Re: But where the heck is the spinor??
- From: Stephen Blake <stebla@xxxxxxxxxxxxxxxx>
- Date: Sat, 19 Apr 2008 16:48:23 +0000 (UTC)
neurop...@xxxxxxxxxxxx wrote:
Hi lads,
I've been reading the other thread "...intrinsic vs orbital
angular momentum..." which talks about spinors, r x p, etc, etc.
It mentioned ch41 of Misner, Thorne & Wheeler's "Gravitation"
book. I've studied all the math there quite thoroughly, no problems
with that. But seems like something's missing...
In Fig 41.6 on p1149 (the one with two concentric spheres, - the inner
sphere connected to the outer by threads, which you're then supposed
to
twist through 2pi or 4pi, and contort the inner sphere around to show
whether you can/can't untwist the threads using only translations of
the inner sphere). OK,... yeah, I get it. I've done the related "Dirac
belt" thing, and I get that 2pi rotation ain't necessarily the same as
4pi. But where the heck is the actual spinor in MTW's diagram??
Later in the chapter MTW rave on about poles and flags, but isn't
that just a combination of a 4-vector and a bivector? Where is the
spinor in "flag+pole"??. If I rotate the flag about its axis through
2pi, the flag returns to its original appearance and I don't see
anything spinor-like until I start trying to move the pole around (as
if the pole was elastic).
So I still have no clue where the spinor is in MTW's diagram. Should
I be thinking instead of an army of tiny flags all the way along an
elastic
flagpole? (That way, rotating only one end of the pole through 2pi/4pi
leaves obviously different orientations of the flags all the way along
the pole, and it's more obvious that all this 2pi-4pi monkey business
has something to do with rotation "here" without a matching rotation
at
"infinity".)
LOL,
Neuropulp.
Dear Neuropulp,
I agree with you, there does not appear to be any connection between a
spinor and Dirac's spanner illustrated by figure 41.6 on page 1148 of
MTW. Dirac's spanner illustrates the fact that the rotation group
SO(3) is not simply connected. The rotations in MTW's figure are
examples of the defining representation of SO(3) which is the spin 1
rep. A spinor rep such as spin 1/2 is not carried by ordinary 3-d
Euclidean space (for example).
Stephen Blake
http://www.stebla.pwp.blueyonder.co.uk
.
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- From: neuropulp
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