If I'm correct, Cherenkov radiation is caused when particle moves faster than light in a certain medium. Can neutrinos, say, travelling through water, cause Cherenkov radiation then? Or is there a property of neutrinos which prevents that?
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"I think I remember reading" is not something we can find ourselves to figure out what you are talking about. If you want to argue that the answer you got is wrong, you're going to have to do better than that.
When a neutrino interacts, the outgoing charged particles can produce Cherenkov radiation.
The first experiment to search for proton decay used a water Cherenkov detector. It never found any proton decay, but it turned out to be very useful for studying neutrinos. Most famously, it detected some of the neutrinos from Supernova 1987a:
You got lucky that someone could make a good guess as to what you were asking about. I was in grad school at Michigan while that experiment was being built. I heard a lot about it when that group gave colloquia to report on their progress, so it came to my mind immediately when I saw "neutrino" and "Cherenkov" together.
The by products of neutrino interactions (charged leptons) emit Cherenkov radiation when they have speed > speed of light in the medium they propagate.
NOW, if you want to discuss Cherenkov radiation in vacuo of apparent superluminal neutrinos (such as those proposed by OPERA some months ago), back then the most promising theoretical idea about why neutrinos should NOT be superluminal, was that if they were they would have emitted "Cherenkov-like radiation" which would shift their spectrum to lower energy values than this of the emission spectrum of theirs (Glashow and Georgi were the authors I think).
Adding up to the others , I would suggest that you try to be more specific:)