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Structural stability of the phase transition in Dicke-like models
The free energy for a general class of Dicke models is computed and expressed simply as the minimum value of a potential function = (E−TS)/N. The function E/N is the image of the Hamiltonian un...

Lie theory and the wave equation in space–time. I. The Lorentz group

J. Math. Phys. 18, 1 (1977); doi:10.1063/1.523130

Issue Date: January 1977

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E. G. Kalnins
Mathematics Department, University of Waikato, Hamilton, New Zealand

W. Miller, Jr.
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
In this article we begin a study of the relationship between separation of variables and the conformal symmetry group of the wave equation psitt −Delta3psi=0 in space–time. In this first article we make a detailed study of separation of variables for the Laplace operator on the one and two sheeted hyperboloids in Minkowski space. We then restrict ourselves to homogeneous solutions of the wave equation and the Lorentz subgroup SO(3,1) of the conformal group SO(4,2). We study the various separable bases by using the methods of integral geometry as developed by Gel'fand and Graev. In most cases we give the spectral analysis for these bases, and a number of new bases are developed in detail. Many of the special function identities derived appear to be new. This preliminary study is of importance when we subsequently study models of the Hilbert space structure for solutions of the wave equation and the Klein–Gordon equation psitt−Delta3psi=lambdapsi. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
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PACS

  • 02.20.Qs
    Mathematical methods in physics Group theory General properties, structure, and representation of Lie groups
  • 03.65.Ge
    Classical and quantum physics; mechanics and fields Quantum theory; quantum mechanics Solutions of wave equations: bound states
  • YEAR: 1977

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
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REFERENCES (21)
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  1. W. Miller Jr., “Symmetry, separation of variables and special functions” in Theory and Application of Special Functions, edited by R. Askey (Academic, New York, 1975).
  2. E. G. Kalnins and W. Miller Jr., J. Math. Phys. 17, 356 (1976).
  3. E. G. Kalnins and W. Miller Jr., J. Math. Phys. 17, 369 (1976).
  4. P. M. Olevski, Mat. Sb. 27, 379–426 (1950).
  5. M. A. Naimark, Linear Representations of the Lorentz Group (Pergamon, New York, 1964).
  6. I. M. Gel'fand, R. A. Minlos, and Z. Ya. Shapiro, Rotation and Lorentz Groups and Their Applications (Pergamon, New York, 1963).
  7. I. M. Gel'fand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions (Academic, New York, 1966), Vol. 5.
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    11. [Sov. Phys. JETP 23, 434–36 (1966)].
  11. V. Bargmann, Ann. of Math. 48, 568–640 (1947).
  12. J. Patera and P. Winternitz, J. Math. Phys. 14, 1130–39 (1973).
  13. E. G. Kalnins and W. Miller Jr., J. Math. Phys. 15, 1263–74 (1974).
  14. These functions are defined in A. Erdélyi et al., Higher Transcendental Functions (McGraw-Hill, New York, 1955), Vol. 3.
  15. E. G. Kalnins, W. Miller Jr., and P. Winternitz, SIAM J. App. Math. 30, 630–64 (1976).
  16. N. Ya. Vilenkin and Ya. A. Smorodinski, Zh. Eksp. Teor. Fiz. 46, 1793 (1964)
    17. [Sov. Phys. JETP 46, 1209 (1964)].
  17. E. G. Kalnins, “Subgroup reductions of the Lorentz group,” Ph.D. thesis, University of Western Ontario (1972).
  18. W. Miller Jr., Symmetry and separation of variables for linear differential equations (Addison-Wesley, Reading, Mass., to appear).
  19. E. G. Kalnins, J. Math. Phys. 14, 1316–19 (1973).
  20. J. Meixner and F. W. Schafke, Mathieushe Funktionen und Spharoidfunktionen (Springer, Berlin, 1965).
  21. N. Ya. Vilenkin, Mat. U.S.S.R. Sb. 3, 109, 21 (1967).

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