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>, 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?