From: baez@galaxy.ucr.edu (John Baez)
Subject: Re: Can there be relativistic anyons?
Date: 16 May 2000 22:49:52 GMT
Newsgroups: sci.physics.research
Summary: [missing]

In article <8fmvio$2ji$1@nnrp1.deja.com>,  <squark@my-deja.com> wrote:

>The spin-statistics theorem is saved (from doom by anyons) via the
>proposition that projective representations of U(1) may be used, i.e.,
>practiacally representations of R (for the rotation group). But, in
>relativistic physics we need representation of the whole Poincare group
>rather than just the rotation group. What happens then? The universal
>cover of the Poincare group in 2+1 dimensions doesn't contain R, does
>it? 

Sure, it does.  Remember, the fundamental group of a Lie group is
always the same as the fundamental group of its maximal compact 
subgroup.  For the Poincare group in 3+1 dimensions the maximal
compact subgroup is just the rotation group SO(3), whose fundamental
group is Z/2.  This means that the universal cover of the Poincare
group is a *double* cover in this case.  But for the Poincare group
in 2+1 dimensions the maximal compact subgroup is the rotation group
SO(2), whose fundamental group is Z.  This means that the universal
cover of the Poincare group in 2+1 dimensions is an *infinite* cover. 
And as you guessed, it contains the universal cover of SO(2), which is
the real line R.  

>Or, if it does, what kind of representations of the Lorentz group
>result (in the massive AND the massless case)?

Hmm, you want me to tell you all the unitary positive-energy 
representations of the Poincare group in 2+1 dimensions?  What
do you think I am, a nerd?