Skip to main content
SpringerLink
Log in

Supercoherent states, super-Kähler geometry and geometric quantization

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Generalized coherent states provide a means of connecting square integrable representations of a semi-simple Lie group with the symplectic geometry of some of its homogeneous spaces. In the first part of the present work this point of view is extended to the supersymmetric context through the study of the OSp(2/2) coherent states. These are explicitly constructed starting from the known abstract typical and atypical representations of osp(2/2). Their underlying geometries turn out to be those of supersymplectic OSp(2/2)-homogeneous spaces. Moment maps identifying the latter with coadjoint orbits of OSp(2/2) are exhibited via Berezin's symbols. When considered within Rothstein's general paradigm, these results lead to a natural general definition of a super-Kähler supermanifold, the supergeometry of which is determined in terms of the usual geometry of holomorphic Hermitian vector bundles over Kähler manifolds. In particular, the supergeometry of the above orbits is interpreted in terms of the geometry of Einstein-Hermitian vector bundles. In the second part, an extension of the full geometric quantization procedure is applied to the same coadjoint orbits. Thanks to the super-Kähler character of the latter, this procedure leads to explicit super-unitary irreducible representations of osp(2/2) in super-Hilbert spaces of superholomorphic square-integrable sections of prequantum bundles of the Kostant type. This work lays the foundations of a program aimed at classifying Lie supergroups' coadjoint orbits and their associated irreducible representations, ultimately leading to harmonic superanalysis. For this purpose a set of consistent conventions is exhibited.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Schrödinger, E.: Der stetige übergang von der Mikro-zur Makromechanik. Naturwissenschaften14, 664 (1926)

    Article  Google Scholar 

  2. Glauber, R.J.: The quantum theory of optical coherence. Phys. Rev.130, 2529 (1963); Coherent and incoherent states of the radiation field. Phys. Rev.131, 2766 (1963)

    Article  Google Scholar 

  3. Perelomov, A.: Generalized Coherent States and Their Applications. Berlin, Heidelberg, New York: Springer, 1986

    Google Scholar 

  4. Klauder, J.R., Skagerstam, B.-S.: Coherent States—Applications in Physics and Mathematical Physics. Singapore: World Scientific, 1985

    Google Scholar 

  5. Daubechies, I.: Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics61, Philadelphia: SIAM, 1992

    Google Scholar 

  6. Berezin, F.A.: General concept of quantization. Commun. Math. Phys.40, 153 (1975)

    Article  Google Scholar 

  7. Onofri, E.: A note on coherent state representations of Lie groups. J. Math. Phys.16, 1087 (1975)

    Article  Google Scholar 

  8. El Gradechi, A.M.: On the supersymplectic homogeneous superspace underlying the OSp(1/2) coherent states. J. Math. Phys.34, 5951 (1993)

    Article  Google Scholar 

  9. El Gradechi, A.M.: Geometric quantization of an OSp(1/2) coadjoint orbit. Lett. Math. Phys.35, 13 (1995)

    Article  Google Scholar 

  10. Zhang, W.M., Feng, D.H., Gilmore, R.: Coherent states: Theory and some applications. Rev. Mod. Phys.26, 867 (1990)

    Article  Google Scholar 

  11. Delbourgo, R.: Minimal uncertainty states for the rotatin and allied groups. J. Phys. A: Math. Gen.10, 1837 (1977); Delbourgo, R., Fox, J.R.: Maximum weight vectors possess minimal uncertainty. J. Phys. A: Math. Gen.10, L235 (1977)

    Article  Google Scholar 

  12. De Bièvre, S., El Gradechi, A.M.: Quantum mechanics and coherent states on the anti-de Sitter spacetime and their Poincaré contraction. Ann. Inst. H. Poincaré57, 403 (1992)

    Google Scholar 

  13. Radcliffe, J.M.: Some properties of spin coherent states. J. Phys. A: Gen. Phys.4, 313 (1971)

    Article  Google Scholar 

  14. Ali, S.T., Antoine, J.-P.: Coherent states of the 1+1-dimensional Poincaré group: Square integrability and a relativistic Weyl transform. Ann. Inst. H. Poincaré51, 23 (1989); Ali, S.T., Antoine, J.-P., Gazeau, J.-P.: De Sitter to Poincaré contraction and relativistic coherent states. Ann. Inst. H. Poincaré52, 83 (1990)

