I know this is really elementary, but it never occurred to me that you could factor a²+b²=(a+bi)(a-bi) so cleanly.
sin²(x)+cos²(x) = 1 isn't 𝑗𝑢𝑠𝑡 Pythagorean theorem on the unit circle It also tells us that a rotation (eⁱˣ) followed by its inverse rotation (e⁻ⁱˣ) is the same as no rotation at allpic.twitter.com/INSZcTxtXr
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lol you work enough with complex numbers you'll see it every time
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You can do it without complex numbers too — just represent a rotation in 2D by the standard SO(2) matrix form, and the same identity follows
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I thought you said without complex numbers
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not for spin 1/2 particles
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SU(2) elements acting on their inverse is still identity
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The argument is somehow circular (sorry for the pun): you need the Pythagorean identity to establish Euler identity in a geometric way. If, instead, you just use the definition of sin and cos given by the convergent series, how do you identify them with the circular functions?
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A complex exponential is taken as a definition, always. You need to define what a complex exponential means first. After that you don’t need Pythagorean theorem to prove Euler’s identity. You can prove it my uniqueness of 2nd order ODE solutions
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The isomorphism U(1)≅SO(2) (x+iy ⇄ (x,y)) that is "slightly" different (but a little sjmilar) to the isomorphism SU(2)/Z2≅SO(3), where Z2= Z/2Z.
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"Spinors" after a 360° rotation change sign and "restore" after 720°: (S→ -S→ S) "half of the angle +360°/2)", while "vectors" don't (V→ V), "the angle+360°" i.e. a "2-to-1 mapping": SU(2) 1 0 0 1 and -1 0 0-1 to SO(3) 1 0 0 0 1 0 0 0 1
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