I think it’s unlikely that they do, although we can’t be completely sure. As you probably heard the OPERA experiment detected neutrinos that seemed to be traveling faster than light, although recently they have suggested that there were some errors in their study which might have made us think that neutrinos are travelling faster than light when they are not.
But what makes me think that the neutrinos are not travelling faster than light is this: We know that neutrinos can interact with electrons, and so if neutrinos go faster than we think they should then electrons should go faster than we expect as well. But we have very precise measurements of the speed of the electron – much more precise than the measurements of the speed of the neutrinos – and electrons travel at exactly the speed they should do.
So probably neutrinos don’t go faster than light.
There was the experiment recently that suggested that neutrinos do travel faster than the speed of light (the OPERA experiment). But those results don’t make sense since there are loads and loads of other experiments that tell us that nothing can travel faster than the speed of light. It now seems that the OPERA experiment made a mistake and that their results are actually wrong, which means that the neutrinos never went faster than the speed of light.
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