| Topic: |
Science > Physics |
| User: |
"Edward Green" |
| Date: |
18 Jul 2006 10:29:14 PM |
| Object: |
SU(2) |
The group (or is it the group representation?) SU(2), wherein a 360
degree rotation is not an identity but a 720 rotation is, is supposed
to be illustrated by various parlor tricks.
In one example, known as the Balinese candle dance, we are asked to
hold something in an upright orientation -- a full cup of coffee or a
candle -- in the palm of our hand as we gradually rotate the object
clockwise or counter clockwise until it has made a 360 rotation about a
vertical axis, and our arm is twisted uncomfortably. Continuing
through another 360 degree rotation in the same sense, we improbably
find our arm untwisted, the object still level, and initial conditions
restored. This is supposed to illustrate that this "720 degree
identity" behavior originates from the connectivity of the rotated
object to its environment.
It makes a nice story, but does this model faithfully capture the
mathematics? I'm not sure. During the first orbit our arm acquires a
twist about its longitudinal axis, while during the second orbit the
twist is undone. The sense of the twist depends on whether the forearm
passes over or under the upper arm. But there's the rub: if we
continued rotating in the same sense but again passed the forearm above
or below, respectively, then instead of undoing the twist, we would
double it. Anatomy prevents it but Rubber Man would have a choice.
If a 360 degree rotation of the palm faithfully corresponded to
multiplication by -1 (a square root of identity), then there should be
no further choice involved. Instead, the rotation seems to correspond
to addition of +/- 1, with an enforcement of alternation of signs by
anatomy.
Do the remaining illustrations suffer from the same defect? Is it
possible to illustrate the group behavior faithfully using connections
in three dimensions, or merely plausibly?
.
|
|
| User: "PD" |
|
| Title: Re: SU(2) |
19 Jul 2006 08:14:30 AM |
|
|
Edward Green wrote:
The group (or is it the group representation?) SU(2), wherein a 360
degree rotation is not an identity but a 720 rotation is, is supposed
to be illustrated by various parlor tricks.
In one example, known as the Balinese candle dance, we are asked to
hold something in an upright orientation -- a full cup of coffee or a
candle -- in the palm of our hand as we gradually rotate the object
clockwise or counter clockwise until it has made a 360 rotation about a
vertical axis, and our arm is twisted uncomfortably. Continuing
through another 360 degree rotation in the same sense, we improbably
find our arm untwisted, the object still level, and initial conditions
restored. This is supposed to illustrate that this "720 degree
identity" behavior originates from the connectivity of the rotated
object to its environment.
It makes a nice story, but does this model faithfully capture the
mathematics? I'm not sure. During the first orbit our arm acquires a
twist about its longitudinal axis, while during the second orbit the
twist is undone. The sense of the twist depends on whether the forearm
passes over or under the upper arm. But there's the rub: if we
continued rotating in the same sense but again passed the forearm above
or below, respectively, then instead of undoing the twist, we would
double it. Anatomy prevents it but Rubber Man would have a choice.
If a 360 degree rotation of the palm faithfully corresponded to
multiplication by -1 (a square root of identity), then there should be
no further choice involved. Instead, the rotation seems to correspond
to addition of +/- 1, with an enforcement of alternation of signs by
anatomy.
Do the remaining illustrations suffer from the same defect? Is it
possible to illustrate the group behavior faithfully using connections
in three dimensions, or merely plausibly?
To see this executed without the over/under problem, do the same thing
but replacing the arm with a belt with one end tied not at the shoulder
but caught in a closed door. Tie the other end of the belt to the
coffee cup (or book) and repeat the exercise.
PD
.
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|
| User: "Ben Rudiak-Gould" |
|
| Title: Re: SU(2) |
19 Jul 2006 10:48:28 AM |
|
|
Edward Green wrote:
[...Balinese candle dance...]
It makes a nice story, but does this model faithfully capture the
mathematics?
