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Action of the conformal group on steady state solutions to Maxwell’s equations and background radiation

  • Bertram KostantEmail author
  • Nolan R. Wallach
Chapter
Part of the Progress in Mathematics book series (PM, volume 257)

Abstract

The representation of the conformal group (PSU(2, 2)) on the space of solutions to Maxwell’s equations on the conformal compactification of Minkowski space is shown to break up into four irreducible unitarizable smooth Fréchet representations of moderate growth. An explicit inner product is defined on each representation. The energy spectrum of each of these representations is studied and related to plane wave solutions. The steady state solutions whose luminosity (energy) satisfies Planck’s Black Body Radiation Law are described in terms of this analysis. The unitary representations have notable properties. In particular they have positive or negative energy, they are of type \(A_{\mathfrak{q}}(\lambda )\) and are quaternionic. Physical implications of the results are explained.

Keywords

Maxwell’s equations Conformal compactification Conformal group Unitary representations Background radiation 

Mathematics Subject Classification

78A25 58Z05 22E70 22E45 

References

  1. [BW]
    A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Second edition. Mathematical Surveys and Monographs, 67. American Mathematical Society, Providence, RI, 2000.Google Scholar
  2. [D]
    P. A. M. Dirac, Wave equations in conformal space, Ann. of Math. 37 (1936), 429–442.Google Scholar
  3. [EW]
    T. Enright and N. Wallach, Embeddings of unitary highest weight representations and generalized Dirac operators, Math. Ann. 307 (1997), 627–646.CrossRefMathSciNetzbMATHGoogle Scholar
  4. [GV]
    I. M. Gelfand and N. Y. Vilenkin, Generalized Functions Vol. 4: Applications of Harmonic Analysis, Academic Press, 1964.Google Scholar
  5. [HBN]
    F. Hoyle, G. Burbidge, J. V. Narlikar, A quasi-steady state cosmological model with creation of matter, The Astrophysical Journal 410 (1993), 437–457.Google Scholar
  6. [GW]
    B. Gross and N. Wallach, On quaternionic discrete series representations, and their continuations, J. Reine Angew. Math., 481(1996), 73–123.MathSciNetzbMATHGoogle Scholar
  7. [HSS]
    M. Hunziker, M. Sepanski, and R. Stanke, The minimal representation of the conformal group and classical solutions to the wave equation, J. Lie Theory 22 (2012), 301–360.Google Scholar
  8. [K]
    B. Kostant, On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected covering group of SL(2, ), Represent. Theory 4 (2000), 181–224.Google Scholar
  9. [PS]
    Stephen M. Paneitz, Irving E. Segal, Analysis in space-time bundles. I. General considerations and the scalar bundle. J. Funct. Anal. 47 (1982), no. 1, 78–142.Google Scholar
  10. [P]
    Roger Penrose, Cycles of Time, Alfred Knopf, New York, 2011.Google Scholar
  11. [SJOPS]
    I. E. Segal, H. P. Jakobsen, B. Ørsted, S. M. Paneitz, B. Speh, Covariant chronogeometry and extreme distances: elementary particles. Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 9, part 1, 5261–5265.Google Scholar
  12. [S1]
    Irving Segal, Radiation in the Einstein universe and the cosmic background, Physical Review D 28(1983), 2393–2402.Google Scholar
  13. [S2]
    Irving Segal, Geometric derivation of the chronometric red shift, Proc. Nat. Acad. Sci. 90 (1993), 11114–11116.Google Scholar
  14. [V]
    M. Vergne, Some comments on the work of I. E. Segal in group representations, J. Funct. Anal. 190 (2002), 29–37.Google Scholar
  15. [VZ]
    D. Vogan and G. Zuckerman, Unitary representations and continuous cohomology, Comp/ Math. 53 (1984), 51–90.Google Scholar
  16. [W1]
    Nolan R. Wallach, Real Reductive Groups I,II, Academic Press, New York, 1988, 1992.Google Scholar
  17. [W2]
    N. Wallach, Generalized Whittaker vectors for holomorphic and quaternionic representations, Comment. Math. Helv. 78 (2003), 266–307.Google Scholar
  18. [W3]
    N. R. Wallach, Lie algebra cohomology and holomorphic continuation of generalized Jacquet integrals, Advanced Studies in Pure Math. 14 (1988), 123–151.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsMITCambridgeUSA
  2. 2.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

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