Weyl CCR Quantization

Encoding canonical commutation relations via Weyl unitaries V(z)=exp(iR(z))

Weyl CCR Quantization

A quantization of $(K,B)$ assigns to each $z\in K$ a selfadjoint operator $R(z)$ so that $V(z)=e^{iR(z)}$ satisfies the Weyl relations

V(z_1)V(z_2)=e^{-iB(z_1,z_2)/2}\,V(z_1+z_2).

Key property (paper use):

Shale requires $V(\cdot)$ to be weakly continuous on each finite-dimensional subspace (regularity).

Example: In 1D, $V(p,q)=e^{i(pP+qQ)}$ with $[P,Q]=i$ .