How does handedness get preserved in this situation?

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In this video it's explained that neutrinos always travel related to the direction of their spin. But neutrinos don't travel at the speed of light, so it's possible to outrun it.

That means from the point of view of earth, a solar neutrino is moving in one direction. But to a higher energy neutrino chasing it, it's moving in the opposite direction.

So does spin change based on frame of reference?
 
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You are mixing several concepts, spin, helicity and chirality. For a massive particle, only spin is conserved while helicity and chirality aren't. Also helicity and chirality don't coincide. For a massless particle, spin is not well defined (as it is angular momentum in the rest frame, which does not exist for a massless particle), while helicity and chirality coincide.
So yes, you can outrun a neutrino and change its helicity.
 
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I'm confused, the video seemed to only mention quantum spin. Was this a mistake? The guys a nuclear phycisist at fermilabs, seems unlikely that he'd make a mistake like that.