Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 064, 28 pages      arXiv:1512.00601      https://doi.org/10.3842/SIGMA.2016.064

Balanced Metric and Berezin Quantization on the Siegel-Jacobi Ball

Stefan Berceanu
National Institute for Physics and Nuclear Engineering, Department of Theoretical Physics, PO BOX MG-6, Bucharest-Magurele, Romania

Received March 03, 2016, in final form June 17, 2016; Published online June 27, 2016

Abstract
We determine the matrix of the balanced metric of the Siegel-Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace-Beltrami operator of this manifold. We discuss several geometric aspects related with Berezin quantization on the Siegel-Jacobi ball.

Key words: Jacobi group; Siegel-Jacobi ball; balanced metric; homogenous Kähler manifolds; Laplace-Beltrami operator; scalar curvature; Ricci form; Berezin quantization.

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References

  1. Arezzo C., Loi A., Moment maps, scalar curvature and quantization of Kähler manifolds, Comm. Math. Phys. 246 (2004), 543-559.
  2. Berceanu S., Coherent states and geodesics: cut locus and conjugate locus, J. Geom. Phys. 21 (1997), 149-168, dg-ga/9502007.
  3. Berceanu S., On the geometry of complex Grassmann manifold, its noncompact dual and coherent states, Bull. Belg. Math. Soc. Simon Stevin 4 (1997), 205-243, dg-ga/9509002.
  4. Berceanu S., Realization of coherent state Lie algebras by differential operators, in Advances in Operator Algebras and Mathematical Physics, Theta Ser. Adv. Math., Vol. 5, Editors F. Boca, O. Bratteli, R. Longo, H. Siedentop, Theta, Bucharest, 2005, 1-24, math.DG/0504053.
  5. Berceanu S., A holomorphic representation of the Jacobi algebra, Rev. Math. Phys. 18 (2006), 163-199, Errata, Rev. Math. Phys. 24 (2012), 1292001, 2 pages, math.DG/0408219.
  6. Berceanu S., A holomorphic representation of the semidirect sum of symplectic and Heisenberg Lie algebras, J. Geom. Symmetry Phys. 5 (2006), 5-13.
  7. Berceanu S., A holomorphic representation of the multidimensional Jacobi algebra, in Perspectives in Operator Algebras and Mathematical Physics, Theta Ser. Adv. Math., Vol. 8, Editors F.-P. Boca, R. Purice, S. Stratila, Theta, Bucharest, 2008, 1-25, math.DG/0604381.
  8. Berceanu S., The Jacobi group and the squeezed states - some comments, AIP Conf. Proc. 1191 (2009), 21-29, arXiv:0910.5563.
  9. Berceanu S., A convenient coordinatization of Siegel-Jacobi domains, Rev. Math. Phys. 24 (2012), 1250024, 38 pages, arXiv:1204.5610.
  10. Berceanu S., Coherent states and geometry on the Siegel-Jacobi disk, Int. J. Geom. Methods Mod. Phys. 11 (2014), 1450035, 25 pages, arXiv:1307.4219.
  11. Berceanu S., Quantum mechanics and geometry on the Siegel-Jacobi disk, in Geometric Methods in Physics, Editors P. Kielanowski, P. Bieliavsky, A. Odesski, A. Odzijewicz, M. Schlichenmaier, T. Voronov, Trends in Mathematics, Birkhäuser, Basel, 2014, 89-98.
  12. Berceanu S., Bergman representative coordinates on the Siegel-Jacobi disk, Romanian J. Phys. 60 (2015), 867-896, arXiv:1409.0368.
  13. Berceanu S., Gheorghe A., Differential operators on orbits of coherent states, Romanian J. Phys. 48 (2003), 545-556, math.DG/0211054.
  14. Berceanu S., Gheorghe A., Applications of the Jacobi group to quantum mechanics, Romanian J. Phys. 53 (2008), 1013-1021, arXiv:0812.0448.
  15. Berceanu S., Gheorghe A., On the geometry of Siegel-Jacobi domains, Int. J. Geom. Methods Mod. Phys. 8 (2011), 1783-1798, arXiv:1011.3317.
  16. Berceanu S., Schlichenmaier M., Coherent state embeddings, polar divisors and Cauchy formulas, J. Geom. Phys. 34 (2000), 336-358, math.DG/9903105.
  17. Berezin F.A., Quantization in complex bounded domains, Dokl. Akad. Nauk SSSR 211 (1973), 1263-1266.
  18. Berezin F.A., Quantization, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1116-1175.
  19. Berezin F.A., General concept of quantization, Comm. Math. Phys. 40 (1975), 153-174.
