×
Berceanu, Stefan

A holomorphic representation of the Jacobi algebra. (English) Zbl 1099.81036

Rev. Math. Phys. 18, No. 2, 163-199 (2006); errata ibid. 24, No. 10, Paper No. 1292001, 2 p. (2012).
The author continues his studies concerning the characterization of the differential action of coherent state (CS-) groups [see S. Berceanu and A. Gheorghe, “Differential operators on orbits of coherent states”, Rom. J. Phys. 48, 545–556 (2003); math.DG/0211054] and references therein).
The present paper is devoted, in particular, to the case of the simplest of the Jacobi groups, that is the group \(G_{1}^{J}\) defined as the semidirect product of the real three-dimensional Heisenberg-Weyl group \(H_1\) with the group SU\((1,1)\). A self-contained organized account of results both new and known is presented, supported by an exhaustive list of references.
After providing the explicit realization of a differential holomorphic representation of the algebra \(\mathfrak{g}_{1}^{J}=\mathfrak{h}_{1} \rtimes \mathfrak{su} (1,1)\) on the associated CS-manifold \(M=\mathbb{C} \times \mathcal{D}_1\), the author details the construction of the reproducing kernel and of the space of functions (the symmetric Fock space) attached to it. Geometrical aspects are also considered by reporting the Kähler potential (logarithm of the reproducing kernel), the fundamental Kähler two-form, the volume form, and the equations of the geodesics on \(M\).
Reviewer: Giulio Landolfi (Lecce)

Cited in 1 Review
Cited in 9 Documents

MSC:

81R30 Coherent states
33C47 Other special orthogonal polynomials and functions
81V80 Quantum optics
32Q15 Kähler manifolds

Keywords:

coherent states; representations of coherent state Lie algebras; Jacobi groups; first-order holomorphic differential operators with polynomial coefficients
PDF BibTeX XML Cite
\textit{S. Berceanu}, Rev. Math. Phys. 18, No. 2, 163--199 (2006; Zbl 1099.81036)
Full Text: DOI arXiv arXiv

References:

[1] DOI: 10.1007/978-1-4612-1258-4
[2] DOI: 10.2307/1969129 · Zbl 0045.38801
[3] DOI: 10.1002/cpa.3160140303 · Zbl 0107.09102
[4] V. Bargmann, Analytic methods in Mathematical Physics, eds. R. P. Gilbert and R. G. Newton (Gordon and Breach, Science Publishers, New York-London-Paris, 1970) pp. 27–63.
[5] Bateman H., Higher Transcendental Functions 2 (1958)
[6] DOI: 10.1063/1.527691 · Zbl 0638.58040
[7] DOI: 10.1063/1.529700 · Zbl 0837.53055
[8] DOI: 10.1063/1.530121 · Zbl 0778.58020
[9] DOI: 10.1016/S0393-0440(99)00075-3 · Zbl 1002.81027
[10] Berceanu S., An. Univ. Timişoara Ser. Mat.-Inform. 39 pp 31–
[11] Berceanu S., Rom. Jour. Phys. 48 pp 545–
[12] S. Berceanu, Recent Advances in Geometry and Topology, eds. Dorin Andrica and Paul A. Blaga (Cluj University Press, 2004) pp. 83–98, arXiv: math.DG/0408233.
[13] S. Berceanu, Advances in Operator Algebras and Mathematical Physics, eds. F. Boca (The Theta Foundation, Bucharest, 2005) pp. 1–24, arXiv: math.DG/0504053.
[14] DOI: 10.1007/BF01609397 · Zbl 1272.53082
[15] DOI: 10.1007/BF01220849 · Zbl 0413.58023
[16] R. Berndt, Seminar on Number Theory (Paris 1982–83), Progress in Mathematics 51 (Birkhäuser Boston, Boston, MA, 1984) pp. 21–32.
[17] DOI: 10.1007/BF02570858 · Zbl 0695.10024
[18] DOI: 10.1007/978-3-0348-0283-3 · Zbl 1235.11046
[19] DOI: 10.1016/0003-4916(71)90012-1
[20] Biedenharn L. C., J. Phys. A 22 pp 4581–
[21] Bogoliubov N. N., Izv. Akad. Nauk. 11 pp 77–
[22] Cahen M., J. Geom. Phys. 7 pp 45–
[23] Dixmier J., Bull. Soc. Math. France 96 pp 209–
[24] DOI: 10.1088/1464-4266/4/1/201
[25] DOI: 10.1007/978-3-662-09645-1
[26] DOI: 10.1007/978-1-4684-9162-3
[27] DOI: 10.1515/9781400882427 · Zbl 0682.43001
[28] DOI: 10.1103/PhysRev.131.2766 · Zbl 1371.81166
[29] Gradšteĭn I. S., Tables of Integrals, Sums, Series and Products (1963)
[30] Hilgert J., J. Reine Angew. Math. 474 pp 67–
[31] DOI: 10.1103/PhysRevD.19.1669
[32] DOI: 10.1103/PhysRev.58.1098 · Zbl 0027.18604
[33] DOI: 10.1007/BF01645755 · Zbl 0152.23204
[34] DOI: 10.1063/1.1665983 · Zbl 0238.17004
[35] DOI: 10.1007/BF02924908 · Zbl 0603.53035
[36] DOI: 10.1007/978-94-009-4728-3_22
[37] Kähler E., Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 16 pp 115–
[38] DOI: 10.1515/9783110905434 · Zbl 1039.01013
[39] DOI: 10.1007/BF01391200
[40] Kirillov A., Éléments de la Théorie des Representations (1974)
[41] DOI: 10.1090/gsm/064
[42] Kirillov A. A., Bull. Amer. Math. Soc. 36 pp 43–
[43] DOI: 10.1088/0305-4470/38/20/011 · Zbl 1067.81060
[44] DOI: 10.2969/jmsj/02040638 · Zbl 0165.40504
[45] DOI: 10.1007/BF01446218 · JFM 12.0292.01
[46] DOI: 10.1016/0034-4877(96)89292-5 · Zbl 0883.22011
[47] DOI: 10.1007/BF02770161
[48] DOI: 10.1103/PhysRev.160.1076
[49] DOI: 10.1515/9783110808148
[50] DOI: 10.1002/prop.2190450204 · Zbl 1144.81496
[51] Pasternak-Winiarski Z., Demonstratio Math. 30 pp 199–
[52] DOI: 10.1007/978-3-642-61629-7
[53] Pyatetskii-Shapiro I. I., Automorphic Functions and the Geometry of Classical Domains (1969)
[54] DOI: 10.1093/qmath/28.4.403 · Zbl 0387.58002
[55] Satake I., Publications of the Mathematical Society of Japan 14, in: Algebraic Structures of Symmetric Domains (1980) · Zbl 0483.32017
[56] DOI: 10.1103/PhysRevA.37.3028
[57] DOI: 10.1007/BF02754988 · Zbl 0197.26401
[58] DOI: 10.1088/1464-4266/2/6/02
[59] DOI: 10.1038/306141a0
[60] DOI: 10.1103/PhysRevA.13.2226
[61] DOI: 10.1103/RevModPhys.62.867
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.