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Understanding quaternions and the Dirac belt trick (2010) (arxiv.org)
59 points by ogogmad 28 days ago | hide | past | favorite | 18 comments

The best demo I ever used to understand quaternions was this editable explorable demo.


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.

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.

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.

Combining the parent post and the dismissive post at #2 the best way you can understand quaternions is to either learn about them or come from a different perspective and extremely dislike them to the point that you understand the simpler or different way.

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

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.

> 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.)

> 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.

yes, i would call `i` a rotation operator. Something like millions of person-years have been based on the pedagogical nightmare of complex numbers as normally treated.

> 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.

Oh, this is lovely! That's the first good visualization I've seen for the issue. Or at least the one that finally clicked for me...

Quaternions enable one to formally prove that SO(3) is not simply connected. Here's how: The path in quaternion space e^{\pi i t} is not a loop, but its image in SO(3) under the covering map is a loop. Any contraction of the image loop would imply a contraction of its pre-image, which is impossible.

yes, but that relies on knowing that the quaternions are such a cover, which is entirely non-obvious. An explanation that starts and ends with SO(3) without roaming into quaternions is necessary for intuition.

I thought this was a better visualization, though he doesn't seem to mention you have to consider the whole system of the cup and his arm https://youtu.be/JDJKfs3HqRg

Another situation where geometric algebra makes everything clearer.

in a sense, but I have some gripes with that too. It's bivectors that help a lot. The 'geometric product' I'm less sure about (this is always my complaint when GA is brought up).

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

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

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