But where the heck is the spinor??

by neuropulp@yahoo.com.au
Tags: heck, spinor
 P: n/a 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.
 P: n/a neurop...@yahoo.com.au 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
P: n/a

But where the heck is the spinor??

a student said:

> I haven't got MTW here, but I do have a bit of a take

Hi salacious student! Thanks for your "take". I like
being taken.

> on looking at spinors (for spin 1/2 systems) - which might
> not be perfectly rigorous (be warned!).

I'm fairly warned, but it's ok. Rigorous or non-rigorous,
I like it both ways.

> First though, it's important to keep in mind that spinors
> are what are acted on by SO(3) (rotations), they are not
> the rotations themselves.

Yep, but did you mean SO(3) or SU(2)? Oh, you're
representing the spinor as (n, +/-), so I guess you really
do mean SO(3).

> .... as a ball in 3-d space, centred at the origin,
> and of radius pi. A point a in the ball then represents
> a right-handed rotation about the a-direction, ....

OK... this is like one of Penrose's other diagrams (the one
that looks a bit like a mostly bald head with some hair-like
comb-overs from one side to the other).

> However, this picture obviously doesn't quite work, as
> nothing happens to the +/- sign of the spinor. The doubly
> connected nature of SO(3) has not shown up. To fix this, one
> needs a *second* 3d ball of radius pi. ....

Ah, and this is a bit different from Penrose. He's only got
one ball with antipodal points identified, which can be
confusing. I like your picture better - two balls plus a
free a tardis trip between corresponding points on their
surfaces. Yep, two balls are definitely better than one.

> However, the second ball is inverted with respect to the
> first.

OK, by "invert" you mean a 69'er with every point swapped
with its antipodal point? Not a conformal-type inversion
where the shell gets exchanged with the center, right? (BTW,
how do your spinors behave under those conformal inversions?
Still a factor of -1? But no, spinors in that case would be
twistors, wouldn't they?)

> The above picture also helps seeing how the rotation group
> SO(3) is connected. Drawing a path from one ball over to the
> other, as in the first 2 pi rotation above (with "virtual"
> portions), one clearly can't shrink this path to a point.
> However, extending such a path to go back to the original
> ball allows a loop to be formed, which can clearly be
> continuously shrunk to a point.

Yes, I've heard shrinkage can be tricky to manage. Homotopy
groups, right? By "shrinking" I guess you mean "shrink while
keeping endpoints fixed", which is only possible if the
endpoints coincide.

LOL.

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