On discrete degenerate representations of the conformal group in space-time

Abstract

The homomorphism between the conformal group in space-time and the noncompact groupSU2.2 is used to derive three new types of unitary irreducible representations of the conformal group. These representations are realized in Hilbert spaces of the so-called harmonic functions. two of them are equivalent (vector) representations, while the third is a projective (spinor) representation. All three are degenerate and of the discrete-series type. The generators of the physical operations of the conformal group are realized as differential operators—some of them explicitly in terms of the variables of the representation space. Matrices of the translation and dilatation operators are calculated between certain harmonic-function states. The dilatation operator is found to connect only states whose values of a certain parameterm differ by two or zero. It is suggested that, should degenerate discrete-series representations of the type described be relevant to the correct classification of fundamental particles, a simple explanation of the experimentally observed discrete mass spectra might be found in this property of the dilatation operator.