From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Why does 3D => fermions and bosons?
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Mon, 27 Jul 92 23:29:16 GMT

In article <allenk.712201931@vex> allenk@vex.ugcs.caltech.edu (Allen Knutson) writes:
>I posted something recently explaining why rotational symmetry causes there
>to be wave-functions that change by only +/- 1 under 360 degree rotations.
>But how does this mix with how the wave-function has to change when you
>take two identical particles and switch them? After all, the two-particle
>system doesn't have the SO(3) symmetry anymore, just O(2).
>
>(And am I thinking right that "two identical particles" means "two
>equivalent irreducible representations of the underlying symmetry group
>of the physics"? And that people discovering additional quantum numbers
>is equivalent to them extending their group? There seem to be so many
>more books out there to explain math to physicists than physics to
>mathematicians : -8 (						Allen K.

And here I thought you were sunning yourself on the beaches of Mexico.

1)  In QM it's standard to describe a system of identical particles by a
Hilbert space on which the symmetric group acts.  This makes sense,
especially if you don't think about it very much.  Then you say the
representation should be unitary - fine.  [Please don't notice that all
one could really expect is a *projective* unitary representation.]  Then
you say that since the particles are *identical*, switching two should
only act by multiplication by a phase.  Well, the symmetric group only
has two reps which satisfy all this, the boson or trivial rep, and the
fermion rep in which the permutation is mapped to its sign.

2)  The above argument is thoroughly bogus.  That's why it breaks down
in dimension < 3 (space).  The real argument is the spin-statistics
theorem, which uses the full glories of relativistic QFT.  The canonical
reference is "Spin, Statistics, PCT, and All That" by Streater and
Wightman.  But if I recall correctly, in that book they simply assume
that the particles are bosons or fermions and show how that must
correlate to integer/half-integer spin... a full treatment also shows
how "parastatistics" (other reps of the symmetric group) may be
reinterpreted as ordinary statistics together with an internal symmetry
group.  Some of the more modern work on this uses the axioms for
algebraic QFT, which are more general than the Garding-Wightman axioms
used in the book cited above.  The names which come to mind are
Doplicher, Haag, and Roberts, Buchholz, and Fredenhagen.   Recent work
by some of the above as well as Frohlich and Gabbiani treats the
low-dimensional cases, where exotic (braid group) statistics arise.

In any event, to provide information rather than merely name-drop, the
point is roughly that using the setup of axiomatic QFT one can relate
the operation of interchanging two particles with an element in the
*complexified* Poincare group.  I gave a brief sketch of the proof on
sci.physics once but I don't have it so if anyone could send it to me or
repost it I'd be thankful.

3)  I have a paper with Ody and Richter on spin and statistics for
nonlinear sigma models which treats the purely *topological* constraints
on spin and statistics as thoroughly as anywhere I have seen.   This
does not touch upon the above spin-statistics theorems (which arise in
the framework of QFT, not quantum mechanics), but it does give one
framework in which it becomes believable that the symmetric group rules
in dimension 3 or more, while the braid group and its generalizations
are important in dimension 2.  This paper also has references to all the
above stuff.  I'll mail you a copy, as well as to anyone else who asks,
with the warning that it's packed with commutative diagrams that you
can't print out unless you have the spiffy commutative diagram package
they have at MIT.  (DON'T ask me to send you this package, 'cause I don't
know how; if you are desperate to draw your own commutative diagrams
with such a package contact richter@math.mit.edu (boy he'll love me for
this); if you just want to read our paper in its full glory ask me for
hardcopy, which will take a while to get to you since I'm moving to
California etc..)

4)  Your definition of "two identical particles" for group theorists is
fine except one wants the tensor product of these two reps to be
equipped with an operator that "switches the two representations".  In
the bosonic case it's just

v x w -> w x v

while in the fermionic case it's

v x w -> -w x v.

The choice of this operator determines the "statistics" of the particle.
In the "anyonic" or "exotic" case it's a Yang-Baxter operator; cf e.g.
my articles on braids and quantization; this quickly throws one into the
world of BRAIDED TENSOR CATEGORIES.  



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