Quantization of canonical isomorphisms and the semiclassical von Neumann theorem

Abstract

We prove the three mutually related theorems: the theorem on the quantizability of canonical isomorphisms, the theorem on the quantizability of classical canonical commutation relations and asemiclassical version of von Neumann's theorem. Although some similar results can be obtained on the basis of the deformation theory (e.g. [16], [15], [10] ) , here we present the proofs which involve only elementary methods and notions. Moreover, in our approach we can easily compute the quantum corrections. Our deformation quantizations (semiclassical algebras) are additionally equipped with the deformation involutions and we study here the algebras of entire functions and of polynomials, instead of frequently used algebras of $C^\infty$ observables.