Advertisement

Communications in Mathematical Physics

, Volume 280, Issue 2, pp 545–562 | Cite as

Orthosymplectic Lie Superalgebras in Superspace Analogues of Quantum Kepler Problems

  • R. B. ZhangEmail author
Article

Abstract

A Schrödinger type equation on the superspace \(\mathbb {R}^{D|2n}\) is studied, which involves a potential inversely proportional to the negative of the osp(D|2n) invariant “distance” away from the origin. An osp(2, D + 1|2n) dynamical supersymmetry for the system is explicitly constructed, and the bound states of the system are shown to form an irreducible highest weight module for this superalgebra. A thorough understanding of the structure of the irreducible module is obtained. This in particular enables the determination of the energy eigenvalues and the corresponding eigenspaces as well as their respective dimensions.

Keywords

Dynkin Diagram Magnetic Monopole Dynamical Symmetry Irreducible Module Kepler Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Al.
    Al-Jaber S.M. (1998). Hydrogen Atom in n Dimensions. Intl. J. Theor. Phys. 37: 1289–1298 zbMATHCrossRefGoogle Scholar
  2. BB.
    Barut A.O. and Bornzin G.L. (1971). SO(4, 2)-formulation of the symmetry breaking in relativistic Kepler problems with or without magnetic charges. J. Math. Physics 12: 841–846 CrossRefADSMathSciNetGoogle Scholar
  3. D1.
    Delbourgo R. (1988). Grassmann wave functions and intrinsic spin. Intl. J. Mod. Phys. A 3(3): 591–602 CrossRefADSMathSciNetGoogle Scholar
  4. D2.
    Delbourgo R. (2006). The flavour of gravity. J. Phys. A 39(18): 5175–5187 ADSMathSciNetGoogle Scholar
  5. D3.
    Delbourgo R. (2006). Flavour mixing and mass matrices via anticommuting properties. J Phys. A39: 14735–14744 ADSMathSciNetGoogle Scholar
  6. DJW.
    Delbourgo R., Jarvis P.D. and Warner R.C. (1994). Schizosymmetry: a new paradigm for superfield expansions. Mod. Phys. Lett. A 9(25): 2305–2313 ADSMathSciNetGoogle Scholar
  7. EHW.
    Enright, T.J., Howe, R., Wallach, N.R.: A classification of unitary highest weight modules. In: Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math. 40, 97–143, (1983)Google Scholar
  8. I.
    Iwai T. (1990). The geometry of the SU(2) Kepler problem. J. Geom. Phys. 7: 507–535 zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. J.
    Jackobsen H.P. (1983). Hermitian symmetric spaces and their unitary highest weight modules. J. Funct. Anal. 52(3): 385–412 CrossRefMathSciNetGoogle Scholar
  10. JG.
    Jarvis P.D. and Green H.S. (1979). Casimir invariants and characteristic identities for generators of the general linear, special linear and orthosymplectic graded Lie algebras. J. Math. Phys. 20(10): 2115–2122 zbMATHCrossRefADSMathSciNetGoogle Scholar
  11. K.
    Kac V.G. (1977). Lie superalgebras. Adv. in Math. 26(1): 8–96 zbMATHCrossRefGoogle Scholar
  12. KLPW.
    Kirchberg A., Länge J.D., Pisani P.A.G. and Wipf A. (2003). Algebraic solution of the supersymmetric hydrogen atom in d dimensions. Ann. Phys. 303: 359–388 zbMATHCrossRefADSGoogle Scholar
  13. M1.
    Meng, G.: MICZ-Kepler problems in all dimensions. J. Math. Physics, 48(3), 032105, 14p. (2007)Google Scholar
  14. M2.
    Meng, G.: To appearGoogle Scholar
  15. MZ.
    Meng, G., Zhang, R.B.: Generalised MICZ-Kepler problems and unitary highest weight modules. http://arxiv.org/list/math-ph/0702086, 2007
  16. MC.
    McIntosh H.V. and Cisneros A. (1970). Degeneracy in the presence of a magnetic monopole. J. Math. Physics 11: 896–916 CrossRefADSMathSciNetGoogle Scholar
  17. S.
    Scheunert, M.: The theory of Lie superalgebras. An introduction. Lecture Notes in Mathematics 716, Berlin: Springer, 1979Google Scholar
  18. Z.
    Zwanziger D. (1968). Exactly Soluble Nonrelativistic Model of Particles with Both Electric and Magnetic Charges. Phys. Rev. 176: 1480–1488CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

Personalised recommendations