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Journal of Mathematical Physics

Generalized Weyl transform for operator ordering: Polynomial functions in phase space

Herbert B. Domingo, Eric A. Galapon

Published Feb 1, 2015• 16 citations
Publisher

Abstract

The generalized Weyl transforms were developed from the Hermiticity condition and the ordering rules were represented by characteristic real-valued functions. The integral transforms give rise to transformation equations between Weyl quantization and differently ordered operators. The transforms also simplify evaluation of commutator and anticommutator of a set of operators following the same ordering rule.

DOI
10.1063/1.4907561

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The generalized Weyl transforms were developed from the Hermiticity condition and the ordering rules were represented by characteristic real-valued functions. The integral transforms give rise to transformation equations between Weyl quantization and differently ordered operators. The transforms also simplify evaluation of commutator and anticommutator of a set of operators following the same ordering rule.
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.
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.
Since the foundation of quantum mechanics, operator-ordering identities for mutual transformation of power of coordinate-momentum operators have been a fundamental and tough topic. To the best of our knowledge, this topic has not been tackled smoothly because there is no elegant and direct way to investigate it. In this paper we report a very concise and novel method to handle this topic, i.e., we employ the generating function of two-variable Hermite polynomial and the characteristics of ordered operators to derive a series of operator-ordering identities for mutual transformation of power of coordinate-momentum operators: they surly possess potential applications. The essence of our method lies in the fact that coordinate-momentum operators can be permutable within ordered product of operators, just as the scenarios in P-Q ordering, Q-P ordering and Weyl ordering. We also derive the integration transformation formula about two-variable Hermite polynomial in phase space. The correspondence relation between operator ordering and quantization recipe is established. The beauty of theoretical physics is embodied extensively in the paper.
We investigate the properties of the Weyl-ordered polynomials Tm,n by the technique of integration within a Weyl-ordered product (IWWOP) and the Weyl quantization rule. The introduction of the Weyl-ordering symbol ⋮⋮ and the IWWOP provide us with a convenient approach for deriving the Weyl-ordered form of the product of Tm,n and other operators. The Weyl-ordered Dirac's δ-function form of the Wigner operator can also directly lead us to the Weyl quantization formalism for the functions of radial vaiables.
Quantization under simplest symmetric ordering of classical monomials with negative integral exponents of momentum has been written in integral form. This integral representation not only proved useful in transforming Weyl quantization to symmetric ordering quantization, and vice versa. They also facilitate derivation of commutation relations among different operators involving inverse powers of momentum operator, relations which can be used in solving quantization problems for classical functions with singularity along the momentum axis.