Suppose you're a little creature living on a Möbius strip.
Then after "walking in a circle" (i.e. 2π radians), you will actually be upside-down.
In order to get back to where you started, you have to go around twice!
Check out my neat gif!
pic.twitter.com/1MMvp83IWG
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The 𝐏𝐚𝐮𝐥𝐢 𝐈𝐝𝐞𝐧𝐭𝐢𝐭𝐲 tells us how to rotate a 2 component 𝒗𝒆𝒄𝒕𝒐𝒓 of complex numbers, called a "spinor", around an axis θ̂.
for spinors, a rotation by 2π yields -1
only a rotation by 4π yields identity
..sound familiar?pic.twitter.com/ECGtLshlH2
13 replies 16 retweets 61 likesShow this thread -
An object that existed in a Möbius-strip-shaped universe would be indistinguishable from its own mirror image
This crab's larger claw switches between left to right with every circulation
(GIF not mine)pic.twitter.com/n3JHJdy0jJ2 replies 2 retweets 18 likesShow this thread -
Replying to @InertialObservr
*Locally* it’s still distinguishable from its own mirror image. So if spacetime turned out to be nonorientable, and you took a trip around the universe along an orientation-reversing path, that would make for a veeery confusing return home!
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Replying to @silvascientist @InertialObservr
Is there a sort of potential field that can possibly result in paths that are globally non-holonomic with respect to a symmetry of Euclidean space? Iirc things like Aharonov Bohm and a spin-field coupled Hamiltonian for fermions only have non-holonomy on SU(2).
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Replying to @lmaogh @InertialObservr
Tbh, I really don't know what any of that means, so ¯\_(ツ)_/¯
In particular I don't know anything at all about holonomy...2 replies 0 retweets 1 like -
Replying to @silvascientist @InertialObservr
Neither do I, so part of your confusion could be that I worded it quite poorly ;) 140 limit also doesn't help with clarification either
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Going out of your comfort zone with terminology and getting corrected is one of the best ways to learn
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