  20. Berezin F.A., Quantization in complex symmetric spaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 363-402.
  21. Berndt R., Sur l'arithmétique du corps des fonctions elliptiques de niveau $N$, in Seminar on Number Theory, Paris 1982-83 (Paris, 1982/1983), Progr. Math., Vol. 51, Birkhäuser Boston, Boston, MA, 1984, 21-32.
  22. Berndt R., Böcherer S., Jacobi forms and discrete series representations of the Jacobi group, Math. Z. 204 (1990), 13-44.
  23. Berndt R., Schmidt R., Elements of the representation theory of the Jacobi group, Progr. Math., Vol. 163, Birkhäuser Verlag, Basel, 1998.
  24. Borel A., Kählerian coset spaces of semisimple Lie groups, Proc. Nat. Acad. Sci. USA 40 (1954), 1147-1151.
  25. Cahen B., Global parametrization of scalar holomorphic coadjoint orbits of a quasi-Hermitian Lie group, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 52 (2013), 35-48.
  26. Cahen B., Stratonovich-Weyl correspondence for the Jacobi group, Commun. Math. 22 (2014), 31-48.
  27. Cahen M., Gutt S., Rawnsley J., Quantization of Kähler manifolds. II, Trans. Amer. Math. Soc. 337 (1993), 73-98.
  28. Calabi E., Isometric imbedding of complex manifolds, Ann. of Math. 58 (1953), 1-23.
  29. Chern S.S., Complex manifolds without potential theory, 2nd ed., Universitext, Springer-Verlag, New York - Heidelberg, 1979.
  30. Chiribella G., Adesso G., Quantum benchmarks for pure single-mode Gaussian states, Phys. Rev. Lett. 112 (2014), 010501, 6 pages, arXiv:1308.2146.
  31. Donaldson S.K., Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), 479-522.
  32. Dorfmeister J., Nakajima K., The fundamental conjecture for homogeneous Kähler manifolds, Acta Math. 161 (1988), 23-70.
  33. Eichler M., Zagier D., The theory of Jacobi forms, Progr. Math., Vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985.
  34. Engliš M., Berezin quantization and reproducing kernels on complex domains, Trans. Amer. Math. Soc. 348 (1996), 411-479.
  35. Gay-Balmaz F., Tronci C., Vlasov moment flows and geodesics on the Jacobi group, J. Math. Phys. 53 (2012), 123502, 36 pages, arXiv:1105.1734.
  36. Helgason S., Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, Vol. 80, Academic Press, Inc., New York - London, 1978.
  37. Hua L.-K., On the theory of automorphic functions of a matrix level. I. Geometrical basis, Amer. J. Math. 66 (1944), 470-488.
  38. Hua L.-K., Harmonic analysis of functions of several complex variables in the classical domains, Amer. Math. Soc., Providence, R.I., 1963.
  39. Hua L.-K., Look K.H., Theory of harmonic functions in classical domains, Scientia Sinica 8 (1959), 743-806.
  40. Hunziker M., Sepanski M.R., Stanke R.J., Global Lie symmetries of a system of Schrödinger equations and the oscillator representation, Miskolc Math. Notes 14 (2013), 647-657.
  41. Jost J., Riemannian geometry and geometric analysis, 5th ed., Universitext, Springer-Verlag, Berlin, 2008.
  42. Kähler E., Raum-Zeit-Individuum, Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 16 (1992), 115-177.
  43. Kähler E., Mathematische Werke/Mathematical works, Walter de Gruyter & Co., Berlin, 2003.
  44. Kobayashi S., Geometry of bounded domains, Trans. Amer. Math. Soc. 92 (1959), 267-290.
  45. Kobayashi S., Nomizu K., Foundations of differential geometry. Vol. I, Interscience Publishers, New York - London, 1963.
  46. Kobayashi S., Nomizu K., Foundations of differential geometry. Vol. II, Interscience Publishers, New York - London - Sydney, 1969.
  47. Kostant B., Quantization and unitary representations. I. Prequantization, in Lectures in Modern Analysis and Applications, III, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, 87-208.
  48. Kramer P., Saraceno M., Semicoherent states and the group ${\rm ISp}(2,{\bf R})$, Phys. A 114 (1982), 448-453.
