β-quantization, Ω-quantization and Weyl quantization of a ray in classical phase space

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