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Chinese Physics B

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

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

Published Jun 1, 2012• 17 citations
Publisher

Abstract

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.

DOI
10.1088/1674-1056/21/6/064204

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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.
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 — and — ordered forms of Ω k ( p , q ) are also derived, which helps us to put the operators into their — and — ordering, respectively.
We introduce the generalized fermionic Wigner operator with an s parameter. Based on its remarkable properties, we establish one-to-one mapping between fermion operators and their s-parametrized pseudo-classical correspondence, which may involve fermionic Weyl pseudo-classical correspondence, P-representation and Q-representation in a unified way. Furthermore, starting with the projector of the fermionic coherent state, we obtain the s-ordered operator expansion formula of fermionic density operators, which includes normally ordered, antinormally ordered and Weyl ordered product of operators for different values of s. Applications in calculating some Fermi operators' s-ordered expansions are presented.