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Re: Dirac String Trick



At 12:25 AM 5/13/01 -0400, Hugh Haskell wrote:
I suspect that the explanation of this trick is closer to quaternions
than fermions.

Hmmmmmm. I don't see how it could be much closer to one than the other.

-- I suspect that quaternions are rather closely related to
Pauli spin matrices.
-- I suspect that Pauli spin matrices have something to do
with the physics of fermions.

Once we have quaternions, we can talk about fermions (and a lot more).

===================

There is an fairly complete geometric explanation of
this type of rotation in the book "Icons and Symmertries," by Simon
L. Altman (London: Oxford University Press, 1992, ISBN:
0-19-855599-7), pp. 55 ff.

OK.

Other sources of information include the following:

A) You can search for
http://www.google.com/search?q=rotation+quaternions+flight+simulator
which turns up things like
http://www.gamasutra.com/features/19980703/quaternions_01.htm
which is a nice discussion of the application of quaternions to something
other than quantum mechanics and parlor tricks.

Here's the general outline of the idea: any rotation can be represented in
terms of quaternions (in exactly two ways). That is, quaternions, SU(2),
give a _double covering_ of the rotation group, SO(3).
-- The fact that it is a covering is obviously useful.
-- The fact that it is a _double_ covering is spooky but harmless.
-- The really special thing is that it is a _continuous_
representation. This comes in super-handy for representing sequences of
maneuvers in D=3 space. This stands in contrast with, say, Euler angles,
which have nasty singularities as you approach the polar axis (even though
the underlying physics is non-singular).

Pedagogical note: This year the US will spend more money on computer games
than on physics research. More of today's students will get jobs writing
flight simulators (for entertainment or for real pilot training) than will
get jobs doing quantum mechanics. So why should they bother with
physics??? Answer: if they pay attention in physics class they might
learn something about how the universe is put together. The ideas, and the
mathematical techniques, often come in handy in seemingly-unrelated fields.


B) You can search for something like
http://www.google.com/search?q=double+covering+continuous+euler+angles+qu
http://www.google.com/search?q=double+covering+continuous+euler+angles+quate
rnions+OR+Pauli
which turns up things like
http://npsnet.org/~bachmann/Research/Duman%20thesis.pdf
which is a master's thesis entitled:
DESIGN, IMPLEMENTATION, AND TESTING OF A REAL-TIME SOFTWARE
SYSTEM FOR A QUATERNION-BASED ATTITUDE ESTIMATION FILTER

The abstract begins:
Human limb segment angle tracking requires a system which can track
through all orientations. The major problem addressed by this research was
to develop a real time inertial motion tracking system based on quaternions
to overcome the singularities of Euler angle filters.
This work involves clarification of the theory behind quaternion
attitude estimation and development of a system capable of determining the
orientation of an object in world coordinates. System sensors were built
using miniature accelerometers, rate sensors, and magnetometers. ....