A New Kind of Integration Transformation in Phase Space Related to Two Mutually Conjugate Entangled-State Representations and Its Uses in Weyl Ordering of Operators

• Published in: Chinese physics letters Vol. 27; no. 5; pp. 5 - 8
• Main Author: 吕翠红 范洪义
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.05.2010
• Subjects: Conjugates; Entangled states; Operators; Order disorder; Phase transformations; Representations; Transformations; Weyl群; 空间变换; 纠缠态
• ISSN: 0256-307X; 1741-3540
• DOI: 10.1088/0256-307X/27/5/050301

Based on the two mutually conjugate entangled state representations ｜ξ〉 and ｜η〉, we propose an integration transformation in ξ - η phase space ∫∫ d^2ξd^2η/π^2e^（ξ-η）（η^＊ -v^＊）-（η-v）（ξ^＊-μ^＊）F（ξ^＊,μ^＊） F（ξ, η）≡D（μ,v）, and its inverse trans- formation, which possesses some well-behaved transformation properties, such as being invertible and the Parseval theorem. This integral transformation is a convolution, where one of the factors is fixed as a special normalized exponential function. We generalize this transformation to a quantum mechanical case and apply it to studying the Weyl ordering of bipartite operators, regarding to （Q1 -Q2） （P1 - P2） ordered and simultaneously （P1 ＋ P2） （Q1＋ Q2） ordered operators.