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

## 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

- [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 - [D]
- [EW]T. Enright and N. Wallach, Embeddings of unitary highest weight representations and generalized Dirac operators,
*Math. Ann.***307**(1997), 627–646.CrossRefMathSciNetzbMATHGoogle Scholar - [GV]I. M. Gelfand and N. Y. Vilenkin,
*Generalized Functions Vol. 4: Applications of Harmonic Analysis*, Academic Press, 1964.Google Scholar - [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 - [GW]B. Gross and N. Wallach, On quaternionic discrete series representations, and their continuations,
*J. Reine Angew. Math.*,**481**(1996), 73–123.MathSciNetzbMATHGoogle Scholar - [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 - [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,*ℝ*),*Represent. Theory***4**(2000), 181–224.Google Scholar - [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 - [P]Roger Penrose,
*Cycles of Time*, Alfred Knopf, New York, 2011.Google Scholar - [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 - [S1]Irving Segal, Radiation in the Einstein universe and the cosmic background,
*Physical Review D***28**(1983), 2393–2402.Google Scholar - [S2]Irving Segal, Geometric derivation of the chronometric red shift,
*Proc. Nat. Acad. Sci.***90**(1993), 11114–11116.Google Scholar - [V]M. Vergne, Some comments on the work of I. E. Segal in group representations,
*J. Funct. Anal.***190**(2002), 29–37.Google Scholar - [VZ]D. Vogan and G. Zuckerman, Unitary representations and continuous cohomology,
*Comp/ Math.***53**(1984), 51–90.Google Scholar - [W1]Nolan R. Wallach,
*Real Reductive Groups I,II*, Academic Press, New York, 1988, 1992.Google Scholar - [W2]N. Wallach, Generalized Whittaker vectors for holomorphic and quaternionic representations,
*Comment. Math. Helv.***78**(2003), 266–307.Google Scholar - [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