Bossavit
Nov8-06, 06:00 AM
As repeatedly asserted here, there is no "conjuring",
in the sense of "cheating", in Dirac's belt trick.
Yet, one may legitimately feel some frustration with
it, for though it "proves" its point (that SO3 is not
simply connected), it does so in a needlessly involved
way.
"Proves", here, actually means, "help visualize".
(Mathematical proof is another issue.) As
illustrated in Greg Egan's display, such visualization
is easy without losing one's pants: Think of a solid
ball in 3D space, centered at some point, dubbed "the
origin" from now on, with pairs of antipodal points
identified. Topologically, this is SO3, the 3D rotations
Lie group. The point to make is that there are two
classes of origin-centered loops in this space, those of
the first class being shrinkable to 0, while the others
are not: Any loop not crossing the surface belongs in
the first class. To get a representative of the second
class, go straight from 0 to the boundary (say, at point
A), reenter from the opposite point (A', say), return
straight to 0. This is a closed loop in SO3, obviously
not reducible to 0 by continuous deformation within SO3.
On the other hand, run along the previous path twice (or
any even number of times), and the resulting loop can
now be shrunk to 0 (provided one correctly interprets
the meaning of "loop": one is supposed to return to 0
at the end of the journey, but the mid-trip passage
through 0 is not mandatory--actually it *must* be avoided
as the loop is deformed).
The latter loop is thus from 0 to A, then jump to A',
go from A' to A, jump to A' again, and return to 0.
How to shrink it to 0, is what Greg Egan's
animation shows. It involves changing direction the
second time around, progressively: Instead of going
from A' to A, go straight from A' to some B
close to A, on the surface (do *not* go through 0
this time), jump to B' opposite to B, then back
home. This new loop, 0AA'BB'0, is "close", in
an obvious sense, to 0AA'AA'0. Now, keeping the first
leg of the journey (from 0 to A aka A') unchanged,
move B (and hence, B', the same point in SO3,
actually) towards A', on the surface. The
deformed loop is now 0AA'A'A0, that is to say, 0A0,
which nicely retracts to 0.
[All this is very easy to draw, though not in ASCII
unfortunately. I tend to think that such a series of
drawings, with comments, may be better as a teaching
aid than an animation, but that's a side issue.
(I did enjoy this animation, as well as others proposed
by Greg Egan, but I always had to first "see things with
the eyes of the mind", as Poirot is suppposed to say
in French translations, before being able to appreciate
them.)]
Now, the belt. Fasten the buckle in some fixed
position (a door knob may do), hold firm the
other end. Pause to show the belt is flat,
i.e., untwisted. Now give it two full (360o) turns
(around some fixed spatial axis, for neatness
--the vertical, say). You used both hands to do
that, presumably. Pause, keep loose end in one
hand, bring attention to the double twist in
the belt. Now, *always keeping this loose end of the
belt (approximately) vertical and not rotating
(not by more than a few degrees)*, pass it behind the
rest of the belt, then back to where it was (notice
you *must change hands* [or perhaps jump above
the belt... never tried that one] to achieve
that; maybe this is what evokes a "conjuring
trick" to some--we'll return to that). See the
belt, miraculously, flatten back to its original
configuration. Make the point that
nothing of this kind could be done after only
*one* full turn (or three, for that matter, if
the belt could stand it). Demonstrate also that
the loose end can be kept very close to the knob-
and-buckle end in this process, with its own frame
almost parallel, always, to the buckle's frame.
