Newsgroups: sci.physics
Subject: Re: Why does 3D => fermions and bosons?
From: mcirvin@husc8.harvard.edu (Mcirvin)
Date: 29 Jul 92 15:24:39 GMT

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).

The particles' state may only have O(2) symmetry (though they might
also be in a spherically symmetric state, depending on the actual
wave function and spin alignment).  But the quantum theory 
on which the state is constructed is still rotationally invariant,
and rotating the whole system in any SO(3) way won't change the
physics.  Therefore, everything you said still applies.
After all, a single-particle state with a nonzero eigenvalue
of spin in some direction isn't rotationally invariant either.

Think about the process of exchanging the particles twice.
This corresponds to some SO(3) rotation that leaves the system
unchanged spatially.  If we factor the wave function into a
wave function for each particle, each particle can pick up a
phase of +1 or -1 from the transformation, in the usual way.
But the two particles are identical, so the overall phase for
the wave function of the whole system, which is the product
of the two wave functions, is necessarily +1.

Therefore, if the particles are exchanged *once*, the phase
from the exchange is a square root of 1; it's either +1 or -1.
We have either fermions or bosons.

The interesting thing that happens is that when you go to
two-dimensional space, the spatial symmetry group
really is only O(2).  Since O(2)'s first homotopy group is
the integers rather than just a two-element group, systems 
can pick up all kinds of fractional phases under a 2pi
rotation, and correspondingly one can have anyons, particles
with fractional exchange statistics.
 
Physically, the effect can be thought of as identical to the
Aharonov-Bohm effect, in which charged particles traveling
around an ideal solenoid (an infinitely long electromagnet
coil, which contains magnetic flux in its interior) pick up a
phase difference depending on the magnitde of the flux,
even though there's no B field outside the solenoid.  Since
the space outside the solenoid has a first de Rham cohomology
equivalent to the real numbers (= since it's topologically
like the interior of a donut), the magnetic vector potential
can point around the solenoid without giving rise to any
magnetic field, and cause the phase change.  Anyons can
be seen as particles in a two-dimensional world that carry
magnetic flux, as if an infinitely long solenoid intersects
the plane at the position of the particle.  Exchanging two
identical particles by physically moving them around each
other gives rise to a fractional phase by the Aharonov-
Bohm effect.
 
>(And am I thinking right that "two identical particles" means "two
>equivalent irreducible representations of the underlying symmetry group
>of the physics"? 

Usually identical particles are thought of as transforming
under the *same* representation of the physics.  I suppose
you could think of them as equivalent irreducible representations.

>And that people discovering additional quantum numbers
>is equivalent to them extending their group? 

If the quantum numbers are conserved, yes.  If not, then the
symmetry group of *part* of the theory may be extended.  For
instance, strangeness is conserved in strong interactions
but not in weak ones, so the strong-interaction part of
the physics has a larger symmetry than the weak part (in
this and several other ways).

The prevailing prejudice among many particle physicists is that
the only really good symmetries, applying for all physics,
are gauge symmetries, which can be applied independently of
position and time.  Each degree of gauge freedom implies
a gauge field, and each simple part of the gauge group has
its own coupling constant.  The idea behind GUTs is to make
the gauge group a simple group so that there's only one
gauge coupling constant.

>There seem to be so many
>more books out there to explain math to physicists than physics to
>mathematicians : -8 (						Allen K.

I think authors assume that you're smarter than we are,
and can pick these things up easily!

-- 
Matt McIrvin  mcirvin@husc.harvard.edu 
Newsgroups: sci.physics