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

Markus Hunziker¹ ; Mark R. Sepanski¹ ; Ronald J. Stanke¹

¹Dept. of Mathematics, Baylor University, Waco, TX 76798-7328, U.S.A.

Journal of Lie Theory, Tome 22 (2012) no. 2, pp. 301-360

Résumé

\def\R{\mathbb{R}} Using an idea of Dirac, we give a geometric construction of a unitary lowest weight representation ${\cal H}^{+}$ and a unitary highest weight representation ${\cal H}^{-}$ of a double cover of the conformal group SO$(2,n+1)_{0}$ for every $n\geq 2$. The smooth vectors in ${\cal H}^{+}$ and ${\cal H}^{-}$ consist of complex-valued solutions to the wave equation $\Box f=0$ on Minkowski space $\R^{1,n}=\R\times \R^{n}$ and the invariant product is the usual Klein-Gordon product. We then give explicit orthonormal bases for the spaces ${\cal H}^{+}$ and ${\cal H}^{-}$ consisting of weight vectors; when $n$ is odd, our bases consist of rational functions. Furthermore, we show that if $\Phi, \Psi\in {\cal S}(\R^{1,n})$ are real-valued Schwartz functions and $u\in {\cal C}^{\infty}(\R^{1,n})$ is the (real-valued) solution to the Cauchy problem $\Box u=0$, $u(0,x)=\Phi(x)$, $\partial_tu(0,x)=\Psi(x)$, then there exists a unique real-valued $v\in {\cal C}^{\infty}(\R^{1,n})$ such that $u+iv\in {\cal H}^{+}$ and $u-iv\in{\cal H}^{-}$.

Détail

Classification : 22E45, 22E70, 35A09, 35A30, 58J70
Mots-clés : Conformal group, minimal representation, wave equation, classical solutions, Cauchy problem