On Equivalence of Two Realizations for a Nonlinear Lie Algebra

Xing, Yang; Xun, D. M.; Rong, X. P.; Shan, Li; Liu, Quanhui

doi:10.1088/0253-6102/57/4/09

Commun. Theor. Phys.

2012

DOI: 10.1088/0253-6102/57/4/09

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On Equivalence of Two Realizations for a Nonlinear Lie Algebra

Yang Xing¹,

D. M. Xun²,

X. P. Rong³

et al.

Abstract: On Equivalence of Two Realizations for a Nonlinear Lie Algebra * YANG Xing ( ´), XUN Da-Mao ( ), RONG Xiao-Ping (Â¢ ), SHAN Lei ( ),

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“…where the occupation numbers are respectively n 1 = a † 1 a 1 and n 2 = a † 2 a 2 , and the eigenvalues of the occupation number operator are n α = 0, 1, 2, .... The unit direction vector N and the geometric momentum Π are respectively [12,28,29],…”

mentioning

confidence: 99%

On Relation Between Geometric Momentum and Annihilation Operators on a Two-Dimensional Sphere

Liu

Shen

Xun

et al. 2013

Int. J. Geom. Methods Mod. Phys.

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With a recently introduced geometric momentum that depends on the extrinsic curvature and offers a proper description of momentum on two-dimensional sphere, we show that the annihilation operators whose eigenstates are coherent states on the sphere take the expected form αx + iβp, where α and β are two operators that depend on the angular momentum and x and p are the position and the geometric momentum, respectively. Since the geometric momentum is manifestly a consequence of embedding the two-dimensional sphere in the three-dimensional flat space, the coherent states reflects some aspects beyond the intrinsic geometry of the surfaces. PACS numbers: 03.65.-w Quantum mechanics; 02.40.-k Differential geometry; 92.60.hc Mesosphere; 73.20.Fz Quantum localization on surfaces and interfacesThe coherent states on two-dimensional sphere, generated by the annihilation operators, were discovered around the turn of present century, independently by Hall [1][2][3][4][5][6] in the Bargmann representation, and by in the position representation, respectively. Once each group of them got to know the work of the other, both soon realized that their coherent states are essentially the same [6,9], and the equivalence was also noted by other group [11]. However, it is puzzling that in the annihilation operators they introduced, there is a fundamental quantity that is represented by a non-hermitian operator and has the same dimension of linear momentum, but it does not bear a transparent physical nor geometric meaning. This article points out that the physical and geometric interpretation of the fundamental quantity is easily available, based on the geometric momentum that is recently introduced to offer a proper description of momentum on sphere [12], whose general form for an arbitrary twodimensional curved surface with M denoting the mean curvature, is given by [12][13][14], *

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mentioning

confidence: 99%

On Relation Between Geometric Momentum and Annihilation Operators on a Two-Dimensional Sphere

Liu

Shen

Xun

et al. 2013

Int. J. Geom. Methods Mod. Phys.

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show abstract

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On Equivalence of Two Realizations for a Nonlinear Lie Algebra

Abstract: On Equivalence of Two Realizations for a Nonlinear Lie Algebra * YANG Xing ( ´), XUN Da-Mao ( ), RONG Xiao-Ping (Â¢ ), SHAN Lei ( ),

Cited by 1 publication

References 7 publications

On Relation Between Geometric Momentum and Annihilation Operators on a Two-Dimensional Sphere

On Relation Between Geometric Momentum and Annihilation Operators on a Two-Dimensional Sphere

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