From cbloom@mail.utexas.edu Thu Dec 12 19:26:25 1996
Path: geraldo.cc.utexas.edu!cs.utexas.edu!howland.erols.net!feed1.news.erols.com!arclight.uoregon.edu!news.bc.net!rover.ucs.ualberta.ca!anderson
From: "Joshua W. Burton"
Newsgroups: sci.physics.research
Subject: Re: candle dances & Pauli exclusion
Date: 10 Dec 1996 07:51:32 GMT
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Phil Fraundorf wrote:
> A newsgroup post I've now lost track of described a four-step
> way to first make wave-function invertability on 360-degree rotation
> plausible for half-integral spin particles, and then to show how
> the Pauli exclusion principle follows therefrom. The post I am
> thinking of had in it a story about blood centrifugation and
> Balinese candle dances at 7200 RPM. If someone has a copy of
> this post, I would be interested because the lack of a simple
> way to make this connection for students working through most
> modern physics texts is quite noticeable.
That was me, sometime last April. I saved a copy; I hope you find
it helpful....
-----
ale2@psu.edu (ale2) wrote:
> can you better Feynman? here is something from Lectures on Physics,
> 4-3:
>
> [why do half-integer spin particles obey Fermi statistics?
> Feynman laments that there appears to be no elementary answer
> in terms accessible to purely physical intuition.]
Interestingly, it was Feynman who finally bettered Feynman, with a
marvelous argument that appears in a Dirac festschrift by him and
Weinberg, but nowhere else in print, so far as I know.
I don't have the patience to do the argument justice, but here is
the outline; go find two half-full coffee cups, and in ten minutes
you should be able to convince yourself that the following is right.
First, note that there is something rum about 360 degree rotations,
namely that in situations where we have to keep track of the
`orientation-entanglement', or the topological relationship
between the object being rotated and the walls at infinity, we
find that 360 degrees is not the same as the identity, but that
720 degrees _is_ the identity. Imagine a coffee cup tied to all
the walls of the room with rubber bands. When you twist the
up through 360 degrees about its vertical axis, the rubber bands
get all twisted, and no amount of fiddling with them will untwist
them while the coffee cup remains in its 360 degree rotated state.
BUT, spin the cup ANOTHER 360 degrees in the same direction.
Now the rubber bands are twice as twisted, but it turns out that
you can pass them over the cup and then under it, and they
magically come untwisted again. It happens that your arm is
properly jointed to demonstrate this (though this is a trivial
fact about human anatomy, more than a deep fact about covering
groups of simple Lie algebras!). It's the Balinese candle-dance
trick, and if you don't know it, go find someone who can show you
how to do it. You hold the coffee cup with your right hand
underneath it, straight out in front of you. Now bring it left,
under your underarm, awkwardly around front with your elbow
straight up in the air. That's 360 degrees, and you're a
pretzel. Keep going around counterclockwise, this time swinging
your arm around over your head. At 720 degrees the coffee cup
is back where it started, unspilt, and your arm is straight
once more. Keep going round and round until you believe it.
OK, what does this have to do with spin-statistics? Well, I will
take as given concepts (1) that 1/2-integer spin wavefunctions
change sign under 360 degree rotations, and (2) that particles
odd under interchange obey the Pauli exclusion principle. So
the logical progression is
1/2 integer <-> odd under 360 <-> odd under swap <-> exclusion,
and I'm only concerned with the middle arrow. For the left
arrow, look at any good QM book, and see how the sigma matrices
work as a representation of the 3D rotation group. In a way
it's an unexpected miracle that complex 2-component spinors
can rotate like real 3-component vectors at all, and it turns
out that they do so in a way that forces us to keep track of
the orientation entanglement, or coffee-cup effect. For the
right arrow, note that the amplitude to find two Fermi-statistics
particles in the same place must be its own negative, hence zero.
All that remains is to bridge that middle arrow.
