From: baez@guitar.ucr.edu (john baez)
Newsgroups: sci.physics,sci.math
Subject: Re: Spin/Statistics Heuristic Argument
Date: 3 Feb 1995 20:29:28 GMT
In article <3gsl9a$7jm@news2.delphi.com> LAURAHELEN@news.delphi.com (LAURAHELEN@DELPHI.COM) writes:
>baez@math.ucr.edu (john baez) writes:
>>In article <3gmgb3$o8j@noc2.drexel.edu>,
>>Jeff Steele wrote:
>>> if you rotate a thing (particle)
>>>with strings all over it, the strings are all twisted after 360
>>>degrees of rotation, but after 720 degrees the strings can be
>>>untangled without moving the object. Similarly, if two things
>>>(particles) connected by lots of strings are interchanged, the strings
>>>are left twisted up exactly as if one particle had been rotated by 360
>>>degrees. So the conclusion is that interchanging two particles is
>>>topologically indistinguishable from a rotation of one particle by 360
>>>degrees - a particle which changes sign after a rotation will be
>>>antisymmetric wrt pairwise interchange.
>This sounds like the "plate trick" -- you can rotate a plate held
>horizontally in your hand through 720 degrees, the first 360 degrees in the
>plane above your upper arm, the second in the plane below your upper arm.
>After the first 360 degrees your arm has one full twist which is undone
>in the second 360 degrees. Is this related? Anyone like to give an
>explanation of how to view this mathematically?
There is a nice picture of the plate trick in Feynman's contribution to:
Elementary particles and the laws of physics : the 1986 Dirac memorial
lectures / Richard P. Feynman and Steven Weinberg ; lecture notes compiled by
Richard MacKenzie and Paul Doust. Cambridge ; New York : Cambridge
University Press, 1987.
and I urge folks to check this out. It is possible thanks to the exact
same math that lets one take an object, connect it to the floor walls
and ceiling of your room by lots of threads, and then rotate it 720
degrees and untangle all the threads. (360 degrees will not work!)
For a picture of *this* to convince the skeptics, try:
Gravitation [by] Charles W. Misner, Kip S. Thorne [and] John Archibald
Wheeler. San Francisco, W. H. Freeman [1973].
in Section 41.5.
In both cases the key is the topology of the group of rotations in 3d
space --- known affectionately by experts as SO(3). Mathematically,
these are 3x3 real matrices which are "orthogonal" and also "special" ---
having determinant equal to one (hence the S and the O). You can
think of these as a subspace of the vector space of all 3x3 matrices.
The process of rotating 2 pi about the z axis traces out a loop
given by
cos t sin t 0
-sin t cos t 0
0 0 1
where t goes from 0 to 2 pi. This loop is not contractible, i.e., it
can't be "pulled tight" to a point. However, if we consider the
"doubled" loop where we let t go from 0 to 4 pi, it *is* contractible.
Proof: the plate trick. The plate traces out the loop corresponding to
a rotation by 4 pi. Your shoulder doesn't move at all, so it traces out
the boring loop that just sits there at one point. Any intermediate
portion of your arm traces out some other loop. As one moves from the
plate to your shoulder, one thus gets a 1-parameter family of loops
continuously interpolating between the 4 pi rotation and none at all.
For more about why this means that particles in 4d spacetime are either
bosons or fermions, try the article by Feynman cited above.
>So, if you had a photon which turns into an electron-positron pair, which
>then reunites to form a photon, this would be represented by a circular
>band? Then if you cut this circular band and attached a ribbon to each
>end of the cut, this would represent what you're talking about.
In my original posts I had ignored photons for simplicity.
I assume you are referring to something like this:
*
*
*
/ \
/ \
/ \
/ \
\ /
\ /
\ /
\ /
*
*
*
or something more topologically fancy such as
*
*
*
/ \
/ \ /\
/ \ \
\ \ /
\ / \/
\ /
*
*
*
or
*
*
*
/ \
/ \ /\
/ \ \
/ / \ \
\ \ / /
\ \ /
\ / \/
\ /
*
*
*
where the *'s represent photon worldlines and the |'s represent
electron/positron worldlines. Yes, indeed, one can consider these
things. (Feynman diagrams in textbook quantum electrodynamics
don't ever draw "twisty" worldlines as above, but that's because those
diagrams are implicitly sums over all possibilities.) Here you can
think of both the electron and photons worldlines as ribbons, but you
only get a phase of -1 for each time two *electron* worldlines cross.
That's because photons are bosons, while electrons are fermions.
> So does the photon have a "topology"?
Not quite sure what that means.
>What would rule out a Moebius strip instead of a circular band?
Well, we're using the ribbon to keep track of 360 degree rotations,
and we're not really interested in 180 degree rotations here, so we
don't use Moebius bands. One could however do a vast variety of
variations on this theme, some of which are mentioned in the file
braids.tex or braids.ascii, available in the directory baez by anonymous
ftp from math.ucr.edu..... and many more of which can be found in
Knots and physics / Louis H. Kauffman. 2nd ed. Singapore ; Teaneck, NJ :
World Scientific, c1993.
Series title: K & E series on knots and everything ; vol. 1.
For related themes one can also try my book with Javier P. Muniain,
volume 4 of the same series, called "Gauge Fields, Knots and Gravity".
>Is what's really happening, that the photon collides with an electron and
>sometimes the electron's wavefunction gets reversed, and this is a way of
>looking at this?
Well, each process gets an "amplitude", of which the phase aspect
discussed above only depends on the topology of the picture, and then
one must sum over all processes. Feynmans' book on QED is great for
that!