The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics (Oxford Landmark Science)

by Roger Penrose

This correspondence is called stereo-graphic projection, and it has many beautiful geometrical properties (e.g. it preserves angles and maps circles to circles). The projection gives us a labelling of the points of the sphere by complex numbers together with ∞, i.e. by the set of possible complex ratios q. A sphere labelled in this particular way is called a Riemann sphere. The significance of the Riemann sphere, for the spin states of an electron, is that the direction of spin given by | ↗〉 = w |↑〉 + z |↓〉 is provided by the actual direction from the centre to the point q = z/w as marked on the Riemann sphere. We note that the north pole corresponds to the state | ↑〉, which is given by z = 0, i.e. by q = 0, and the south pole to | ↓〉, given by w = 0, i.e. by q = ∞. The rightmost point is labelled by q = 1, which provides the state |→ 〉 = |↑ 〉 + |↓〉, and the leftmost point by q = –1, which provides |←〉 = |↑〉 – |↓〉. The farthest point around the back of the sphere is labelled q = i, corresponding to the state |↑〉 + i|↓〉 where the spin points directly away from us, and the nearest point, q = –i, corresponds to |↑〉 – i|↓〉, where the spin is directly towards us. The general point, labelled by q, corresponds to |↑〉 + q |↓〉.