What has thus been "proved"? With respect to the
knob and buckle, each segment t of the belt, t
going from 0 to 1, has a translation vector v(t)
in 3D vector space V3, and a rotation s(t), element
of SO3, with respect to the buckle's frame. So
the belt materializes a path, t --> {v(t, s(t)},
in the product space V3 x SO3, i.e., in the Lie group
D3 of (direct) 3D displacements, the unit element
of which corresponds to the {knob, buckle's frame}
pair. [The algebraic structure of D3, as a
semi-direct product, is ignored here--perhaps
not a good thing.] Since buckle and loose end can
be kept very close and almost parallel,
this path is an honest approximation to a *loop*,
a loop in D3. Flat belt and doubly twisted belt
materialize two such loops, which are shown to be
homotopic by performing the trick. So what is
addressed, really, is the non-trivial homotopic
structure *of D3*, not SO3.
From which, of course, the same about SO3 is
easily *derived*. But let's not be fooled. The
Dirac's trick is an involved one, which achieves
more than was bargained for. Too much, perhaps.
Arguably, it takes too much classroom time,
if compared to the sequence-of-drawings process.
But also, and more of a concern, it tends to elicit
further questions which the teacher may not feel
ready to address.
For example, we noticed that the path in D3 could
be considered, for what it's worth, as a loop. So
why not make it a real, physical loop? Take the belt,
give it two full turns, now *fasten the belt*. Any
way to disentangle it? You bet. The inspired remark
that "if the belt's material was this kind of magic
stuff that can intersect itself, then the belt could
untwisted (after 2n turns, and not after 2n + 1)"
may not win applause.
And yet, this is it, for a part: D3 can be
considered as a fibered space, fiber SO3, base V3
(or rather, more accurately, base A3, the affine
3D space), and while homotopies in D3 do
generate homotopies in SO3, they don't project
to homotopies in A3. This is why the belt must
intersect itself to untwist, or be left unbuckled
in order to leave room for the manipulation (which
by the way could not be achieved if the loose end
was always kept in the same hand). The suspicion
of a "conjuring trick", though unfair, can thus be
understood, if not endorsed.
The "plate trick", though its analysis is involved
too, looks a little more transparent, but discussing
this would take too long. More interestingly perhaps,
it would be nice if topologists could explain to us,
more thoroughly and accurately than suggested here,
the mathematics behind the belt trick, beyond the
basic fact that SO3 is doubly connected.
in the sense of "cheating", in Dirac's belt trick.
Yet, one may legitimately feel some frustration with
it, for though it "proves" its point (that SO3 is not
simply connected), it does so in a needlessly involved
way.
"Proves", here, actually means, "help visualize".
(Mathematical proof is another issue.) As
illustrated in Greg Egan's display, such visualization
is easy without losing one's pants: Think of a solid
ball in 3D space, centered at some point, dubbed "the
origin" from now on, with pairs of antipodal points
identified. Topologically, this is SO3, the 3D rotations
Lie group. The point to make is that there are two
classes of origin-centered loops in this space, those of
the first class being shrinkable to 0, while the others
are not: Any loop not crossing the surface belongs in
the first class. To get a representative of the second
class, go straight from 0 to the boundary (say, at point
A), reenter from the opposite point (A', say), return
straight to 0. This is a closed loop in SO3, obviously
not reducible to 0 by continuous deformation within SO3.
On the other hand, run along the previous path twice (or
any even number of times), and the resulting loop can
now be shrunk to 0 (provided one correctly interprets
the meaning of "loop": one is supposed to return to 0
at the end of the journey, but the mid-trip passage
through 0 is not mandatory--actually it *must* be avoided
as the loop is deformed).
The latter loop is thus from 0 to A, then jump to A',
go from A' to A, jump to A' again, and return to 0.
How to shrink it to 0, is what Greg Egan's
animation shows. It involves changing direction the
second time around, progressively: Instead of going
from A' to A, go straight from A' to some B
close to A, on the surface (do *not* go through 0
this time), jump to B' opposite to B, then back
home. This new loop, 0AA'BB'0, is "close", in
an obvious sense, to 0AA'AA'0. Now, keeping the first
leg of the journey (from 0 to A aka A') unchanged,
move B (and hence, B', the same point in SO3,
actually) towards A', on the surface. The
deformed loop is now 0AA'A'A0, that is to say, 0A0,
which nicely retracts to 0.
