𝔓-quantization, 𝔔-quantization and Weyl quantization of a ray in classical phase space

Abstract

By examining three quantization schemes of a ray function in classical phase space (a geometric ray is expressed by δ(x-λq-νp)), we find that the Weyl quantization scheme can reasonably demonstrate the correspondence between classical functions and quantum mechanical operators, since δ(x-λq-νp) really maps onto the operator δ(x-λQ-νP), where [Q, P] = iℏ, and δ(x-λQ-νP) represents a pure state (the coordinate-momentum intermediate representation), while 𝔓-ordered, 𝔔-ordered quantization schemes δ(x-λq-νp) to two different Fresnel integration kernels in Weyl-ordered form. Thus, Weyl quantization is more reasonable and preferable.