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

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