Some Continuous Field Quantizations, Equivalent to the C*-Weyl Quantization

Abstract

Starting from a (possibly infinite dimensional) pre-symplectic space $(E,\sigma )$, we study a class of modified Weyl quantizations. For each value of the real Planck parameter $\hslash$, we have a C*-Weyl algebra $\mathcal{W}(E,{\hslash }_{\sigma })$, which altogether constitute a continuous field of C*-algebras, as discussed in previous works. For $\hslash =0$, we construct a Fréchet–Poisson algebra, densely contained in $\mathcal{W}(E,0)$, as the classical observables to be quantized. The quantized Weyl elements are decorated by so-called quantization factors, indicating the kind of normal ordering in specific cases. Under some assumptions on the quantization factors, the quantization map may be extended to the Fréchet–Poisson algebra. It is demonstrated to constitute a strict and continuous deformation quantization, equivalent to the Weyl quantization, in the sense of Rieffel and Landsman. Realizing the C*-algebraic quantization maps in regular and faithful Hilbert space representations leads to quantizations of the unbounded field expressions.