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Chinese Physics B

S -parameterized Weyl transformation and the corresponding quantization scheme

Ji‐Suo Wang, Xiang-Guo Meng, Hong-Yi Fan

Published Jan 1, 2015• 6 citations
Publisher

Abstract

By extending the usual Weyl transformation to the s -parameterized Weyl transformation with s being a real parameter, we obtain the s -parameterized quantization scheme which includes P – Q quantization, Q – P quantization, and Weyl ordering as its three special cases. Some operator identities can be derived directly by virtue of the s -parameterized quantization scheme.

DOI
10.1088/1674-1056/24/1/014203

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By extending the usual Weyl transformation to the s -parameterized Weyl transformation with s being a real parameter, we obtain the s -parameterized quantization scheme which includes P – Q quantization, Q – P quantization, and Weyl ordering as its three special cases. Some operator identities can be derived directly by virtue of the s -parameterized quantization scheme.
By extending the usual Wigner operator to the s-parameterized one as 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, antinormal 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.
Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator ?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 Ωk(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 Ωk(p,q) are also derived,which helps us to put the operators into their P- and Q- ordering, respectively.