[All this is very easy to draw, though not in ASCII
unfortunately. I tend to think that such a series of
drawings, with comments, may be better as a teaching
aid than an animation, but that's a side issue.
(I did enjoy this animation, as well as others proposed
by Greg Egan, but I always had to first "see things with
the eyes of the mind", as Poirot is suppposed to say
in French translations, before being able to appreciate
them.)]
Now, the belt. Fasten the buckle in some fixed
position (a door knob may do), hold firm the
other end. Pause to show the belt is flat,
i.e., untwisted. Now give it two full (360o) turns
(around some fixed spatial axis, for neatness
--the vertical, say). You used both hands to do
that, presumably. Pause, keep loose end in one
hand, bring attention to the double twist in
the belt. Now, *always keeping this loose end of the
belt (approximately) vertical and not rotating
(not by more than a few degrees)*, pass it behind the
rest of the belt, then back to where it was (notice
you *must change hands* [or perhaps jump above
the belt... never tried that one] to achieve
that; maybe this is what evokes a "conjuring
trick" to some--we'll return to that). See the
belt, miraculously, flatten back to its original
configuration. Make the point that
nothing of this kind could be done after only
*one* full turn (or three, for that matter, if
the belt could stand it). Demonstrate also that
the loose end can be kept very close to the knob-
and-buckle end in this process, with its own frame
almost parallel, always, to the buckle's frame.
What has thus been "proved"? With respect to the
knob and buckle, each segment t of the belt, t
going from 0 to 1, has a translation vector v(t)
in 3D vector space V3, and a rotation s(t), element
of SO3, with respect to the buckle's frame. So
the belt materializes a path, t --> {v(t, s(t)},
in the product space V3 x SO3, i.e., in the Lie group
D3 of (direct) 3D displacements, the unit element
of which corresponds to the {knob, buckle's frame}
pair. [The algebraic structure of D3, as a
semi-direct product, is ignored here--perhaps
not a good thing.] Since buckle and loose end can
be kept very close and almost parallel,
this path is an honest approximation to a *loop*,
a loop in D3. Flat belt and doubly twisted belt
materialize two such loops, which are shown to be
homotopic by performing the trick. So what is
addressed, really, is the non-trivial homotopic
structure *of D3*, not SO3.
From which, of course, the same about SO3 is
easily *derived*. But let's not be fooled. The
Dirac's trick is an involved one, which achieves
more than was bargained for. Too much, perhaps.
Arguably, it takes too much classroom time,
if compared to the sequence-of-drawings process.
But also, and more of a concern, it tends to elicit
further questions which the teacher may not feel
ready to address.
For example, we noticed that the path in D3 could
be considered, for what it's worth, as a loop. So
why not make it a real, physical loop? Take the belt,
give it two full turns, now *fasten the belt*. Any
way to disentangle it? You bet. The inspired remark
that "if the belt's material was this kind of magic
stuff that can intersect itself, then the belt could
untwisted (after 2n turns, and not after 2n + 1)"
may not win applause.
And yet, this is it, for a part: D3 can be
considered as a fibered space, fiber SO3, base V3
(or rather, more accurately, base A3, the affine
3D space), and while homotopies in D3 do
generate homotopies in SO3, they don't project
to homotopies in A3. This is why the belt must
intersect itself to untwist, or be left unbuckled
in order to leave room for the manipulation (which
by the way could not be achieved if the loose end
was always kept in the same hand). The suspicion
of a "conjuring trick", though unfair, can thus be
understood, if not endorsed.
The "plate trick", though its analysis is involved
too, looks a little more transparent, but discussing
this would take too long. More interestingly perhaps,
it would be nice if topologists could explain to us,
more thoroughly and accurately than suggested here,
the mathematics behind the belt trick, beyond the
basic fact that SO3 is doubly connected.