  49. Lee M.H., Theta functions on Hermitian symmetric domains and Fock representations, J. Aust. Math. Soc. 74 (2003), 201-234.
  50. Lisiecki W., Coherent state representations. A survey, Rep. Math. Phys. 35 (1995), 327-358.
  51. Loi A., Regular quantizations of Kähler manifolds and constant scalar curvature metrics, J. Geom. Phys. 53 (2005), 354-364.
  52. Loi A., Mossa R., Berezin quantization of homogeneous bounded domains, Geom. Dedicata 161 (2012), 119-128, arXiv:1106.2510.
  53. Loi A., Mossa R., Some remarks on homogeneous Kähler manifolds, Geom. Dedicata 179 (2015), 377-383, arXiv:1502.00011.
  54. Lu Q.-K., On Kähler manifolds with constant curvature, Acta. Math. Sin. 66 (1966), 269-281.
  55. Lu Q.-K., Holomorphic invariant forms of a bounded domain, Sci. China Ser. A 51 (2008), 1945-1964.
  56. Lütkepohl H., Handbook of matrices, John Wiley & Sons, Ltd., Chichester, 1996.
  57. Maass H., Die Differentialgleichungen in der Theorie der Siegelschen Modulfunktionen, Math. Ann. 126 (1953), 44-68.
  58. Mandel L., Wolf E., Optical coherence and quantum optics, Cambridge University Press, Cambridge, 1995.
  59. Marmo G., Michor P.W., Neretin Yu.A., The Lagrangian Radon transform and the Weil representation, J. Fourier Anal. Appl. 20 (2014), 321-361, arXiv:1212.4610.
  60. Molitor M., Gaussian distributions, Jacobi group, and Siegel-Jacobi space, J. Math. Phys. 55 (2014), 122102, 40 pages, arXiv:1409.7917.
  61. Moroianu A., Lectures on Kähler geometry, London Mathematical Society Student Texts, Vol. 69, Cambridge University Press, Cambridge, 2007.
  62. Neeb K.-H., Holomorphy and convexity in Lie theory, de Gruyter Expositions in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2000.
  63. Nomizu K., Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33-65.
  64. Perelomov A., Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986.
  65. Quesne C., Vector coherent state theory of the semidirect sum Lie algebras ${\rm wsp}(2N,{\bf R})$, J. Phys. A: Math. Gen. 23 (1990), 847-862.
  66. Rawnsley J., Cahen M., Gutt S., Quantization of Kähler manifolds. I. Geometric interpretation of Berezin's quantization, J. Geom. Phys. 7 (1990), 45-62.
  67. Rawnsley J.H., Coherent states and Kähler manifolds, Quart. J. Math. Oxford Ser. (2) 28 (1977), 403-415.
  68. Satake I., Algebraic structures of symmetric domains, Kanô Memorial Lectures, Vol. 4, Iwanami Shoten, Tokyo, Princeton University Press, Princeton, N.J., 1980.
  69. Sepanski M.R., Stanke R.J., Global Lie symmetries of the heat and Schrödinger equation, J. Lie Theory 20 (2010), 543-580.
  70. Shuman K.L., Complete signal processing bases and the Jacobi group, J. Math. Anal. Appl. 278 (2003), 203-213.
  71. Siegel C.L., Symplectic geometry, Amer. J. Math. 65 (1943), 1-86.
  72. Takase K., A note on automorphic forms, J. Reine Angew. Math. 409 (1990), 138-171.
  73. Takase K., On unitary representations of Jacobi groups, J. Reine Angew. Math. 430 (1992), 139-149.
  74. Takase K., On Siegel modular forms of half-integral weights and Jacobi forms, Trans. Amer. Math. Soc. 351 (1999), 735-780.
  75. Vinberg E.B., Gindikin S.G., Kähler manifolds admitting a transitive solvable group of automorphisms, Math. USSR Sb. 3 (1967), 333-351.
  76. Yang J.-H., The method of orbits for real Lie groups, Kyungpook Math. J. 42 (2002), 199-272, math.RT/0602056.
  77. Yang J.-H., Invariant metrics and Laplacians on Siegel-Jacobi space, J. Number Theory 127 (2007), 83-102, math.NT/0507215.
  78. Yang J.-H., A partial Cayley transform of Siegel-Jacobi disk, J. Korean Math. Soc. 45 (2008), 781-794, math.NT/0507216.
  79. Yang J.-H., Invariant metrics and Laplacians on Siegel-Jacobi disk, Chin. Ann. Math. Ser. B 31 (2010), 85-100, math.NT/0507217.
  80. Yang J.-H., Yong Y.-H., Huh S.-N., Shin J.-H., Min G.-H., Sectional survatures of the Siegel-Jacobi space, Bull. Korean Math. Soc. 50 (2013), 787-799.

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