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 Exercise [22.33] 
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Joined: 12 Jul 2010, 07:44
Posts: 154
Post Exercise [22.33]
From basic geometrical considerations, one can show that the (south-pole) stereographic projection of a complex number p onto the Riemann sphere (in \left(x,y,z\right) coordinates) is given by:


x = \frac{2 Re\{p\}}{1+\left|p\right|^2}


y = \frac{2 Im\{p\}}{1+\left|p\right|^2}


z = \frac{1-\left|p\right|^2}{1+\left|p\right|^2}



Now if we consider the projection of another point p^\prime = -1/\bar{p} \; = -p/\left|p\right|^2, then plugging into the above formula gives


x^\prime = -x \; \; \; \; \; \; \; \; y^\prime = -y \; \; \; \; \; \; \; \; z^\prime = -z


i.e. the points p and -1/\bar{p} project to diametrically opposite directions on the Riemann sphere.


We now recall from page 555 that the spin direction of a spinor with components \left\{w,z\right\} is the stereographic projection of the ratio u = z/w. For our two spinors \xi^A and \eta^A, these ratios are, respectively:



u = \xi^1/\xi^0 = e^{-i\phi}tan\left(\theta/2\right)


u^\prime = \eta^1/\eta^0 = -e^{-i\phi}\left[tan\left(\theta/2\right)\right]^{-1}




It is easy to see that u^\prime = -1/\bar{u}, and that therefore \xi^A and \eta^A represent diametrically opposite (antipodal) spin directions.



\blacksquare


03 Mar 2013, 18:32
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