It does, provided you recognize that two configurations of your arm are
considered equivalent if you can deform one into the other without changing
the orientation of your hand. If your arm were flexible enough you could
rotate 360 degrees over or under twice in a row and then untwist your arm
without moving your hand. But after a single 360-degree rotation it's
impossible to untwist your arm, no matter how flexible.
By the way, you needn't restrict yourself to rotations around the vertical
axis. If you experiment you'll find that a 360-degree rotation of your hand
around any axis will get you into the same twisted configuration, and
another 360-degree rotation around any other axis will get you out. In
principle. As PD said, it's easier with a belt.
-- Ben
.
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|
|
| User: "Edward Green" |
|
| Title: Re: SU(2) |
19 Jul 2006 06:49:18 PM |
|
|
Ben Rudiak-Gould wrote:
Edward Green wrote:
[...Balinese candle dance...]
It makes a nice story, but does this model faithfully capture the
mathematics?
It does, provided you recognize that two configurations of your arm are
considered equivalent if you can deform one into the other without changing
the orientation of your hand. If your arm were flexible enough you could
rotate 360 degrees over or under twice in a row and then untwist your arm
without moving your hand. But after a single 360-degree rotation it's
impossible to untwist your arm, no matter how flexible.
By the way, you needn't restrict yourself to rotations around the vertical
axis. If you experiment you'll find that a 360-degree rotation of your hand
around any axis will get you into the same twisted configuration, and
another 360-degree rotation around any other axis will get you out. In
principle. As PD said, it's easier with a belt.
My subjective prior on the correctness of what you wrote is high based
on the fact that Ben Rudiak-Gould wrote it. But I also have a strong
prior on the correctness of what I wrote, based on the fact that I
tried it, in thought and practice. Are we talking about the same
thing?
Say we replace the arm and the belt with a gooseneck, which can be
bent, but not twisted. The gooseneck is fixed rigidly at one end to a
face of a cube, and at the other to a wall via a bearing, which allows
the neck to turn freely about its longitudinal axis. A line is drawn
along the top of the neck, initially straight, which allows us to track
the rotation of the neck at the wall.
Rotate the cube 180 degrees about a vertical axis. The neck must
either pass over or under the cube at this point, as the point of
connection is on the far side, and it can be seen that the neck has
also rotated 180 in its bearing on the wall. It can also be seen that
as the block passes through this extreme point of the rotation the neck
maintains a given sense of rotation in the bearing, and returns to its
starting orientation when the cube returns to its starting orientation.
However, at the end of one such rotation the neck has rotated 2pi, and
would have had a twist in it except for the bearing. Furthermore, the
neck will continue to rotate in this same sense in the bearing if we
constantly pass it over (or respectively, under) the cube during the
rotation. In order to untwist the neck, we must first pass it one way
and then the other way as we continue to rotate the cube in the same
sense.
So, no, I don't see how the over under problem goes away, and yes, I
did try it with a book and a tie (instead of a belt), and this is
exactly what I observed: if I continually passed the tie over the same
side of the book as I rotated it, the tie became progressively more
twisted, and could only be untwisted, keeping the same rotation sense
of the book, by passing it on the other side. Maybe the rigidity of a
stiff belt somehow discretely enforces the rule that the belt pass on
opposite sides of the book on progressive rotations, creating the
illusion that this is a mathematical necessity?
If the electron is joined to its environment by a virtual arm, then its
an arm which can only twist so much, and is contrained to twist first
in one sense, then the other, as the electron pirouettes full circle
twice. I notice that the real human arm doesn't give us any choice
about whether to wind and then unwind, or vice versa, as we rotate in
one sense: the choice is determined by the handedness of the arm.
Since "handedness" is a good concept to have around any time one is
speaking of something like rotation, this makes a similar connectivity
for the electron not totally implausible! Maybe the analogy with the
candle dance is deeper than we think.
.
