Some continuous field quantizations, equivalent to the \(C^*\)-Weyl quantization (Q2386884)

Summary: 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 \(\hbar\), we have a \(C^*\)-Weyl algebra \({\mathcal W}(E, \hbar \sigma)\), which altogether constitute a continuous field of \(C^*\)-algebras, as discussed in previous works. For \(\hbar =0\), 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.