# Wave equation: predicting geometric dispersion with group theory

Context

The wave equation

$$\partial_{tt}\psi=v^2\nabla^2 \psi$$

describes waves that travel with frequency-independent speed $$v$$, ie. the waves are dispersionless. The character of solutions is different in odd vs even number of spatial dimensions, $$n$$. A point source in odd-$$n$$ creates a disturbance that propagates on the light cone and vanishes elsewhere: if the point source is a flash of light, an observer sees darkness, then a flash, then darkness. When $$n$$ is even, a disturbed media never returns to rest: the observer sees darkness, then brightness that lingers for all $$t$$. This phenomena is known as geometric dispersion.

Question

Is it possible to show that geometric dispersion is predicted by the wave equation, using group theory? For a point source at the origin, we would be searching for spherically symmetric solutions, and the rotation group $$SO(n)$$ has a different structure depending on whether $$n$$ is odd or even. In particular, I am interested in doing this without actually solving the wave equation. Unfortunately, I don't know enough group theory to know if this is even possible.

What I know

I can 'show' geometric dispersion by solving the wave equation with an initial condition, or computing the Green's function for the wave equation and noting that it is either supported only on the light cone (odd $$n$$), or everywhere within the light cone (even $$n$$).

I know some group theory 'for physicists'.

Related

This unanswered question is similar. I think my question is more specific: I'm asking about a way to predict (rather than explain) geometric dispersion using group theory.

• I would love to see an answer to this. Group theory is one of my weakest areas though, so I won't be able to produce one.... Oct 27, 2021 at 13:11
• Following the comments in the linked question, I arrived at Theorem 4.3.1 in the book "Nonabelian harmonic analysis" of Howe and Tan (thank you Alp Uzman). That theorem contains exactly the answer to this question. Unfortunately, it uses heavy machinery from representation theory and I cannot fully understand it. Jun 25 at 15:40
• @GiuseppeNegro Thank you, that reference seems to answer precisely my question- I do not understand it yet either
– Sal
Jun 25 at 16:35
• I wonder if we can @AlpUzman to notify them. It's them, not me, who gave that useful reference Jun 25 at 17:00
• @GiuseppeNegro I had tried studying the Howe-Tan account of the Huygens' principle about five years ago and at the time I too could not handle the representation theory involved (I'm afraid I still can't). I don't really have any useful insights, as such I can't claim any credits, though thank you for acknowledging my comment elsewhere. Jun 25 at 17:35