Journal of Lie Theory

no. 2
The Minimal Representation of the Conformal Group and Classical Solutions to the Wave Equation
Journal of Lie Theory, Volume 22 (2012) no. 2, pp. 301-360
\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 $u-iv\in{\cal H}^{-}$.
DOI: 10.5802/jolt.671
Classification: 22E45, 22E70, 35A09, 35A30, 58J70
Keywords: Conformal group, minimal representation, wave equation, classical solutions, Cauchy problem 
@article{JOLT_2012_22_2_a0,
     author = {M. Hunziker and M. R. Sepanski and R. J. Stanke},
     title = {The {Minimal} {Representation} of the  {Conformal} {Group} and {Classical} {Solutions} to the {Wave} {Equation}},
     journal = {Journal of Lie Theory},
     pages = {301--360},
     year = {2012},
     volume = {22},
     number = {2},
     doi = {10.5802/jolt.671},
     zbl = {1250.22015},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.671/}
}
TY  - JOUR
AU  - M. Hunziker
AU  - M. R. Sepanski
AU  - R. J. Stanke
TI  - The Minimal Representation of the  Conformal Group and Classical Solutions to the Wave Equation
JO  - Journal of Lie Theory
PY  - 2012
SP  - 301
EP  - 360
VL  - 22
IS  - 2
UR  - https://jolt.centre-mersenne.org/articles/10.5802/jolt.671/
DO  - 10.5802/jolt.671
ID  - JOLT_2012_22_2_a0
ER  - 
%0 Journal Article
%A M. Hunziker
%A M. R. Sepanski
%A R. J. Stanke
%T The Minimal Representation of the  Conformal Group and Classical Solutions to the Wave Equation
%J Journal of Lie Theory
%D 2012
%P 301-360
%V 22
%N 2
%U https://jolt.centre-mersenne.org/articles/10.5802/jolt.671/
%R 10.5802/jolt.671
%F JOLT_2012_22_2_a0
M. Hunziker; M. R. Sepanski; R. J. Stanke. The Minimal Representation of the  Conformal Group and Classical Solutions to the Wave Equation. Journal of Lie Theory, Volume 22 (2012) no. 2, pp. 301-360. doi: 10.5802/jolt.671

Cited by Sources: