Dualities and the Unitary Representations of Non - compact Groups and Supergroups : Wigner versus Dirac

I review the relationship between AdS/CFT ( anti-de Sitter / conformal field theory) dualities and the general theory of positive energy unitary representations of non-compact space-time groups and supergroups. I show , in particular, how one can go from the manifestly unitary compact basis of the lowest weight ( positive energy) representations of the conformal group ( Wigner picture) to the manifestly covariant coherent state basis ( Dirac picture). The coherent states labelled by the space-time coordinates correspond to covariant fields with a definite conformal dimension. These results extend to higher dimensional Minkowskian spacetimes as well as generalized spacetimes defined by Jordan algebras and Jordan triple systems. The second part of my talk discusses the extension of the above results to conformal supergroups of Minkowskian superspaces as well as of generalized superspaces defined by Jordan superalgebras. The (super)-oscillator construction of generalized (super)-conformal groups can be given a dynamical realization in terms of generalized (super)-twistor fields. Invited talk presented at the 6th International Wigner Symposium in Istanbul, Turkey (16-22 August, 1999). To appear in Turkish Journal of Physics. Work supported in part by the National Science Foundation under Grant Number PHY-9802510. e-mail: [email protected]