Re: Belt trick for spin 3/2
- From: "Cl.Massé" <retour@xxxxxxxxxxxxxxxxxxx>
- Date: Sat, 6 Jan 2007 17:13:33 +0100
As I am discriminated on sci.physics.research by the moderators Herr Phillip
Helbig and Mr. Igor
Khakine, I answer on this free group, which isn't controlled by Herr
Mautsch's chums:
"Thomas Mautsch" <mautsch@xxxxxxx> a écrit dans le message
news:456231a6@xxxxxxxxxxxxx:
^^^^^^^^^^^^^^ ^^^^^^^^^^^That is a quite rigorous reasoning, based on attentive observation.
^^^^^ ??? Observation of what?
^^^^^^^^^^The "belt trick" has never been intended
to be wholly mathematically equivalent,
"Equivalent"?! - Equivalent to what?
^^^^^^^^^^^^^^but to give a rough picture to lay persons.
The audience that the belt trick is shown
to, usually consists of physics students. -
I wouldn't call them "lay persons".
^^^^^ Proof of what?A true mathematical proof exists,
and it's all what is needed by a physicist.
I wish you good luck with this philosophy of yours! :-I
I wrote:
Like the conjuring trick of the cord that seems to be winded two times,
but that is made loose because the second time was in the other
direction.
The first time, the buckle is passed beneath, the second time it is
passed above. All what have been done is reversing the direction of
rotation with
^^^^
^^^^^^^^^^^^^^^^^^^respect to the belt, stealthily because it is supple. The axis isn't
fixed,
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^it is stealthily changed by a small amount, that has a big topological
consequence because the belt is very thin. To see that, it suffices to
make a continuous deformation, such that the buckle remains in a plane
perpendicular to the axis of the belt, which is kept as straight as
^^^^^^^^possible.
The parts I underlined indicate that he
wants to keep the middle line of the belt fixed,
even wants to keep it on a(n almost(?)) straight line.
This way the body of the belt can not move freely in space. -
He is basically only allowing the belt to describe
a path in SO(2). And since, unlike SO(3), there are
no elements of order two in the fundamental group pi_1(SO(2)),
he can not observe what the belt trick is about.
So, Herr Mautsch, like an authoritarian teacher, underlines in red some
phrases and indulge in derogatory comments. But who is he, and does he know
who I am? Unlike a teacher, I don't xerox whole passages of books and give
it as unquestionable food to my pupils. I relies on my own reasoning power,
and am able to spot misconceptions. In few words, education is different
from intelligence.
And now to the topic:
The belt is a one dimensional object, it can therefore only illustrate
SO(2), that's why the trick can't work. Actually it is intended to
illustrate a particular case in SO(3) SU(2), the one where a 360° rotation
with a fixed axis is performed, which corresponds to a subgroup SO(2) U(1).
Well, you can twist the belt by an angle of 4*Pi as described
above - keeping the axis of the belt fixed, but
you can never untwist it if you keep the middle line of the belt fixed.
You have to use a conjuring trick in order to make believe the subsequent
360° rotation is in the same direction, so untwisting rather than twisting
again. The magic is here: 360°-360° = 0°!
He does not seem to understand, that
the real belt trick is a demonstration that you can
untwist the belt from the "4Pi-twisted" position as above
when the ends are not allowed to be rotated
with respect to their respective positions, but are still
allowed to be moved around in space by parallel-translations.
Parallel in witch frame or reference? It is obvious that we have the same
process if it is seen in a rotating frame. The translation of the ends
isn't parallel in that frame. Let's take for configuration of departure the
belt merely folded in two equal parts on itself, and the frame that makes a
360° rotation around the belt axis in the appropriate direction during the
whole process. In it, we see that during the operation, the moving end
isn't passed behind but keep its location, while each end is 360° rotated
around the belt axis in the same direction, but actually in the opposite
direction because of the fold.
It isn't the trick I described above (the real trick for spin 1/2, not for
SO(3) or Spin(3)), this one should rather be called the tie trick. But it
also uses the stealthy inversion of the axis of rotation, in addition to a
frame conjuring trick to further muddy the waters.
Actually, the three dimensional space had to be introduced, but that changes
the topology, and the proof isn't about SO3 any longer. We now have the
fibre bundle SO3 x R3. In the attempt to change a one dimensional object to
a three dimensional one, we also changed the manifold, and so the nature of
the problem. In the conjuring trick of the women cut in two parts, it is
analogous to the introduction of a second women and the bending of the first
one.
Herr Mautsch, at the sole sound of "belt trick", jumped to the paragraph
where it is mentioned, without noticing or knowing that the end "spin 3/2"
indicates clearly that it isn't that trick, but its spin-off from Feynman
(The Spinor Spanner, E.D.Bolker, Am. Math. Monthly, 80(1973)977-84.), as
anxious to show off his knowledge as he is. So behave the idiots and the
pedantic.
^^^^^^^^^^The "belt trick" has never been intended
to be wholly mathematically equivalent,
"Equivalent"?! - Equivalent to what?
^^^^^^^^^^^^^^but to give a rough picture to lay persons.
The audience that the belt trick is shown
to, usually consists of physics students. -
I wouldn't call them "lay persons".
Herr Mautsch deduces that since the belt trick is given as "heuristical"
proof in schools, therefore is it a proof in the mathematical sense, not
aware that mathematics aren't made in classrooms where the teacher is always
right by definition, but in institutes. From which his confusion between
"intended to" and "given to". Dirac aimed at lay persons, but his bigot
followers took it for granted because it was Dirac. Authority arguments are
another clue of mind impairment.
--
~~~~ clmasse on free F-country, aka the Surprise Guest.
Knowledge is nothing without imagination.
.
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