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Chinese Physics Letters

A Family of Generalized Wigner Operators and Their Physical Meaning as Bivariate Normal Distribution

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

Published Oct 1, 2011• 2 citations
Publisher

Abstract

By extending the usual Wigner operator to the s-parameterized one, we find that in the process of the generalized Weyl quantization the s parameter plays the role of correlation between two quadratures Q and P . This can be exposed by comparing the normally ordered form of Ω s with the standard form of the Gaussian bivariate normal distribution of random variables in statistics. Three different expressions of Ω s and the quantization scheme with use of it are presented.

DOI
10.1088/0256-307x/28/10/104209

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By extending the usual Wigner operator to the s-parameterized one, we find that in the process of the generalized Weyl quantization the s parameter plays the role of correlation between two quadratures Q and P . This can be exposed by comparing the normally ordered form of Ω s with the standard form of the Gaussian bivariate normal distribution of random variables in statistics. Three different expressions of Ω s and the quantization scheme with use of it 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.
Combining quantum mechanics and the normal distribution in statistics we study the coherent state from the point of view of statistics and by using the integration method within ordered product of operators. We find that the pure coherent state |z >z| exhibits a bivariate normal distribution of randon variables in (q,p) phase space, z=(q+ip)/√2, with a real k-parameter which is related to the quantization scheme, and the correlation coefficient is ik. For k=±1, |z >z| respectively is arranged as P-ordering (all P stand on the left of all Q) and Q-ordering (all Q stand on the left of all P), while in the case of k=0, |z >z| is arranged as the Weyl-ordering. In the cases of P-ordering and Q-ordering, in the classical correspondence function of |z >z||z=(q+ip)/√2 the bivariates (q,p) are correlated, only in the case of Weyl correspondece, (q,p) are independent. In other words, the Weyl ordering of operators is liable to decouple the correlation in bivariates.
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.