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   holes.  First of all, why do we really need a represenation of S_n - in   
   quantum mechanics a projective representation is good enough!  Secondly,   
   if one considers representations where S_n does not act as scalars but   
   as an "internal symmetry group" one gets even more possibilities.  These   
   were investigated under the name of parastatistics.  Anyway, one can   
   come up with a better argument, the spin-statistics theorem, in   
   relativistic quantum field theory, and that, together with the fact that   
   parastatistics can be redescribed as fermions and bosons in disguise,   
   seems to give a solid explanation for why all we see is bosons and   
   fermions.  (Though I couldn't say I'm very familiar *myself* with the   
   whole story.)   
      
   Now for the catch: the spin-statistics theorem only holds for spacetimes   
   of dimension 4 and up.  You could just say "thank heavens! that just   
   happens to apply to *our* universe!" and leave it at that, or you could   
   note that it's occaisionally possible to *simulate* universes of lower   
   dimension.  Take, for example, a thin 2-dimensional layer of stuff: this   
   can act like a little 3-dimensional spacetime.  Similarly, filaments can   
   act like 2-dimensional universes.  These days condensed matter theorists   
   delight in the odd processes that occur in these contexts, and it was   
   only a matter of time before someone noted that one can, at least in   
   principle, arrange to get particles that are neither bosons or fermions.   
   Wilczek is generally credited with taking the idea of these "anyons"   
   seriously, though it had occured to others earlier.   
      
   Here's how it goes in its most primitive form.  Say we have some   
   tubes of magnetic flux moving around.  (One can play with these   
   flux tubes using superconductors, for example).   As long as these tubes   
   stay parallel the problem is essentially a 2-dimensional one: pick a   
   plane perpendicular to the flux tubes and just pay attention to their   
   intersection with that plane.  Each tube intersects the plane in a spot   
   which we will regard as a "particle".  In each spot there is a B-field   
   perpendicular to the plane, and going around each spot is an A-field   
   whose curl is the B-field.  If you like you can think of each spot as a   
   "vortex" of the A-field.  Now suppose - and here I don't know if this   
   has ever been experimentally achieved - that each tube is   
   electrically charged.   In our planar picture then, we've got these   
   "particles" which are charged, each also being a vortex of the A-field.   
   Let's assume that all these particles are identical.  Now let's see what   
   happens when we interchange two of them.  Recall that when you move a   
   particle with charge e through an A-field, its phase changes by   
      
   exp(i theta),   
      
   where theta is the line integral of the A-field along the path traversed   
   by the particle.  Thus when we interchange two of our particles - and   
   here I mean you physically "grab" them and move them around each other   
   so that they trade places! - the wave vector of the system is changed by   
   a phase.  I'll let you calculate it.  The point is, depending on the   
   charge and the strength of the magnetic flux tube, one can arrange for   
   this phase to be whatever one likes - any complex number of magnitude   
   one!  If this number equals 1 one has bosons' if its -1 one has   
   fermions, but otherwise one has ANYONS.   
      
   One can have fun playing with this idea.  Many people have.  Wilczek is   
   a big proponent of an anyonic theory of high-temperature   
   superconductivity, although recent experiments, demonstrating an   
   apparent absense of spontaneous parity violation that one would expect   
   in this theory, seem to rule it out.  It's not 100% dead, though, and in   
   my opinion it's so beautiful that someone should try to make a   
   superconductor based on this principle just for the glory of it.   
   Something that anyone who has followed me so far can have fun doing, is   
   to see what sort of particle a bound state of anyons acts like.  It's   
   well-known that two fermions together act like a boson, two bosons act   
   like a boson, and a boson and a fermion act like a fermion ... extend   
   this to anyons.   
      
   A question for the real physicists out there: has anybody ever REALLY   
   MADE anyons and played with them yet?   
      
   Now anyons have a lot to do with braids because, as you may have   
   noticed, I have covertly stopped thinking of the the operation of   
   interchanging identical particles as an abstract "switching" - modelled   
   by the symmetric group -- and started thinking of it as moving one   
   particle around another.  If one draws the worldlines of some anyons   
   as one moves them around each other this way, one has -- a braid!  I.e.,   
   out with the symmetric group, in with the braid group!   
      
   The high-handed manner in which I've thrown out the symmetric group and   
   started working with braid group statistics *should* disturb you, but   
   again I can cite fancy mathematical physics papers which should allay   
   your fears.  A very nice one is "Local Quantum Theory and Braid Group   
   Statistics" by Froehlich and Gabbiani, which gives a proof of the   
   generalized spin-statistis theorem that holds in 2 and 3 dimensions.   
      
   So now we see a close relation between quantum theory - to be precise,   
   "statistics" in quantum theory - and the braid group.   Looking back at   
   Kaufmann's original insight into the relation of knots and quantum   
   mechanics, it's not blindingly obvious what *that* has to do with   
   *this*!  Nonetheless it's all part of one story.  (A story which I   
   strongly feel is far from over.)  More later.   
      
      
   Polyakov's Model   
      
   Some people have written saying they enjoy these "Braids and   
   quantization" articles, so I'll keep 'em coming.  Some also wrote   
   saying that the best known explanation for the mysterious "fractional   
   quantum Hall effect" involves anyons, and referred me to a paper of   
   Frohlich and Zee, "Larges Scale Physics of the Quantum Hall Fluid," Nuc   
   Phys B364, 517-540.  The Hall effect, recall, is fact that if one   
   applies a magnetic field perpendicular to current in a wire, this makes   
   the electrons wnat to veer off to one side (recall the force is   
   proportional to v x B), and they do until the increased charge density   
   on that side creates enough electric field to keep more from crowding   
   over to that side.  One observes this by measuring the electric field.   
   The "quantum Hall effect" refers to the charming fact that this effect   
   is quantized for a sufficiently good (cold) conductor: as one increases   
   the magnetic field the resulting electric field goes up like a STEP   
   FUNCTION; in the right units, the effect takes a jump when the magnetic   
   field is an integer.  This effect is reasonably well understood, they   
   say (though I've heard murmurs of discontent occaisionally).  I know   
   that Belissard has written nicely about the relationship with   
   noncommutative differential geometry a la Connes.  The *fractional*   
   quantum Hall effect refers to the fact that not only at inegers, but   
   also at some *fractions*, one sees a jump.  I will enjoy learning how   
   this could result from fractional (i.e., anyonic) spin and statistics.   
      
   Now, though, I feel like rambling on a bit about Polyakov's   
   construction of anyons by adding a Hopf term to the Lagrangian of a   
   certain nonlinear sigma model.  This is actually used in the   
   anyonic theory of high-Tc superconductivity, but even if that theory is   
   a bunch of baloney, Polyakov's idea is a charming bit of mathematical-physical   
   speculation.  It's a nice introduction to solitons, topological quantum   
      
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