# Can Gaussian measure be characterized by unitary representations?

It is well known that Fourier transform switches positive-definite functions with positive measures on a (locally compact topological) group. Further, the positive definite functions can be characterized as coefficients of the unitary representation of the group.

My question is: Is it possible to characterize Gaussian measures using this representation theory framework (even for obvious case like $$\mathbb{R}$$) for measures defined on the dual space of nuclear spaces? I thought about this for a while, but did not get anywhere. I also looked up standard reference book like Gelfand & Vilenkin and did not see any discussion there. I suspect the answer is something obvious and trivial. But I still do not know. In particular I am willing to restrict to LCA groups just to make matters simpler.

• Can someone explain the downvote? – Bombyx mori Feb 4 at 17:04
• Wasn't me. However, it's not clear what kind of characterization you would like. In the notation of Dima's answer in the link you gave, being Gaussian means that $\log \langle \pi(t)h,h\rangle$ is quadratic in $t$. I don't know right off the bat what this means representation theoretically. Perhaps the corresponding Lie algebra representation can be expressed by creation and annihilation operators as with the harmonic oscillator. – Abdelmalek Abdesselam Feb 4 at 17:26
• @AbdelmalekAbdesselam: I mean this: forget about finite dimensional projections, can we characterize Gaussian measure on the LCA group using a certain number of axioms that is satisfied by the unitary representations? I would love to know if my question is trivial; but so far I still do not understand how to answer it. And it seems nuclear spaces is beyond Math.SE level. But I could be wrong. – Bombyx mori Feb 4 at 21:50
• I think you can probably work this out on your own by, on the the contrary, examining the finite dimensional case first. Write $\pi(t)=e^{itH}$ for some self-adjoint operator $H$ on $L^2(V)$ with $V$ finite dimensional. Then ask what must $H$ satisfy so for all $h\in V$, $t\mapsto \log\langle e^{itH} h,h\rangle$ is quadratic. – Abdelmalek Abdesselam Feb 5 at 15:32
• @AbdelmalekAbdesselam: I understand. I think this is a good point. I take a look at your web-page; may I ask how relevant is the Gaussian measure related to quantum field theory (for path integrals, maybe)? I saw this appearing a lot in literature, but I do not know any physics. So I cannot really digest the material. I will try to delete the post - I did not realize it was trivial. – Bombyx mori Feb 5 at 23:14