Mutual transformations between the P–Q,Q–P, and generalized Weyl ordering of operators

Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator Ω_k(p,q) with a real k parameter and can unify the P–Q, Q–P, and Weyl ordering of operators in k=1,-1,0, respectively, we find the mutual transformations between δ(p-P)δ(q-Q), δ(q-Q)δ(p-P), and Ω_k(p,q), which are, respectively, the integration kernels of the P–Q, Q–P, and generalized Weyl quantization schemes. The mutual transformations provide us with a new approach to deriving the Wigner function of quantum states. The P- and Q- ordered forms of Ω_k(p,q) are also derived, which helps us to put the operators into their P- and Q- ordering, respectively.