The minimal representation of the conformal group and classical solutions to the wave equation (Q2892858)

Using an idea of Dirac, the authors give a geometric construction of a unitary lowest weight representation \({\mathcal H}^+\) and a unitary highest weight representation \({\mathcal H}^-\) of a double cover of the conformal group \(SO(2,n+1)_0\) for every \(n\geq 2.\) The smooth vectors in \({\mathcal H}^+\) and \({\mathcal H}^-\) consist of complex-valued solutions to the wave equation NEWLINE\[NEWLINE \square f=-f_t^2 +\sum_{i=1}^n f_{x_i x_i}=0 NEWLINE\]NEWLINE on the Minkowski space \({\mathbb R}^{1,n}={\mathbb R}\times{\mathbb R}^n\) and the invariant product is the usual Klein-Gordon product. Explicit orthonormal bases are given for the spaces \({\mathcal H}^+\) and \({\mathcal H}^-\) consisting of weight vectors; when \(n\) is odd the bases consist of rational functions. It is shown that if \(\Phi,\Psi\in {\mathcal S}({\mathbb R}^{1,n})\) are real-valued Schwartz functions and \(u\in {\mathcal C}^\infty({\mathbb R}^{1,n})\) is the (real-valued) solution to the Cauchy problem NEWLINE\[NEWLINE \square u=0,\quad u(0,x)=\Phi(x),\quad \partial_t u(0,x)=\Psi(x) NEWLINE\]NEWLINE then there exists a unique real-valued \(v\in {\mathcal C}^\infty({\mathbb R}^{1,n})\) such that \(u+iv\in {\mathcal H}^+\) and \(u-iv\in {\mathcal H}^-\).