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|
|
| User: "James Dolan" |
|
| Title: Re: SU(2) |
20 Jul 2006 12:27:46 AM |
|
|
in article <1153352958.767789.62290@s13g2000cwa.googlegroups.com>,
edward green <spamspamspam3@netzero.com> wrote:
|Ben Rudiak-Gould wrote:
|> Edward Green wrote:
|> > [...Balinese candle dance...]
|> >
|> > It makes a nice story, but does this model faithfully capture the
|> > mathematics?
|>
|> It does, provided you recognize that two configurations of your arm are
|> considered equivalent if you can deform one into the other without changing
|> the orientation of your hand. If your arm were flexible enough you could
|> rotate 360 degrees over or under twice in a row and then untwist your arm
|> without moving your hand. But after a single 360-degree rotation it's
|> impossible to untwist your arm, no matter how flexible.
|>
|> By the way, you needn't restrict yourself to rotations around the vertical
|> axis. If you experiment you'll find that a 360-degree rotation of your hand
|> around any axis will get you into the same twisted configuration, and
|> another 360-degree rotation around any other axis will get you out. In
|> principle. As PD said, it's easier with a belt.
|
|My subjective prior on the correctness of what you wrote is high based
|on the fact that Ben Rudiak-Gould wrote it. But I also have a strong
|prior on the correctness of what I wrote, based on the fact that I
|tried it, in thought and practice. Are we talking about the same
|thing?
|
|Say we replace the arm and the belt with a gooseneck, which can be
|bent, but not twisted. The gooseneck is fixed rigidly at one end to
|a face of a cube, and at the other to a wall via a bearing, which
|allows the neck to turn freely about its longitudinal axis. A line
|is drawn along the top of the neck, initially straight, which allows
|us to track the rotation of the neck at the wall.
|
|Rotate the cube 180 degrees about a vertical axis. The neck must
|either pass over or under the cube at this point, as the point of
|connection is on the far side, and it can be seen that the neck has
|also rotated 180 in its bearing on the wall. It can also be seen
|that as the block passes through this extreme point of the rotation
|the neck maintains a given sense of rotation in the bearing, and
|returns to its starting orientation when the cube returns to its
|starting orientation.
|
|However, at the end of one such rotation the neck has rotated 2pi,
|and would have had a twist in it except for the bearing.
|Furthermore, the neck will continue to rotate in this same sense in
|the bearing if we constantly pass it over (or respectively, under)
|the cube during the rotation. In order to untwist the neck, we must
|first pass it one way and then the other way as we continue to rotate
|the cube in the same sense.
|
|So, no, I don't see how the over under problem goes away, and yes, I
|did try it with a book and a tie (instead of a belt), and this is
|exactly what I observed: if I continually passed the tie over the
|same side of the book as I rotated it, the tie became progressively
|more twisted, and could only be untwisted, keeping the same rotation
|sense of the book, by passing it on the other side. Maybe the
|rigidity of a stiff belt somehow discretely enforces the rule that
|the belt pass on opposite sides of the book on progressive rotations,
|creating the illusion that this is a mathematical necessity?
|
|If the electron is joined to its environment by a virtual arm, then
|its an arm which can only twist so much, and is contrained to twist
|first in one sense, then the other, as the electron pirouettes full
|circle twice. I notice that the real human arm doesn't give us any
|choice about whether to wind and then unwind, or vice versa, as we
|rotate in one sense: the choice is determined by the handedness of
|the arm. Since "handedness" is a good concept to have around any
|time one is speaking of something like rotation, this makes a similar
|connectivity for the electron not totally implausible! Maybe the
|analogy with the candle dance is deeper than we think.
the balinese dance trick _is_ a legitimate demonstration of what it
purports to show: namely, that a specific element y in the so-called
"fundamental group" of the space x = so(3) has the property that y^2
is equal to the identity element.
however, there are several confounding factors to the trick that can
cause confusion (especially when piled on top of each other). these
confounding factors are often swept under the rug because in the final
analysis they don't de-legitimize the trick, and because the very
theatricality of the trick naturally promotes the spirit of sweeping
things under the rug.
