Highest weight unitary modules for non-compact groups and applications to physical problems

TL;DR

This paper reviews the classification of highest weight unitary modules for non-compact Lie groups, compares different methods, and explores applications to physics, including conformal and de Sitter algebras.

Contribution

It unifies Jakobsen's and Enright-Howe-Wallach's methods for unitarity classification and extends the analysis to new unitarity regions and physical applications.

Findings

Unified classification of unitary modules using multiple methods

Identification of new unitarity regions for non-compact groups

Applications to conformal and de Sitter space wave equations

Abstract

The study of unitarization of representations for non compact real forms of simple Lie Algebras has been achieved in the past decade by Jakobsen (JA81, JA83) and by Enright, Howe and Wallach (EH83) following different paths but arriving at the same final results. In order to discuss unitarity we need to introduce a scalar product. This is done in sections II and III introducing a sesquilinear form on the universal enveloping algebra (GL90). Such a sesquilinear form was introduced by Harish-Chandra (HC55), Gel'fand and Kirillov (GK69) and Shapovalov (SH72). The new developments given to Jakobsen method in GEL90 are contained in sections IV and V. In section VI we summarize the principal results due to Enright Howe and Wallach (EHW method). In section VII we give the possible places for unitarity including those for which the reduction level can't be higher than one and that were not considered in GEL90. We see also in this section and in a explicit way how the Jakobsen method and the EHW method give the same final results. This type of representations have found applications in physics for a long time (see GK75, 82; GL83, 89; LO86, 89 and references contained therein) In sections VIII and IX we consider the conformal and de Sitter algebras. In particular the construction of wave equations for conformal multispinors and the wave equations in de Sitter space are reviewed in sections X and XI. In the Appendix we give the representations of the classical algebras in the subspace Omega extending the results obtained in LG84 for the algebra A_l.