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


SIGMA 3 (2007), 055, 84 pages      math-ph/0703080      https://doi.org/10.3842/SIGMA.2007.055

Eigenfunction Expansions of Functions Describing Systems with Symmetries

Ivan Kachuryk a and Anatoliy Klimyk b
a) Khmel'nyts'kyy National University, Khmel'nyts'kyy, Ukraine
b) Bogolyubov Institute for Theoretical Physics, 14-b Metrologichna Str., Kyiv-143, 03143 Ukraine

Received March 02, 2007; Published online March 28, 2007

Abstract
Physical systems with symmetries are described by functions containing kinematical and dynamical parts. We consider the case when kinematical symmetries are described by a noncompact semisimple real Lie group G. Then separation of kinematical parts in the functions is fulfilled by means of harmonic analysis related to the group G. This separation depends on choice of a coordinate system on the space where a physical system exists. In the paper we review how coordinate systems can be chosen and how the corresponding harmonic analysis can be done. In the first part we consider in detail the case when G is the de Sitter group SO0(1,4). In the second part we show how the corresponding theory can be developed for any noncompact semisimple real Lie group.

Key words: representations; eigenfunction expansion; special functions; de Sitter group; semisimple Lie group; coordinate systems; invariant operators.

pdf (751 kb)   ps (434 kb)   tex (71 kb)