    Google Scholar 

  15. Ali, S.T., Antoine, J.-P., Gazeau, J.-P.: Square integrability of group, representations on homogeneous spaces. I. Reproducing triples and frames. Ann. Inst. H. Poincaré55, 829 (1991); II. Coherent and quasi-coherent states. The case of the Poincaré group. Ann. Inst. H. Poincaré55, 857 (1991)

    Google Scholar 

  16. Kirillov, A.: Eléments de la théorie des représentations. Moscou: Editions Mir, 1974; The method of orbits in representation theory. In: Lie Groups and Their Representations. Gelfand, I.M. (ed.), Budapest: Akadémiai Kiadó, 1975, p. 219

    Google Scholar 

  17. Kostant, B.: Quantization and unitary representation In: Lecture Notes in Mathematics170, Berlin, Heidelberg, New York: Springer, 1970, p. 87

    Google Scholar 

  18. Souriau, J.M.: Structure des systèmes dynamiques. Paris: Dunod, 1970

    Google Scholar 

  19. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. Ann. Phys. (N.Y.)111, 61 (1978)

    Article  Google Scholar 

  20. Freund, P.G.O.: Introduction to Supersymmetry. Cambridge: Cambridge University Press, 1986

    Google Scholar 

  21. Berezin, F.A.: The Method of Second Quantization. New York: Academic Press 1966

    Google Scholar 

  22. Berezin, F.A.: Introduction to Superanalysis. Dordrecht: Reidel, 1987; The mathematical basis of supersymmetric field theories. Soviet J. Nucl. Phys.29, 857 (1979)

    Google Scholar 

  23. Cornwell, J.F.: Group Theory in Physics Vol. 3-Supersymmetries and Infinite Dimensional Algebras. London: Academic Press 1989

    Google Scholar 

  24. Scheunert, M.: The Theory of Lie Superalgebras—An Introduction. Lecture Notes in Mathematics716, Berlin, Heidelberg, New York: Springer, 1979

    Google Scholar 

  25. Kac, V.: Representations of classical Lie superalgebras. In: Lecture Notes in Mathematics676, Berlin: Springer-Verlag, 1978, p. 597

    Google Scholar 

  26. Leites, D.A.: Introduction to the theory of supermanifolds. Russ. Math. Surv.35, 1 (1980)

    Google Scholar 

  27. DeWitt, B.S.: Supermanifolds. 2nd Edition, Cambridge: Cambridge, University Press, 1992

    Google Scholar 

  28. Kostant, B.: Graded manifolds, graded Lie theory and prequantization. In: Conference on Differential Geometric Methods in Mathematical Physics. Bleuler, K., Reetz, A.: eds., Lecture Notes in Mathematics570, Berlin, Heidelberg, New York: Springer 1977, p. 177

    Google Scholar 

  29. Manin, Y.I.: Gauge Field Theory and Complex Geometry. Berlin: Springer, 1988

    Google Scholar 

  30. Bartocci, C., Bruzzo, U., Hernández-Ruipérez, D.: The Geometry of Supermanifolds. Dordrecht: Kluwer Academic Publishers, 1991

    Google Scholar 

  31. Tuynman, G.M.: Supermanifolds: A Geometric Approach. Book in preparation

  32. Scheunert, M., Nahm, W., Rittenberg, V.: Irreducible representations of the osp(2, 1) and spl(2, 1) graded Lie algebras. J. Math. Phys.18, 155 (1977)

    Article  Google Scholar 

  33. Nishiyama, K.: Characters and super-characters of discrete series representations for orthosymplectic Lie superalgebras. J. Algebra141, 399 (1991)

    Article  Google Scholar 

  34. Balantekin, A.B., Schmitt, H.A., Barrett, B.R.: Coherent states for the harmonic oscillator representations of the orthosymplectic supergroup OSp(1/2N, R). J. Math. Phys.29, 1634 (1988)

    Article  Google Scholar 

  35. Balantekin, A.B., Schmitt, H.A., Halse, P.: Coherent states for the noncompact supergroups OSp(2/2N, R). J. Math. Phys.30, 274 (1989)

    Article  Google Scholar 

  36. Fatyga, B.W., Kostelecký, V.A., Nieto, M.M., Truax, D.R.: Supercoherent states. Phys. Rev.D43, 1403 (1991)

    Article  Google Scholar 

  37. Duval, C., Horváthy, P.A.: On Schrödinger superalgebras. J. Math. Phys.35, 2516 (1994)

    Article  Google Scholar 

  38. Rothstein, M.: The structure of supersymplectic supermanifolds. In: Differential Geometric Methods in Mathematical Physics, Bartocci, C., Bruzzo, U., Cianci, R. eds., Lecture Notes in Physics375, Berlin, Heidelberg, New York: Springer, 1991, p. 331

