General ordering problem and its combinatorial roots

Shähandeh, Farid; Bazrafkan, M. R.

doi:10.1088/1751-8113/45/15/155204

Cited by 12 publications

(13 citation statements)

References 25 publications

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“…[11][12][13] When Θ(x) is set to unity, this integral is simply the usual Weyl transform of p m q n . Indeed, generalized Weyl transform (14) arises as a consequence of Hermiticity condition.…”

Section: Generalized Weyl Transformmentioning

confidence: 99%

“…The solution is straightforward from direct use of (14) to the application of Baker-Campbell-Hausdorff formula e A+B = e B e 1 2 [A,B] e A for operators A = i(αp + βq) and B = i(ρp + σq). The commutator and anticommutator are then…”

Section: General Commutation and Anticommutation Relationsmentioning

confidence: 99%

“…(20) made possible the transformation between any ordering rule and the Weyl-ordering rule, i.e., A m, n were expressed as some combination of T m ′ , n ′ and vice versa. We shall then consider the transformations between ordering rules as a second consequence of integral form (14) of A m, n . We apply them in Sec.…”

Section: E Examplesmentioning

confidence: 99%

“…These transformations not only pave the way towards explicit expressions for the commutation and anticommutation relations between similarly ordered operators, especially non-Weyl operators but also address the problem of transformation between two ordering rules in terms of integrals that generally diverge. 14 We show here that the Weyl-ordered forms T m, n , for instance, can be expressed in terms of differently ordered operators if its integral representation shall be treated in the distributional sense.…”

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confidence: 90%

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

Domingo

Galapon

2015

Journal of Mathematical Physics

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Wick calculus using the technique of integration within an ordered product of operators Am.Hermite polynomial expansions of the error function and related F 0 (w) integralThe 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. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4907561] I. INTRODUCTIONQuantization remains to be the most available method for forming quantum operators corresponding to classical observables. Its usefulness is well-known. 1,2 Likewise, there is a widespread acquaintance with the different quantization rules 3-5 as well as with obstructions to quantization. [6][7][8] We want, in this paper, to direct our attention towards the different expressions of quantization rules and away from the multitude of obstruction issues, that is, to quantization as a practical tool and not to quantization as an object of investigation.A simplistic approach to quantization can be operated by replacing the canonical variables by the canonical operators, a procedure which is not without defect. Any product of classical position and momentum variables can be replaced by a seeming infinity of possible ordering of corresponding operators. The search for the appropriate quantum image of a classical function is then tantamount to finding the best ordering rule for operators. Meanwhile, the choice for an ordering rule also depends on the algebra of the resulting operators, i.e., on the commutation and anticommutation relations.An immediate impression gleaned from Refs. 9-13 would seem to suggest that an ordering rule can be mapped to a definite class of functions. To every known ordering rule, there exists an associated real-valued function Θ(x) such that Θ(0) = 1. Weyl ordering corresponds to the unit function Θ(x) = 1, the simplest symmetrization ordering rule corresponds to the cosine function Θ(x) = cos(x/2), and the Born-Jordan ordering rule corresponds to Θ(x) = (2/x) sin(x/2). These examples indicate how meaningful it is to devise an ordering rule by merely specifying a certain Θ(x). The associated functions Θ(x) appear as the possible kernels of a generalized Weyl transform, which was introduced in Refs. 11-13 as the kind of mapping that characterizes each operator ordering.Under what assumptions can this association of ordering rules with ordinary functions hold? According to Refs. 12 and 13, if quantization is defined to be a mapping from the algebra of q, p-polynomials to the algebra of q, p-polynomials, four things are necessary.(a) The classical variables q, p and the unit function in phase space are mapped into the canonical operators q, p and the identity operator I, respectively. a)

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Section: Generalized Weyl Transformmentioning

confidence: 99%

Section: General Commutation and Anticommutation Relationsmentioning

confidence: 99%

Section: E Examplesmentioning

confidence: 99%

mentioning

confidence: 90%

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

Domingo

Galapon

2015

Journal of Mathematical Physics

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“…[3] However, in a recent work the authors have introduced the most general theorem regarding orderings of operators which will be referred to as the general ordering theorem or just GOT. [4] In this regard, one may t-order any multiplicative sequence of s j -ordered functions with j ∈ {2, 3, . .…”

Section: Introductionmentioning

confidence: 99%