...that inspired our page on
candle-dances and atoms.A newsgroup post by Joshua Burton on the connection between...
...and the 2π rotated electron...
...that inspired our page on
candle-dances and atoms.
From an e-mail by Achim Rosch to Phil Fraundorf:
You are probably looking for the post which I have attached at the end. At that time I also asked John Burton by E-Mail for a reference on this subject, it is printed in
Feynman, Richard Phillips.
Elementary particles and the laws of physics : the 1986 Dirac
memorial lectures
Cambridge ; New York : Cambridge University Press, 1987.
x, 110 p. : ill.; 20 cm.
LC CALL NUMBER: QC793.28 .F49 1987
Regards
Achim
>From - Sun Apr 28 15:42:06 1996 Newsgroups:
sci.physics.research
From: jburton@nwu.edu (Joshua W. Burton)
Subject: Re: Challenge
Date: Fri, 26 Apr 1996 18:36:41 GMT
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?]
Interestingly, it was Feynman who finally bettered Feynman, with a marvelous argument that appears in a festschrift by him and Dirac, 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 cup 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, unspilled, 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.
Joshua W Burton
(847)677-3902
jburton@nwu.edu|
From: Joshua W. Burton <jburton@nwu.edu>
Date: Sun, 28 Apr 96 10:58:51 -0500
To: Achim Rosch <achim2@tkm.physik.uni-karlsruhe.de>
Subject: Re: Feynman/ Challenge
> Dear Mr. Burton, Can you give me some hint (better a reference) where this argument has been published?
It's a little red book with two essays, one by Feynman and one (I think) by Weinberg. This looks like it's probably it:
Feynman, Richard Phillips. Elementary particles
and the
laws of physics : the 1986 Dirac memorial lectures /
Cambridge ; New York : Cambridge University Press, 1987.
x, 110 p. : ill.; 20 cm.
LC CALL NUMBER: QC793.28 .F49 1987
If not, then the only other thing in the Library of Congress catalog that looks remotely right is:
Dirac and Feynman : pioneers in quantum
mechanics
New Delhi: Wiley Eastern, 1993. viii, 214 p. ; 25 cm.
LC CALL NUMBER: QC174.26.W28 D39 1993
Joshua W. Burton
(847)677-3902
jburton@nwu.edu
Note 19aug2013 by pf: Proposed Czech translation of this page by Marina Stepanenko.
Note 02feb1999 by pf: Related references from
Martin Gardner's New Ambidextrous Universe (Freeman,
1990) include
his discussion of the "Dirac scissors trick" on page
329,
and possibly as well:
• Proc. Symp. Pure Math. 48 (1988) 317-328.
• J. Math. Phys. 8 (1987) 345-366.
• Quantum Gravity (Oxford U. Press, 1975).
• Fiber Bundles, Sci. Amer. (July 1981).