A generalized Weyl–Wigner quantization scheme unifying P–Q and Q–P ordering and Weyl ordering of operators

By extending the usual Wigner operator to the s-parameterized one as (1/4π²)∫_-∞^∞ dyduexp≤iu≤(q-Q) + iy≤(p-P) + i(s/2)yu with s being a real parameter, we propose a generalized Weyl quantization scheme which accompanies a new generalized s-parameterized ordering rule. This rule recovers P-Q ordering, Q-P ordering, and Weyl ordering of operators in s=1,-1,0 respectively. Hence it differs from the Cahill-Glaubers' ordering rule which unifies normal ordering, anti-normal ordering, and Weyl ordering. We also show that in this scheme the s-parameter plays the role of correlation between two quadratures Q and P. The formula that can rearrange a given operator into its new s-parameterized ordering is presented.