    Google Scholar 

  39. Monterde, J.: A characterization of graded symplectic structures. Differential Geom. Appl.2, 81 (1992)

    Article  Google Scholar 

  40. Frappat, L., Sciarrino, A., Sorba, P.: Structure of basic Lie superalgebras and of their affine extensions. Commun. Math. Phys.121, 457 (1989), and references therein

    Article  Google Scholar 

  41. Sternberg, S., Wolf, J.: Hermitian Lie algebras and metaplectic representations. Trans. Am. Math. Soc.238, 1 (1978)

    Google Scholar 

  42. Green, P.: On holomorphic graded manifolds. Proc. Am. Math. Soc.85, 587 (1982)

    Google Scholar 

  43. Rothstein, M.: Deformations of complex supermanifolds. Proc. Am. Math. Soc.95, 255 (1985)

    Google Scholar 

  44. Batchelor, M.: The structure of supermanifolds. Trans. Am. Math. Soc.253, 329 (1979)

    Google Scholar 

  45. Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Princeton, NJ: Princeton University Press, 1987

    Google Scholar 

  46. Woodhouse, N.M.J.: Geometric Quantization. Oxford: Clarendon Press, 1980

    Google Scholar 

  47. Kirillov, A.A.: Geometric quantization. In: Dynamical Systems IV, Arnol'd, V.I., Novikov, S.P. eds., Berlin, Heidelberg, New York: Springer, 1990, p. 137

    Google Scholar 

  48. Tuynman, G.M.: Geometric quantization of the BRST charge. Commun. Math. Phys.150, 237 (1992)

    Article  Google Scholar 

  49. Schaller, P., Schwarz, G.: Anomalies from geometric quantization of fermionic field theories. J. Math. Phys.31, 2366 (1990)

    Article  Google Scholar 

  50. Borthwick, D., Klimek, S., Lesniewski, A., Rinaldi, M.: Super Toeplitz operators and nonperturbative quantization of supermanifolds. Commun. Math. Phys.153, 49 (1993)

    Article  Google Scholar 

  51. Tuynman, G.M.: Quantization: Towards a comparison between methods. J. Math. Phys.28 2829 (1987)

    Article  Google Scholar 

  52. Renouard, P.: Variétés symplectiques et quantification. Thèse, Universié d'Orsay, 1969

  53. Schmitt, T.: Supergeometry and hermitian conjugation. J. Geom. Phys.7, 141 (1990)

    Article  Google Scholar 

  54. Nagamachi, S., Kobayashi, Y.: Hilbert superspace. J. Math. Phys.33, 4274 (1992)

    Article  Google Scholar 

  55. El Gradechi, A.M.: On the super-unitarity of discrete series representations of orthosymplectic Lie superalgebras. Preprint CRM-2279. Submitted

  56. Beckers, J., Gagnon, L., Hussin, V., Winternitz, P.: Superposition formulas for nonlinear superequations. J. Math. Phys.31, 2528 (1990)

    Article  Google Scholar 

Download references

Author information

Author notes

  1. Luis M. Nieto

    Present address: Dpto. de Física Teórica, Universidad de Valladolid, 47011, Valladolid, Spain

Authors and Affiliations

  1. Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128-centre-ville, H3C 3J7, Montréal, (Québec), Canada

    Amine M. El Gradechi & Luis M. Nieto

  2. Department of Mathematics and Statistics, Concordia University, H4B 1R6, Montréal, (Québec), Canada

    Amine M. El Gradechi

Authors
  1. Amine M. El Gradechi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luis M. Nieto
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by S.-T. Yau

Address after October 1, 1995: Faculté des sciences “Jean Perrin”, Université d'Artois, rue Jean Souvraz, S.P. 18, F-62307 Lens, France. E-mail: amine@gat.univ-lillel.fr.

Rights and permissions

Reprints and permissions

About this article

Cite this article

El Gradechi, A.M., Nieto, L.M. Supercoherent states, super-Kähler geometry and geometric quantization. Commun.Math. Phys. 175, 521–563 (1996). https://doi.org/10.1007/BF02099508

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02099508

Keywords

Search

Navigation