to counter the resulting tendency of the trick to create unwarranted
suspicion about its own legitimacy, i will in this post take the
opposite of the usual approach, by sweeping as many things as possible
_out_ from under the rug, to try to show how harmless they are.
the general principle that the trick is based on is this:
living in the universal covering space of a space x is just like
living in x, except that you're like a dog on a leash. that means
that in order to specify your location in the universal covering
space, you have to specify not only your location in x, but also an
additional piece of information describing "the state of your
leash", meaning how your leash is threaded through "wormholes" in x.
for example if there's a tree in your backyard then going around the
tree is effectively a wormhole, and a single location in x is
effectively many different possible locations to your dog, depending
on how many times (and in which direction) his leash is threaded
through the wormhole.
given a point p in x, the extra information describing "the state of
your leash", needed to specify your location in the universal
covering space when you already know that your location in x is p,
is precisely an element of the fundamental group of x at the
basepoint p.
the first confounding factor that we must face here, though, applies
quite generally: the above general principle is _not valid_ unless
your leash is a magical leash that not only has unlimited
stretchabiility but can also pass through itself. there's a big
difference between a leash getting tangled with itself and a leash
getting threaded through a wormhole, and officially here we're only
interested in the latter phenomenon. by requiring your leash to be
magical we ensure that it can't get tangled with itself.
the second confounding factor that we have to face here is specific
rather than general: the space x = so(3) whose universal covering
space (which is called "su(2)") we're interested in exploring is
accessible to our intuition most directly not as a "physical" space
(like the space of points in your backyard) but rather as the
"configuration space" of a physical object, namely of a rigid 3d
object with only rotational degrees of freedom.
our task is then to imagine an object living "on a leash" in this
configuration space. and that turns out to be not as hard to imagine
as you might at first fear: take the object to be a cup of coffee,
take the leash to be your arm, and take the basepoint that the leash
is staked to to be your shoulder. this works because each point along
your arm has a sense of what its own rotational configuration is, and
your arm gets noticeably unhappy when these configurations fail to
vary continuously as a function of position along the arm. (being
normally stiff-jointed is here seen to be a feature rather than a
bug.)
now watch in your mind's eye as the trick is performed: first the
coffee cup performs a 360-degree clockwise loop (and its leash goes
along for the ride, and after the maneuver is finished the state of
the leash amounts to a space-like historical record of the time-like
motion performed by the coffee cup). then the coffee cup is asked to
perform _the exact same maneuver_ a second time. and it does, though
you might worry at first that it's performing a slightly different
maneuver, because if the cup went over your arm the first time then
(nine times out of ten) it'll go under your arm the second time. but
in the relevant configuration space this difference between "over" and
"under" has no meaning! the "difference" is registered only in
certain extra coordinate dimensions that have no official meaning in
the conceptual context of the trick. these extra coordinate
dimensions function in the trick only as a convenience to help us
simulate the "magicalness" of the leash (that it can pass through
itself, simulated by sidestepping into a harmless and inessential
extra dimension).
once you realize that this is what's going on you should be able to
reassure yourself of the legitimacy of the trick without even having
to work through all the details of how the conventional steps of the
trick simulate aspects of the magicalness of the leash.
(this is not to say that you couldn't somehow take your intuition that
says that the second maneuver isn't an exact repetition of the first
maneuver and somehow turn it into a legitimate objection to some
different version of the trick that purported to show that a
repetition of the first maneuver cancels out the first maneuver even
when the leash is granted some lesser degree of magical power.)
--
jdolan@math.ucr.edu
.
|
|
|
| User: "Paul Abbott" |
|
| Title: Re: SU(2) |
20 Jul 2006 01:44:57 AM |
|
|
In article <e9n48i$ab3$1@glue.ucr.edu>, (James Dolan)
wrote:
the balinese dance trick _is_ a legitimate demonstration of what it
purports to show: namely, that a specific element y in the so-called
"fundamental group" of the space x = so(3) has the property that y^2
is equal to the identity element.
