# Representation theoretic explanation of Huygens' Principle

Huygens' principle -- Wave equations in $$\mathbb{R}^n$$ seem similar for different $$n$$, but behave quite differently depending on the parity of $$n$$: waves in odd dimensional spaces never look back, while waves in even dimensional spaces linger. A more vivid depiction can be found in this related MSE post

.. when you switch a light bulb on and then off (in 3D), there will be a light wave traveling with the speed on light and behind the wave, there will be total darkness. But when you throw a rock into a pond (with a 2D surface), there will be lots of waves traveling outwards from where the rock hit the water and, in theory, the water will never a still again.

There are several ways to explain the difference. One is by looking at the solutions to the PDE directly, one uses complex analysis, one interprets waves in a case as a "projection" of another case..

### Question

In this post, I'm looking for a representation theoretic explanation. Indeed, the wave equations look pretty similar for all $$n$$, and indeed all properties of waves are entailed in the equations. The only (hopeful) difference I notice is the symmetries that involve: the representation theory of $$\frak{so}(n,1)$$ is quite different based on the parity of $$n$$! There's already a hint in this answer without details. I hope someone would point out a more detailed account that show Huygens' principle alone this line. To me, this will be more satisfactory as an intuitive explanation.

• In odd spatial dimensions the fundamental solution of the wave operator is supported on the light-cone, while in even spatial dimensions its support is the solid cone. This can be explained in terms of intertwinings among principal series repns of the relevant orthogonal groups, yes, if one wants...? Commented Aug 12, 2021 at 18:34
• @paulgarrett if one wants...? Commented Aug 12, 2021 at 19:11
• The complex analysis involved in meromorphic continuation of intertwining operators given by integrals is very similar to meromorphic continuation of corresponding homogeneous distributions/functions... So I myself would just talk about distributions with homogeneity/invariance conditions, rather than characterizing them as "spherical vectors" in principal series. I think the latter possible viewpoint does not gain much in this case. Commented Aug 12, 2021 at 19:17
• See also Howe & Tan's book "Non-Abelian Harmonic Analysis - Applications Of $\text{SL}(2,\mathbb{R})$", Chapter IV.4. Commented Feb 16, 2022 at 6:13
• @AlpUzman: Thanks a lot. Very interesting reference. Commented Jun 25, 2022 at 15:37