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

Using an idea of Dirac, we give a geometric construction of a unitary lowest weight representation H + and a unitary highest weight representation 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 solutions to the wave equation f = 0 on Minkowski space R 1,n = R R n and 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 , S (R 1,n ) are real-valued Schwartz functions and u C (R 1,n ) is the (real-valued) solution to the Cauchy problem u = 0, u(0, x) = (x) , t u(0, x) = (x) , then there exists a unique real-valued v C (R 1,n ) such that u + iv H + and uiv H -.