Now take TWO coffee cups, one in each hand. Swap the two
particles by crossing your arms. Uh-oh, you only swapped the
particles, not their orientation-entanglements (your shoulders).
Better fix that: while holding the coffee cups motionless in
space, walk around behind them so that you are facing the other
wall, and your arms are uncrossed. Oh, dear, one of your arms
is twisted 360 degrees now! (Which one depends on which way
you walked around.) So it looks like when you transposed the
two coffee cups, you really put one of them through a 360 degree
rotation with respect to the other. When you take the twisted
arm through 360 degrees to get clean with the universe again,
the two-fermion wavefunction will change sign, if the coffee
cups happen to be spin-1/2 particles that care about 360 rotations.
So odd under 360 <-> odd under swaps. QED.
A few more thoughts about orientation-entanglement. First, here
is a nice way (which I have never seen in print) to see that 720
really has to be the same as zero. If you take a fixed vector
and parallel-transport it around the earth at a given latitude,
it changes orientation by a natural angle equal to the solid angle
enclosed. This is Berry's phase, well known from Foucault's
pendulum and elsewhere. At 90 degrees north, we can send the
thing around the earth on a parallel of latitude without moving
an inch, so no solid angle is subtended by our path, and the
vector stays put. At 48.6 degrees north, we go around a circle
that encloses pi/2 steradians, and sure enough our pendulum swings
through pi/2 in the course of a day. At 30 degrees north, our circle
subtends pi radians, and our pendulum goes through a half-circle.
At the equator, the solid angle subtended is a whole 2 pi hemisphere,
so the vector describes a full circle in space as it goes once around
the earth. This is very different physically from what the vector
at the north pole does, namely nothing! But what happens at the
SOUTH pole? Our vector subtends a solid angle of 4 pi to the north,
the entire earth...or is it an angle of zero to the south? If the
two are going to be equivalent in all ways, then the behavior of a
physical object under 4 pi rotations must in ALL respects, including
orientation-entanglement, be the same as the identity. The equator
(360 degrees subtended) is different from the poles, but the poles
(0, or 720, degrees subtended) are just like each other.
One more goody, and I'm done. When you do the coffee cup trick,
your hand is describing a 720-degree rotation, but your shoulder is
doing nothing. In other words, there exists a homotopy from the
4 pi rotation to the identity, and your arm is that homotopy: the
rotation can be smoothly contracted to the identity, and each point
on the length of your arm describes one of the intermediate motions
in that smooth contraction. Now, when you donate platelets, they put
a tube in one arm to take the blood out, and one in the other arm to
put it back in. The blood goes through a centrifuge, and comes back
without the platelets, cooled to what feels like room temperature,
and they put an electric blanket on you and it doesn't help and you're
still cold as hell, because you're getting cooled on the INSIDE by
your own damn blood. The problem: the whole assembly has to be
sterile, just for you, and in fact there is a bag inside the
centrifuge that they use for your blood and then throw away, so
nothing else ever touches your blood. One hose goes into that bag,
and one comes out, and they are sealed without joints, and they spin
at several thousand rpm. *How the foxtrot uniform charlie kilo do
the two hoses avoid getting tangled up?*
I asked a nurse about this near the beginning of the two-hour ordeal.
It took me about forty minutes to convince her that there was a
fundamental problem, but it was worth it. She drove her whole
department insane about it for the next few weeks, and when I came
back the next time, they had taken a centrifuge and disassembled it
so they could see what was going on. Sure enough, the two hoses go
into a bracket which passes over, and under, and over, and under, at
exactly half the rotational speed of the centrifuge, because of the
way it is rigidly geared to the rotational motion. A Balinese candle
dance, at 7200 revolutions per minute.
``You can't make an omelette without +----------------------------------+
breaking eggs...but it is amazing how | Joshua W Burton (847)677-3902 |
many eggs you can break without making | jburton@nwu.edu |
a decent omelette.'' -- C. P. Issawi +----------------------------------+