Since it has not been mentioned in this thread, I should add that it is
also known as the "Philippine wine dance" and the "Dirac String Trick".
See
http://www.meru.org/coast/DiracPC71997.pdf
and "Air on the Dirac Strings" at
http://www.evl.uic.edu/hypercomplex/html/dirac.html
Cheers,
Paul
_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul
.
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|
|
| User: "Edward Green" |
|
| Title: Re: SU(2) |
20 Jul 2006 07:57:52 AM |
|
|
Paul Abbott wrote:
In article <e9n48i$ab3$1@glue.ucr.edu>, (James Dolan)
wrote:
the balinese dance trick _is_ a legitimate demonstration of what it
purports to show: namely, that a specific element y in the so-called
"fundamental group" of the space x = so(3) has the property that y^2
is equal to the identity element.
Since it has not been mentioned in this thread, I should add that it is
also known as the "Philippine wine dance" and the "Dirac String Trick".
See
http://www.meru.org/coast/DiracPC71997.pdf
and "Air on the Dirac Strings" at
http://www.evl.uic.edu/hypercomplex/html/dirac.html
[Being a sophisticate is not bad, but being an unaffected simpleton
also has its compensations! If I were a sophisticate, I might engage
James Dolan (whose word hoard may even exceed my own) on his own
ground. If I were an affected simpleton, I might shrink from admitting
that everything he wrote wasn't trivial to me, and so allow a proof by
intimidation. Being an unaffected simpleton, I will instead focus on
what I do understand rather than on what I don't.
First, some questions:
Given an inner sphere attached via a ribbon to an outer sphere, and
giving the inner sphere a 360 degree rotation about the axis of the
ribbon, then the twist of the ribbon is:
(b) indeterminate up to +/- n720 degrees.
This can be understood by considering that there are two operations at
work: a rotation of the inner sphere, which gives the ribbon a twist
identical to the sphere's rotation, and a maneuver in which the ribbon
loops behind the inner sphere, independent of rotation of the sphere.
This latter maneuver twists or untwists the ribbon in 720 degree
increments.
If the inner sphere is rotated about an axis perpendicular to the
ribbon axis, then to unwrap the ribbon from around the sphere it must
be passed on one side or the other in what amounts to half the ribbon
maneuver, arriving in one of two configuration separated by a twist of
720 degrees -- the "over/under" ambiguity.
In order to describe an operation in our model, it is not sufficient to
given a rotation to the inner sphere, and let the ribbon come along for
the ride. A behavior must be specified for the ribbon also.
Faithful Representations of a Proper Square Root of the Identity
Every term in this heading is probably clear, but formally, say we have
a set of elements x,y,z,..., and a set of operations A,B,C,... which
can be applied to elements of the first set. An identity I is an
operation s.t. Ix = x for all x. A proper square root of the identity
is an operation A s.t. AA = I, but A /= I. Now we are ready for our
second question:
Does the model of nested spheres connected by a ribbon contain a
faithful representation of a proper square root of the identity?
(c) it depends
It depends in particular on exactly how we define the operation. To be
a square root, then each application of the operation in the square
must be identical. At this point, you may expect me to argue that
"passing the ribbon over the first time and under the second is clearly
not identical, so this is not a faithful representation of a square
root". However, I'm trying to be open minded. Nothing stops us from
defining a semantically more complex operation (for the case where
rotation is about the ribbon axis):
"Rotate the inner sphere, and, if the ribbon is originally untwisted,
allow it to acquire a 360 degree twist. However, if the ribbon is
already twisted in the same sense, combine the rotation of the sphere
with a crossing behind of the ribbon to remove this twist".
Applying this same rule twice in succession returns the starting state
-- at least if we are careful to arrange things so more complicated
states of ribbon twist are never encountered -- and so faithfully
represents a square root of the identity operation.
