But where the heck is the spinor??


by neuropulp@yahoo.com.au
Tags: heck, spinor
neuropulp@yahoo.com.au
#1
Apr20-08, 05:00 AM
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.

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Stephen Blake
#2
Apr20-08, 05:00 AM
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

a student
#3
Apr21-08, 05:00 AM
P: n/a
On Apr 19, 3:52 am, 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??


I haven't got MTW here, but I do have a bit of a take on looking at
spinors (for spin 1/2 systems) - which might not be perfectly rigorous
(be warned!). 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.

You can think of spinors as a pair of objects: a unit direction in
space, and a +1 or -1 sign. So, the spinor can be represented as a
pair (n, +/-), where n.n=1.

What then is important is to ask is how the rotation group acts on a
spinor. For that, you need a picture of the rotation group, and how
it is doubly connected. Intuitively, one would think *at first* that
a good way of representing rotations is 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, (i.e., in
the direction a/|a|), by an amount equal to the distance of a from the
origin (i.e., by an angle equal to |a|). Since a rotation by pi about
some direction a is equivalent to a rotation by -pi about the
direction -a, one has to identify opposite points on the surface of
this ball. The effect of a rotation R_a, represented by point a of
the ball, would then change a spinor (n,+/-) to the spinor (R_a n,
+/-).

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. One can sit this ball in a different space to the first, but it
is mentally convenient to think of the second ball as sitting close
by. However, the second ball is inverted with respect to the first.
In particular, a point b in the second ball corresponds to a *left-
handed* rotation about the direction b/|b|, by an angle |b|. Now,
instead of identifying antipodal points of the first ball with each
other, one instead simply identifies the surface of the first ball
with the surface of the second ball (for essentially the same reason:
a right-handed rotation of pi about a given direction is equivalent to
a left-handed rotation of pi about the negative direction). This
gives us what we need.

A rotation is now characterised by a point a in *one* of the two
balls. If it is a point in the first ball, the effect on the spinor
(n,+/-) is to map it to the spinor (R_a n, +/-) as before. However,
if it is in the second ball, then the effect on the spinor is to map
it to the spinor (R'_a n, -/+), i.e., the second component of the
spinor is reversed in sign. Here, R'_a denotes a left-handed rotation
by angle |a|.

To see the 4 pi effect, consider the action on a spinor as one moves
through rotation space, starting at the centre of the first ball
(i.e., no rotation). First, move up along the z-axis to the north
pole. This is the same as the north pole is of the second ball, so
move over to the second ball's north pole (you can imagine a "virtual
trajectory" through the intervening 3d space if you like). Second,
move down the z-axis of the second ball to its centre. What has
happened to a spinor (n,+) ? Well, you ended up in the centre of the
second ball, so R'_0 is the identity matrix, and n is mapped to n - no
change there. But, you are in the second ball, so the second
component of n changes sign. Hence, (n,+) is mapped to (n,-).

Physically, what has happened? Well, first you rotated to the right
by pi, corresponding to moving up to the north pole of the first
ball. Then you hopped over to the second north pole, as you had
equivalently rotated to the left by pi. But then you undid the
rotation to the left by pi, by moving down to the centre of the second
ball, which is the same as rotating to the right by pi again. Hence,
overall, you rotated by 2 pi, but the spinor flipped its sign.

To get back to the original spinor, (n,+), one can imagine keeping
moving down to the south pole of the 2nd ball, then over to the south
pole of the 1st ball (draw another virtual trajectory between the two
balls), and up to the origin of the first ball. This corresponds
rotating by another 2 pi angle, i.e., 4 pi altogether, and the spinor
flips back from (n,-) to (n,+) again.

Hope this picture helps. The important thing is remember that the
balls aren't spinors, they are ways of visualising the action of the
rotation group on spinors.

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.


neuropulp@yahoo.com.au
#4
Apr22-08, 05:00 AM
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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