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Modern Physics Letters A

QUANTIZATION SCHEME FOR FERMIONIC SYSTEM AND s-ORDERED OPERATOR EXPANSION FORMULA OF FERMIONIC DENSITY OPERATORS

Ye‐Jun Xu, Jun Song, Hong‐Chun Yuan, Hong-Yi Fan, Qiuyu Liu

Published Mar 28, 2011• 3 citations
Publisher

Abstract

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.

DOI
10.1142/s0217732311035213

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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.
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.
s-parametrized quantization is essential to phase space theory of quantum mechanical. Based on s-ordered Wigner operator, we examine the classical correspondence of the s2-parametrized Wigner operator through the s1-parametrized quantization scheme, and establish the mutual transformation relation between different s-parametrized quantization schemes. It turns out that the s-parametrized Wigner operator's s-ordering is just the Dirac delta function, which seems to be a new result. As applications, we derive the s-ordered form of the density operator of thermal states and some new generating function formula of Hermite polynomials.
We point out that there is a close connection between the Weyl correspondence and coherent state correspondence. By virtue of differential and integral method within normal ordered products we demonstrate that the essence of the Weyl correspondence is just coherent state correspondence in both boson and fermion cases. Furthermore, coherent state form and explicit operator form for the Wigner function are given. With these forms we can derive some important operator formulas from the Weyl correspondence. The Wigner function's zero-point value is also discussed.