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