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 -
Replying to @InertialObservr
Fun fact: Spinor calculus already contains tensor calculus. Every tensor has a spinor analogue. A spin-vector multiplied by its complex conjugate produces a null vector.
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Replying to @zimtyzimtbart
by null vector you mean SU(N) invariant? Then i agree. What's more is that you can construct all tensor reps of SU(N) by suitable direct products of the fundamental representation!
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Replying to @InertialObservr
I'm coming from GR here. There is a isomorphism between the space of real [1,1]-index spinors and the tangent space of 4d space-time. (By (1,1) I mean 1 index from a spin-vector space and one from it's complex conjugate). 1/~
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Replying to @zimtyzimtbart
yea! we're saying the same thing! What you're calling a [1,1] spinor is actually just the (1/2, 1/2) representation of the Poincare group!
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Replying to @InertialObservr
nice! I should really take a look at rep theory :D
1 reply 0 retweets 1 like
yes! it's a beautiful field and you'll make a lot of connections (no put intended!) coming from GR
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Replying to @InertialObservr @zimtyzimtbart
Quite affine connection indeed.
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