
Understanding quaternions and the Dirac belt trick (2010) - ogogmad
https://arxiv.org/abs/1001.1778
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cheschire
The best demo I ever used to understand quaternions was this editable
explorable demo.

[https://eater.net/quaternions/video/intro](https://eater.net/quaternions/video/intro)

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thechao
I think the best way to understand quaternions is to spend about a month
curled up with a good book or two on Grasmannian/Clifford algebras.

~~~
dvt
This post is a bit highfalutin, but it has a good point. You understand
quaternions by _doing math_ with them (the same way you understand complex
numbers, really). There's really nothing magical about them, and I'd really
push back against the idea that a 3b1b video really made you "understand" them
-- other than on a very cursory and shallow semi-intuitive basis.

~~~
esperent
A cursory understanding is a good starting point, even if it's not 100%
technically correct. This is especially true for people who don't consider
themselves mathematicians but still have to deal with quaternions or similar
entities. The biggest stumbling block to understanding math (for many people)
is a lack of confidence combined with feelings of frustration. If a simple
video can get you over that first step, you are on a good path to developing a
deeper understanding.

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thedirt0115
Here's a GIF of the belt trick they're talking about in the paper:
[https://www.gregegan.net/images/DiracAnimation.gif](https://www.gregegan.net/images/DiracAnimation.gif)

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ajkjk
I am pretty sure I have studied every interpretation and explanation of
spinors and the non-simply-connectedness of SO(3) rotations, and I eventually
finally understood it -- but, by ignoring explanations like the 'belt trick',
rather than embracing them. I don't think this article makes it any better. As
long as you're stuck on quaternions you're not going to be able to see what's
going on, and the belt metaphor just adds complexity as well.

The version that currently makes the most sense to me is this:

* Quaternions are a complete distraction and should be ignored. i, j, and k are (xy), (yz), (zx) bivector rotation operators (up to a factor of -i or something like that). Pauli matrices are the same and should also be ignored.

* The factors of "theta/2" in the exponents that are used in representations of rotations using quaternions (rotating vectors with e^(R θ/2) v e^(-R θ/2)) are distracting and should be ignored.

* The best way to see what is meant by "the space of rotations in SO(3) is not simply connected", you need to think about paths in _rotation_ space carefully. More carefully than I did as an undergrad! (Although I'm sure this is obvious to people who have studied the approach math in a course?)

The wrong approach -- which I was stuck on for years -- is to think about a
point on a sphere in R^3 moving around. A vector that starts at, say, +z in
R^3 and is rotated by 2pi in the xz plane ends up where it started exactly.

The key insight is that 'non-simply-connectedness' refers to _paths of
rotations_, rather than paths of what the rotations act on. So imagine
gradually rotating that vector +z in the (xz) plane. If you go around by 2pi,
you've made a path that can be modeled as an exponential operation: e^(2pi
(z^x)).

The question is: can this path be deformed to the identity path? It seems like
it can -- you just change from rotating +z all the way around a great circle,
to a smaller and smaller circular path until it is just rotating in place (and
xy rotation). But somehow when you collapse it, you still have a 2pi rotation!
It's just that now it's a 2pi rotation in xy rather than in zx. No matter what
you do, if you collapse the rotation path to be the identity on the +z vector,
the resulting path rotates _some_ vector by 2pi, instead of keeping it at the
identity.

So in physics, it's not that there is a negative phase factor _per se_ that
matters for physics. It's that there are two physically distinguishable states
(identity rotations and anti-identity rotations) -- so two different electron
wave functions can't be identified as the same electron, because they are a
2pi rotation apart just due to how they got there. And the fact that we
_model_ this is as a negative sign is entirely an artifact of our obfuscating
choices of mathematics for the situation.

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Koshkin
> _i, j, and k are ... bivector rotation operators_

Well, that’s like calling the imaginary unit “a rotation operator.” (I mean,
sure, you can do things like that with abstract algebraic constructs, marvel
for a few moments, but then you move on, searching for the _true_ meaning -
which never happens to lie with any particular representation or an
application.)

~~~
madhadron
> you move on, searching for the true meaning

Which turns out to show up when you go beyond three dimensions. In two and
three dimensions, numerical coincidence let you identify different grades of
operation. In 3D space, i, j, and k are unit vectors...and unit bivectors. As
soon as you hit 4D you have to separate that out, and things get a lot
clearer. This is why people keep harping about geometric algebra.

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fulafel
The argument against (in 3d graphics/games): "Let's remove Quaternions from
every 3D Engine" by Marc ten Bosch,
[https://marctenbosch.com/quaternions/](https://marctenbosch.com/quaternions/)

(And its HN threads
[https://news.ycombinator.com/item?id=22200260](https://news.ycombinator.com/item?id=22200260)
&
[https://news.ycombinator.com/item?id=18365433](https://news.ycombinator.com/item?id=18365433))

