Timeline for Can Gaussian measure be characterized by unitary representations?
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11 events
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| Feb 7, 2019 at 14:51 | comment | added | Abdelmalek Abdesselam | I don't see a reason to delete the question. I asked a few questions which remained unanswered, but there is a chance someone knowledgeable finds one and contributes an interesting answer even years later. BTW, there has been quite some research activity related to the theme of your question. You should look up the works of Palle Jorgensen, Karl-Hermann Neeb and Gestur Olafsson. See for example arxiv.org/abs/1802.09037 | |
| Feb 7, 2019 at 1:09 | comment | added | Bombyx mori | @AbdelmalekAbdesselam: I understand. I will leave the question open for a few days. If there is no answer I will delete it. Thanks for the comment. | |
| Feb 6, 2019 at 14:36 | comment | added | Abdelmalek Abdesselam | I don't think it is trivial. I upvoted your question. What I mentioned in my previous comments is just an outline of how I would go about trying to answer your question. My general feeling is that the representation theoretic counterpart of the measure being Gaussian is related to the en.wikipedia.org/wiki/Oscillator_representation but that's where my expertise ends. | |
| Feb 6, 2019 at 3:02 | history | undeleted | Bombyx mori | ||
| Feb 6, 2019 at 2:51 | history | deleted | Bombyx mori | via Vote | |
| Feb 5, 2019 at 23:14 | comment | added | Bombyx mori | @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. | |
| Feb 5, 2019 at 15:32 | comment | added | Abdelmalek Abdesselam | 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. | |
| Feb 4, 2019 at 21:50 | comment | added | Bombyx mori | @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. | |
| Feb 4, 2019 at 17:26 | comment | added | Abdelmalek Abdesselam | 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. | |
| Feb 4, 2019 at 17:04 | comment | added | Bombyx mori | Can someone explain the downvote? | |
| Feb 4, 2019 at 5:48 | history | asked | Bombyx mori | CC BY-SA 4.0 |