The minimal representation of the conformal group and classic solutions to the wave equation

TL;DR

This paper constructs a uniform minimal representation of the conformal group on Minkowski space, providing explicit bases and linking classical wave solutions to this representation, revealing rational functions for odd dimensions.

Contribution

It introduces a new uniform realization of the minimal conformal group representation, explicitly constructs an orthonormal basis, and connects classical solutions to this representation using Fourier analysis.

Findings

Explicit orthonormal basis for the minimal representation

All basis functions are rational functions for odd dimensions

Every classical wave solution corresponds to a unique element in the representation

Abstract

We give a uniform realization of the minimal representation of a double cover of the conformal group SO(2,n+1)_0 in the kernel of the wave operator on flat Minkowski space as a positive energy representation H^+ for n even and odd. Using this realization, we obtain an explicit orthonormal basis for H^+ that is well behaved with respect to energy and angular momentum. Of special note, for n odd, all functions in our basis are rational functions. Finally, using Fourier analysis with respect to this basis, we prove that every classical real-valued solution to the wave equation is the real part of a unique continuous element in the representation H^+.