Smooth Solutions and Discrete Imaginary Mass of the Klein-Gordon Equation in the de Sitter Background

TL;DR

This paper uses Lie algebra methods to classify all smooth solutions of the Klein-Gordon equation on 4D de Sitter space, revealing a discrete, imaginary mass spectrum related to Casimir eigenvalues.

Contribution

It provides a complete characterization of smooth solutions and establishes the quantized, imaginary nature of the scalar mass in de Sitter spacetime.

Findings

Mass spectrum is discrete and proportional to -N(N+3).

All smooth solutions are classified via Lie algebra techniques.

Mass m is imaginary, with m^2 related to Casimir eigenvalues.

Abstract

Using methods in the theory of semisimple Lie algebras, we can obtain all smooth solutions of the Klein-Gordon equation on the 4-dimensional de Sitter spacetime (dS^4). The mass of a Klein-Gordon scalar on dS^4 is related to an eigenvalue of the Casimir operator of so(1,4). Thus it is discrete, or quantized. Furthermore, the mass m of a Klein-Gordon scalar on dS^4 is imaginary: m^2 being proportional to -N(N+3), with N >= 0 an integer.