Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 011, 23 pages      arXiv:1001.4810
Contribution to the Proceedings of the 5-th Microconference Analytic and Algebraic Methods V

Krein Spaces in de Sitter Quantum Theories

Jean-Pierre Gazeau a, Petr Siegl a, b and Ahmed Youssef a
a) Astroparticules et Cosmologie (APC, UMR 7164), Université Paris-Diderot, Boite 7020, 75205 Paris Cedex 13, France
b) Nuclear Physics Institute of Academy of Sciences of the Czech Republic, 250 68 Rez, Czech Republic

Received October 19, 2009, in final form January 15, 2010; Published online January 27, 2010

Experimental evidences and theoretical motivations lead to consider the curved space-time relativity based on the de Sitter group SO0(1,4) or Sp(2,2) as an appealing substitute to the flat space-time Poincaré relativity. Quantum elementary systems are then associated to unitary irreducible representations of that simple Lie group. At the lowest limit of the discrete series lies a remarkable family of scalar representations involving Krein structures and related undecomposable representation cohomology which deserves to be thoroughly studied in view of quantization of the corresponding carrier fields. The purpose of this note is to present the mathematical material needed to examine the problem and to indicate possible extensions of an exemplary case, namely the so-called de Sitterian massless minimally coupled field, i.e. a scalar field in de Sitter space-time which does not couple to the Ricci curvature.

Key words: de Sitter group; undecomposable representations; Krein spaces; Gupta-Bleuler triplet; cohomology of representations.

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