Lie theory and the wave equation in space-time. I. The Lorentz group

In this article we begin a study of the relationship between separation of variables and the conformal symmetry group of the wave equation ψ_n -Δ₃ψ = 0 in space-time. In this first article we make a detailed study of separation of variables for the Laplace operator on the one and two sheeted hyperboloids in Minkowski space. We then restrict ourselves to homogeneous solutions of the wave equation and the Lorentz subgroup SO(3,1) of the conformal group SO(4,2). We study the various separable bases by using the methods of integral geometry as developed by Gel'fand and Graev. In most cases we give the spectral analysis for these bases, and a number of new bases are developed in detail. Many of the special function identities derived appear to be new. This preliminary study is of importance when we subsequently study models of the Hilbert space structure for solutions of the wave equation and the Klein-Gordon equation ψ_n -Δ₃ψ = λψ.