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

Now if we consider the projection of another point , then plugging into the above formula gives

i.e. the points and project to diametrically opposite directions on the Riemann sphere.

We now recall from page 555 that the spin direction of a spinor with components is the stereographic projection of the ratio . For our two spinors and , these ratios are, respectively:

It is easy to see that , and that therefore and represent diametrically opposite (antipodal) spin directions.

03 Mar 2013, 18:32
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