References

  1. Fushchych W.I., Nikitin A.G., Symmetries of equations of quantum mechanics, Allerton Press, New York, 1994.
  2. Sitenko G.A., Scattering theory, Springer, Berlin, 1985.
  3. Vilenkin N.Ja., Klimyk A.U., Group representation and special functions, Vol. II, Kluwer, Dordrecht, 1993.
  4. Nikiforov A.F., Suslov S.K., Uvarov V.B., Classical orthogonal polynomials of discrete variables, Nauka, Moscow, 1985 (in Russian).
  5. Smirnov Yu.F., Shitikov K.V., K-harmonic method and the model of shells, Sov. J. Part. Nucl. 8 (1977), 847-910.
  6. Filippov G.F., Ovcharenko V.I., Smirnov Yu.F., Microscopic theory of collective exitations of atomic nuclears, Naukova Dumka, Kiev, 1981 (in Russian).
  7. Vilenkin N.Ja., Klimyk A.U., Group representation and special functions, Vol. I, Kluwer, Dordrecht, 1991.
  8. Basu D., Wolf K.B., The unitary irreducible representations of SL(2,R) in all subgroup reductions, J. Math. Phys. 23 (1982), 189-208.
  9. Vilenkin N.Ja., Smorodinsky Ja.A., Invariant expansion of relativistic amplitudes, Soviet Physics JETP 19 (1964), 1209-1218.
  10. Winternitz P., Smorodinsky Yu.A., Sheftel M.R., Poincaré and Lorentz invariant expansions of relativistic amplitudes, Yadernaya Fizika 7 (1968), 1325-1338.
  11. Winternitz P., Two variable expansions based on the Lorentz and conformal groups, in Lectures in Theoretical Physics, Editors A.O. Barut and W.E. Brittin, Colorado University Press, Boulder, 1971.
  12. Kalnins E.G., Patera J., Sharp R.T., Winternitz P., Two variable Galilei-group expansion of nonrelativistic scattering amplitudes, Phys. Rev. D 8 (1973), 2552-2572.
  13. Winternitz P., Poincaré groups, its little subgroups and their applications in particle physics, Lectures at the Summer School on Group Representations and Quantum Theory, Dublin, 1969.
  14. Winternitz P., Lucac I., Smorodinsky Yu.A., Quantum numbers in the little groups of the Poincaré group, Yadernaya Fizika 7 (1968), 192-201.
  15. Warner G., Harmonic analysis on semisimple Lie groups, Vols. 1, 2, Springer, Berlin, 1972.
  16. Helgason S., Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York, 1978.
  17. Klimyk A.U., Kachurik I.I., Computation methods in theory of group representations, Vyshcha Shkola, Kiev, 1986 (in Russian).
  18. Vilenkin N.Ja., Klimyk A.U., Group representation and special functions, Vol. III, Kluwer, Dordrecht, 1992.
  19. Vilenkin N.Ja., Klimyk A.U., Group representation and special functions. Recent advances, Kluwer, Dordrecht, 1995.
  20. Erdelyi A., Magnus W., Oberheittinger F., Tricomi F., Higher transcendental functions, Vol. I, McGraw-Hill, New York, 1953.
  21. Erdelyi A., Magnus W., Oberheittinger F., Tricomi F., Higher transcendental functions, Vol. II, McGraw-Hill, New York, 1954.
  22. Boyer C.P., Matrix elements for the most degenerate principal series of representations of SO0(p,1), J. Math. Phys. 12 (1971), 1599-1603.
  23. Varlamov V.V., Spherical functions on the de Sitter group, J. Phys. A: Math. Teor. 40 (2007), 163-201, math-ph/0604026.
  24. Gel'fand I.M., Graev M.I., Applications of the method of orispheres to spectral analysis of functions on real and imaginary Lobachevsky spaces, Trudy Moscow Math. Soc. 11 (1962), 243-308.
  25. Vilenkin N.Ja., Special functions and the theory of group representations, Amer. Math. Soc., Providence, RI, 1968.
  26. Thomas L.H., On unitary representations of the group of de Sitter space, Ann. Math. 51 (1941), 113-126.
  27. Newton T.D., A note on the representations of the de Sitter group, Ann. Math. 60 (1950), 730-733.
  28. Dixmier J., Representations integrable du groupe de De Sitter, Bull. Soc. Math. France 89 (1961), 9-41.
  29. Takahashi R., Sur les representations unitaire des groupes de Lorentz generalises, Bull. Soc. Math. France 91 (1963), 289-433.
  30. Hirai T., On irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad. 38 (1962), 258-262.
  31. Gurseiy F., Group theoretical concepts and methods in elementary particle physics, Gordon and Breach, New York, 1964.
  32. Fronsdal C., Elementary particles in a curved space, Rev. Mod. Phys. 37 (1965), 221-224.
  33. Malkin I.A., Man'ko V.I., Dynamical symmetries and coherent states of quantum systems, Nauka, Moscow, 1979 (in Russian).
  34. Barut A.O., Bohm A., Dynamical groups and mass formulas, Phys. Rev. B 139 (1965), 1107-1112.
  35. Kachuryk I.I., Invariant expansions of solutions of 5-dimensional Klein-Gordon equation, Ukrainian Phys. J. 21 (1976), 1853-1862
  36. Hirai T., The Plancherel formula for the Lorentz group of n-th order, Proc. Japan Acad. 42 (1966), 323-326.
  37. Limic N., Niederle J., Raczka R., Continuous degenerate representations of noncompact rotation groups, J. Math. Phys. 7 (1966), 2026-2035.
  38. Maurin K., General eigenfunction expansions and unitary representations of topological groups, PWN, Warszawa, 1968.
  39. Gel'fand I.M., Graev M.I., Vilenkin N.Ja., Integral geometry and related problems in the theory of representations, Vol. 5, Generalized functions, Academic Press, New York, 1966.
  40. Gradshtein I.S., Ryzhik I.M., Table of integrals, series and products, Academic Press, New York, 1980.
  41. Ablamowitz M., Stegun I.A. (Editors), Handbook on special functions, National Bureau of Standards, New York, 1964.
  42. Zhelobenko D.P., Compact Lie groups and their representations, Amer. Math. Soc., Providence, RI, 1973.
  43. Knapp A.W., Representation theory of semisimple Lie groups. An overview based on examples, Princeton Univ. Press, Princeton, NJ, 1986.
  44. Klimyk A.U., Representation matrix elements and Clebsch-Gordan coefficients, Naukova Dumka, Kiev, 1979 (in Russian).
  45. Berger M., Les espaces symmetriques non compacts, Ann. Ecole Norm. Sup. 74 (1957), 85-177.
  46. Oshima T., Sekiguchi J., Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980), 1-81.
  47. Muracami S., Sur la classification des algebres de Lie reelles et simples, Osaka J. Math. 2 (1965), 291-307.
  48. Bruhat F., Sur les representations induites des groupes de Lie, Bull. Soc. Math. France 84 (1956), 97-207.
  49. Matsuki T., The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Jap. 31 (1981), 331-357.
  50. Helgason S., A duality for symmetric spaces, with applications to group representations, Adv. Math. 5 (1970), 1-154.
  51. Helgason S., Geometric analysis on symmetric spaces, Amer. Math. Soc., Providence, RI, 1994.
  52. Harish-Chandra, Harmonic analysis on the real reductive groups, Ann. Math. 104 (1976), 117-201.

Previous article   Next article   Contents of Volume 3 (2007)