Polynomial Weyl Representations

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.

Further Representations of the Canonical Commutation Relations

“…In this section we restrict our attention to the computationally simpler canonical transformations of arbitrary but ®nite degree. These are the counter-parts in our approach to the polynomial representations of [30]. Note, however, that due to the boundedness assumptions made in [30], which do not need to be made here, we shall be discussing a larger class of representations than does [30].…”

“…These are the counter-parts in our approach to the polynomial representations of [30]. Note, however, that due to the boundedness assumptions made in [30], which do not need to be made here, we shall be discussing a larger class of representations than does [30]. In any case, the questions treated below are not addressed in [30]; at the cost of the additional technicalities involved in working in®nitesimally with representations of the CCR, one gains a more detailed computational power than one apparently can attain when working globally from the outset.…”

“…Many classes of representations of the CCR for in®nitely many degrees of freedom have been rigorously studied in the literature; we mention, in particular, the Fock [7], coherent [31], quasifree [27] (or symplectic), in®nite product [15], and, more recently, quadratic [26,22,29] and polynomial representations [30]. In this paper, we wish to construct and prove certain mathematically and physically relevant properties of a broad class of representations of the CCR containing all of the above (except possibly the in®nite product representations) as special cases.…”

“…with all coef®cients totally symmetric in the indices. (In point of fact, the polynomial representations of the CCR constructed in [30] were constructed globally and not via their in®nitesimal generators as is done in this paper.) The canonical transformations of general degree that we construct in this paper are of the form q k U 3 q k ; p k U 3 p k m; k 1 ;...; k m l kk 1 ... k m : q k 1 .…”

“…Many classes of representations of the CCR for infinitely many degrees of the Fock [7], coherent [31], quasifree [27] (or symplectic), infinite product [15], and, more recently, quadratic [26][22] [29] and polynomial representations [30].…”