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Note that the last property 𝑅ⁿ(θ) = 𝑅(𝑛θ) is eerily similar to the "de Moivre property" .
This is indeed no accident
The underlying relationship is that the groups SO(2) and U(1) are 𝒊𝒔𝒐𝒎𝒐𝒓𝒑𝒉𝒊𝒄pic.twitter.com/KUFh4cRQN0
Here comes the real beauty
Define the matrix 𝐿 as the derivative of 𝑅(θ) about θ=0
𝑅(θ) can be written as a matrix exponential of 𝐿 (check it!)
We then obtain a beautiful matrix relation reminiscent of Euler's identity
This identity tells us that exp(𝑖𝐿θ) = 𝑅(θ)pic.twitter.com/zis6198nZq
Yes, and this is how you rotate a spinor. I believe E. Cartan, Dirac and Weyl had a fair bit of playing around with that one too.
Wait is L the derivative of R(θ) at θ=0 or is it -i times the derivative of R(θ) at θ=0? I think there’s a conflict between what you typed above and what’s in the picture...
It’s -i Times the derivative.. of you look closely since I raised e to +I they cancel out. You could argue that this is pointless, but the reason for that is so that L is hermitian as opposed to antihermitian.. this is a matter of convention
Giving us the infinitesimal generator of the rotations. It is remarkable how that idea extends to more general Lie groups (and other structures).
This is exactly Euler's identity, but applied to complex numbers embedded into 2x2 matrices
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