Why two Turns for Spin 1/2?

Why does a spin 1/2 particle need 2 complete revolutions to get back to its starting position? And what the hell can be meant by a 'starting position' if the particle is actually something pointlike?
- Dipanjali (age 16)
Siliguri, West Bengal, India

Those are great tough questions. I can at least answer the second one.

Say you have a neutron (for example) and send it through a partial reflector, so half its quantum state travels one path and half travels another path before they get recombined. (Yes, that sounds impossible classically, but that's how the quentum world works.) Do this with a lot of neutrons and you get some sort of interference pattern at the detectors, showing where the two paths are in phase and out of phase with each other. Now put a magnetic field in just one path. It rotates the neutrons, meaning it changes the quantum phases of different components of their spin states. That changes the interference pattern. How much do you have to rotate the neutrons to get back to the original interference pattern? Not once around but twice around! Very strange, but that's how it is. (I think these experiments and other related ones are described in Sam Werner's book:  

OK, now for your second question, why is it that way? And if it can be that weird, how much weirder could it be? There's a theorem,"spin-statistics", that shows that in a spatially 3-D world obeying special relativity there are only two possibilities. The quantum state must be returned to its original value after either one turn or two. So why doesn't nature stick with just the common-sense possibility- one turn? Somebody else maybe could give a much deeper answer, but one way to think of it is that nature seems to try out all the possibilities that aren't actually ruled out.

Perhaps another weird thing to add is that if there were a universe with only spin-0, spin-1, etc. partcles it would consist entirely of "bosons". (That's what the spin-statistics theorem says.) Bosons are fundamentally unsuited to be the ingredients of chemicals and other complicated structures. So if there is such a universe, nobody is in it to ask where are the missing half-integer spins.

Mike W.

(published on 11/06/2013)