🚀 go-pugleaf

In article <92315.002515CCB104@psuvm.psu.edu> <CCB104@psuvm.psu.edu> writes:
>I heard it said by a former physics grad student that a certain professor
>of his said that it is *obvious* why there is "no spin in 2 dimensions"
>(i.e. in a 2-dimensional differentiable manifold of some sort?). What might
>they have meant, please? Any ideas?
>Thanks,
>Carey

Well, I'm not sure what he's saying is exactly right, but if he meant
1+1 dimensions (one space, one time) I can imagine one reason why it's
"obvious" - things can't turn around in 1-dimensional space!  That is,
the rotation group O(1) is disconnected - just Z_2.  (And the element
that's not the identity is really reflection, or parity.)

Of course if one treat spin relativistically in 1+1 dimensions one gets
the group O(1,1) -- the connected component of which is just the group
of real numbers, R.  These parametrize "boosts" - aka Lorentz
transformations.  Particles can transform nontrivially under boosts and
the "spin" is parametrized by a real-valued parameter since the
irreducible unitary representations of R are just  t -> exp(iat)  where
a is a real-valued parameter.

Hope this isn't painfully abstruse.  It boils down to the fact that
spins describe what happens to particles when you rotate them, and
rotating particles in 1+1 dimensions is relatively boring.