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Lie theory and the wave equation in space–time. 5. R-separable solutions of the wave equation psitt−Delta3psi=0
A detailed classification is made of all orthogonal coordinate systems for which the wave equation tt−3=0 admits an R-separable solution. Only those coordinate systems are given which are not co...

Lie theory and the wave equation in space–time. 4. The Klein–Gordon equation and the Poincaré group

J. Math. Phys. 19, 1233 (1978); doi:10.1063/1.523819

Issue Date: June 1978

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E. G. Kalnins and W. Miller, Jr.
Mathematics Department, University of Waikato, Hamilton, New Zealand
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
A detailed classification is made of all orthogonal coordinate systems for which the Klein–Gordon equation in space–time, psitt−Delta3psi=lambdapsi, admits a separation of variables. We show that the Klein–Gordon equation is separable in 261 orthogonal coordinate systems. In each case the coordinate systems presented are characterized in terms of three symmetric second order commuting operators in the enveloping algebra of the Poincaré group. This paper also consitutes an important step in the study of separation of variables for the wave equation in space–time psitt−Delta3psi=0, and its relation to the underlying conformal symmetry group O(4,2) of this equation. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
Permalink: http://link.aip.org/link/?JMAPAQ/19/1233/1
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PACS

  • 03.65.Ge
    Classical and quantum physics; mechanics and fields Quantum theory; quantum mechanics Solutions of wave equations: bound states
  • 03.65.Fd
    Classical and quantum physics; mechanics and fields Quantum theory; quantum mechanics Algebraic methods
  • 02.20.Rt
    Mathematical methods in physics Group theory Discrete subgroups of Lie groups
  • YEAR: 1978

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
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REFERENCES (14)
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  1. E. G. Kalnins and W. Miller, Jr., “Lie theory and the wave equation in space-time. 1. The Lorentz group,” J. Math. Phys. 18, 1 (1977).
  2. E. G. Kalnins and W. Miller, Jr., “Lie theory and the wave equation in space-time. 2. The group SO(4, [script C]),” SIAM J. Math. Anal. (to appear).
  3. E. G. Kalnins and W. Miller, Jr., “Lie theory and the wave equation in space-time. 3. Semisubgroup coordinates,” J. Math. Phys. 18, 271 (1977).
  4. L. P. Eisenhart, “Separable systems of Stackel,” Ann. Math. 35, 284—305 (1934).
  5. S. Schweber, Relativistic Quantum Field Theory (Harper, New York, 1961).
  6. P. Winternitz, “The Poincare group, its little groups and their applications in particle physics,” Preprint RPP-T3, Rutherford Laboratory, England (1969).
  7. H. Joos, “Zur darstellungs theorie der inhomogenen Lorentz gruppe als Grundlage quantenmechanischer Kinematik,” Fortschr. Phys. 10, 65—146 (1962).
  8. E. P. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Ann. Math. 40, 149—204 (1939).
  9. E. G. Kalnins and W. Miller, Jr., “Lie theory and the wave equation in space-time. 4,” University of Waikato Research Report, No. 37 (1976).
  10. W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, Reading, Mass., 1977).
  11. J. Meixner and F. W. Schafke, Mathieusche Funktionen and Spharoidfunktionen (Springer-Verlag, Berlin, 1954).
  12. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966).
  13. E. G. Kalnins and W. Miller, Jr., “Lie theory and separation of variables. 9. Orthogonal R-separable coordinate systems for the wave equation Psitt−Delta2Psi = 0,”J. Math. Phys. 17, 331—355 (1976).
  14. C. P. Boyer, E. G. Kalnins, and W. Miller, Jr., “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J. 60, 35—80 (1976).

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