This may seem like a trick, and indeed, any illustration -- however
cunningly executed by computer graphics -- of a ribbon twisting and
untwisting itself as if that were the only geometric possibility, is
misleading. The ribbon _can_ untwist ...
Never mind. :-) I guess the operative word is "can". If we take any
configuration of the ribbon which can be reached by continuous
deformation of the ribbon -- without moving the spheres -- as
equivalent, then the ribbon indeed comes back to an equivalent
configuration after a 720 degree rotation of the inner sphere, whether
or not we bother unwrapping it.]
Edward Green wrote:
[...Balinese candle dance...]
It makes a nice story, but does this model faithfully capture the
mathematics?
It does, provided you recognize that two configurations of your arm are
considered equivalent if you can deform one into the other without changing
the orientation of your hand. If your arm were flexible enough you could
rotate 360 degrees over or under twice in a row and then untwist your arm
without moving your hand. But after a single 360-degree rotation it's
impossible to untwist your arm, no matter how flexible.
Aha. I have it now.
.
|
|
|
| User: "Lee Rudolph" |
|
| Title: Re: SU(2) |
20 Jul 2006 08:16:07 AM |
|
|
"Edward Green" <spamspamspam3@netzero.com> writes:
....
Given an inner sphere attached via a ribbon to an outer sphere, and
giving the inner sphere a 360 degree rotation about the axis of the
ribbon, then the twist of the ribbon is:
(b) indeterminate up to +/- n720 degrees.
This can be understood by considering that there are two operations at
work: a rotation of the inner sphere, which gives the ribbon a twist
identical to the sphere's rotation, and a maneuver in which the ribbon
loops behind the inner sphere, independent of rotation of the sphere.
This latter maneuver twists or untwists the ribbon in 720 degree
increments.
On general principles, this can be transformed with no loss of the
properties you're interested in to the case of two spheres in 3-space
each exterior to the other; which is easier to see, maybe.
It can also be gussied up with references to "braid groups of the
2-sphere" (said groups being proper quotients of the corresponding
braid groups of the plane, the quotienting coming from being able
to "slip across the point at infinity"), in particular the 2-string
braid group (the two strings being the opposite hems of your ribbon).
Lee Rudolph
.
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|
|
| User: "Edward Green" |
|
| Title: Re: SU(2) |
21 Jul 2006 10:44:06 AM |
|
|
Lee Rudolph wrote:
"Edward Green" <spamspamspam3@netzero.com> writes:
...
Given an inner sphere attached via a ribbon to an outer sphere, and
giving the inner sphere a 360 degree rotation about the axis of the
ribbon, then the twist of the ribbon is:
(b) indeterminate up to +/- n720 degrees.
This can be understood by considering that there are two operations at
work: a rotation of the inner sphere, which gives the ribbon a twist
identical to the sphere's rotation, and a maneuver in which the ribbon
loops behind the inner sphere, independent of rotation of the sphere.
This latter maneuver twists or untwists the ribbon in 720 degree
increments.
On general principles, this can be transformed with no loss of the
properties you're interested in to the case of two spheres in 3-space
each exterior to the other; which is easier to see, maybe.
That's the picture I was using.
It can also be gussied up with references to "braid groups of the
2-sphere" (said groups being proper quotients of the corresponding
braid groups of the plane, the quotienting coming from being able
to "slip across the point at infinity"), in particular the 2-string
braid group (the two strings being the opposite hems of your ribbon).
Thanks for the additional info.
Ben Rudiak-Gould nailed my question right off, though I couldn't
immediately see it. The hint came from watching the video linked by
Paul Abbott, which however suffered from the defect associated with the
wine glass trick, that it made the periodic unwinding seem mechanically
inevitable, which it is not. Except, that is, for the human arm.
Maybe this is fact a feature, not a defect.
The video suffered from another defect, imho. The narrator mentioned
something about an observer, explaining what the particle was linked to
in the outer sphere. That's fine, but all particles have observers,
while not all particles behave this way. There is some special
structural feature of the fermions at work.
