The Minimal Representation of the Conformal Group and Classical Solutions to the Wave Equation

摘要

Using an idea of Dirac, we give a geometric construction of a unitary lowest weight representation H~+ and a unitary highest weight repre-sentation H~- of a double cover of the conformal group SO(2, n + 1)_0 for every n ≥ 2. The smooth vectors in H~+ and H~- consist of complex-valued solu-tions to the wave equation □f = 0 on Minkowski space R~(1,n) = R x R~nand the invariant product is the usual Klein-Gordon product. We then give explicit orthonormal bases for the spaces H~+ and H~- consisting of weight vectors, when n is odd, our bases consist of rational functions. Furthermore, we show that if φ,ψ,∈ y(R~(1,n)) are real-valued Schwartz functions and u ∈ ∮∞ (R~(1,n) is the (real-valued) solution to the Cauchy problem □u = 0, u(0, x)=φ(x), θ_tu(0, x) = ψ(x), then there exists a unique real-valued v ∈ ∮∞(R~(1,n) such that u +iv∈ H~+ and u - iv ∈ H~- .