Krein Spaces in de Sitter Quantum Theories

TL;DR

This paper explores the mathematical framework of Krein spaces in de Sitter quantum theories, focusing on scalar representations and their implications for quantizing massless minimally coupled fields in curved spacetime.

Contribution

It provides a detailed mathematical analysis of Krein structures in de Sitter group representations, highlighting their role in quantum field theory in curved spacetime.

Findings

Krein structures are relevant for de Sitter scalar representations

Undecomposable cohomology impacts quantization of fields

Mathematical tools for de Sitter quantum theories are developed

Abstract

Experimental evidences and theoretical motivations lead to consider the curved space-time relativity based on the de Sitter group $SO_0(1,4)$ or $Sp(2,2)$ as an appealing substitute to the flat space-time Poincare 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.