My thanks also to Jos=E9 Carlos Santos for the references.
.
|
|
|
| User: "Mark L. Fergerson" |
|
| Title: Re: SU(2) |
24 Jul 2006 09:17:16 AM |
|
|
Edward Green wrote:
Lee Rudolph wrote:
"Edward Green" <spamspamspam3@netzero.com> writes:
Given an inner sphere attached via a ribbon to an outer sphere, and
giving the inner sphere a 360 degree rotation about the axis of the
ribbon, then the twist of the ribbon is:
(b) indeterminate up to +/- n720 degrees.
This can be understood by considering that there are two operations at
work: a rotation of the inner sphere, which gives the ribbon a twist
identical to the sphere's rotation, and a maneuver in which the ribbon
loops behind the inner sphere, independent of rotation of the sphere.
This latter maneuver twists or untwists the ribbon in 720 degree
increments.
On general principles, this can be transformed with no loss of the
properties you're interested in to the case of two spheres in 3-space
each exterior to the other; which is easier to see, maybe.
That's the picture I was using.
It can also be gussied up with references to "braid groups of the
2-sphere" (said groups being proper quotients of the corresponding
braid groups of the plane, the quotienting coming from being able
to "slip across the point at infinity"), in particular the 2-string
braid group (the two strings being the opposite hems of your ribbon).
Thanks for the additional info.
Ben Rudiak-Gould nailed my question right off, though I couldn't
immediately see it. The hint came from watching the video linked by
Paul Abbott, which however suffered from the defect associated with the
wine glass trick, that it made the periodic unwinding seem mechanically
inevitable, which it is not. Except, that is, for the human arm.
Maybe this is fact a feature, not a defect.
BTW the trick can be done twisting one's arm either way,
"left-handed" or "right-handed". I've seen somewhere a video of a candle
dancer doing both hands opposite ways. Dizzying, it was.
The video suffered from another defect, imho. The narrator mentioned
something about an observer, explaining what the particle was linked to
in the outer sphere. That's fine, but all particles have observers,
while not all particles behave this way. There is some special
structural feature of the fermions at work.
For giggles, think about what may be meant by "observer" here. If
you're the one doing the candle/wineglass trick, you're the observer and
there are two non-equivalent "same" orientations for the artifact.
But you're standing on a planet, and anyone also standing on the
planet sees the same thing.
Remember your question about sitting in a swivel chair and trying to
change your orientation by swinging an arm around?
Now try the trick while standing on a lazy susan (assume frictionless
bearings). What happens?
Mark L. Fergerson
.
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|
| User: "Ben Rudiak-Gould" |
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| Title: Re: SU(2) |
21 Jul 2006 10:38:36 AM |
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Paul Abbott wrote:
http://www.meru.org/coast/DiracPC71997.pdf
http://www.evl.uic.edu/hypercomplex/html/dirac.html
Here's an animation showing the belt, the equivalent path in SO(3), and the
spinor corresponding to the end of the belt:
http://gregegan.customer.netspace.net.au/APPLETS/21/21.html
and an explanation of it:
http://gregegan.customer.netspace.net.au/APPLETS/21/DiracNotes.html
-- Ben
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| User: "=?ISO-8859-1?Q?Jos=E9_Carlos_Santos?=" |
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| Title: Re: SU(2) |
19 Jul 2006 04:07:10 AM |
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On 19-07-2006 4:29, Edward Green wrote:
Do the remaining illustrations suffer from the same defect? Is it
possible to illustrate the group behavior faithfully using connections
in three dimensions, or merely plausibly?
Suggested reading:
Bolker, Ethan D.
The spinor spanner.
American Mathematical Monthly 80, 977-984 (1973)
and also, perhaps,
Hartung, R. W.
Pauli principle in Euclidean geometry
American Journal of Physics 47, 900-910 (1979)
Best regards,
Jose Carlos Santos
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