[HN Gopher] Understanding quaternions and the Dirac belt trick (...
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Understanding quaternions and the Dirac belt trick (2010)
Author : ogogmad
Score : 27 points
Date : 2020-07-15 20:01 UTC (2 hours ago)
(HTM) web link (arxiv.org)
(TXT) w3m dump (arxiv.org)
| ajkjk wrote:
| 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 th/2) v e^(-R th/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.
| ogogmad wrote:
| 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.
| moron4hire wrote:
| 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
| [deleted]
| cheschire wrote:
| The best demo I ever used to understand quaternions was this
| editable explorable demo.
|
| https://eater.net/quaternions/video/intro
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