Lie theory and the wave equation in space-time. 4. The Klein-Gordon equation and the Poincaré group

A detailed classification is made of all orthogonal coordinate systems for which the Klein-Cordon equation in space-time, ψ_tt - Δ₃ψ = Λψ, admits a separation of variables. We show that the Klein-Gordon equation is separable in 261 orthogonal coordinate systems. In each case the coordinate systems presented are characterized in terms of three symmetric second order commuting operators in the enveloping algebra of the Poincaré group. This paper also consitutes an important step in the study of separation of variables for the wave equation in space-time ψ_tt - Δ₃ψ = 0, and its relation to the underlying conformal symmetry group O(4,2) of this equation.