lead to representations of the canonical commutation relations. We provide conditions entailing the irreducibility of such representations, show explicitly that our class of representations subsumes previously studied classes, and give necessary and sufficient conditions for our representations to be unitarily equivalent, or just quasi-equivalent, with Fock, coherent or quasifree representations. 1991 Mathematics Subject Classification: 81S05, 46L60, 81T05.