Quantum Mechanics for Mathematicians

For some insight into this construction, consider first the analog for real
numbers, where (R^2 − 0 )/R∗can be thought of as the space of all lines in the
plane going through the origin.

(x 1 ,x 2 )

(−x 1 ,−x 2 )

(1,0)

(0,1)

(− 1 ,0)

(0,−1)

R^2

RP^1

identify

Figure 7.1: The real projective lineRP^1.

One sees that each such line hits the unit circle in two opposite points, so
this set could be parametrized by a semi-circle, identifying the points at the two
ends. This space is given the nameRP^1 and called the “real projective line”. In
higher dimensions, the space of lines through the origin inRnis calledRPn−^1
and can be thought of as the unit sphere inRn, with opposite points identified
(recall from section 6.2.3 thatSO(3) can be identified withRP^3 ).
What we are interested in is the complex analogCP^1 , which is quite a bit
harder to visualize since in real terms it is a space of two dimensional planes
through the origin of a four dimensional space. A standard way to choose
coordinates onCP^1 is to associate to the vector
(
z 1
z 2

)

∈C^2

the complex numberz 1 /z 2. Overall multiplication by a complex number will
drop out in this ratio, so one gets different values for the coordinatez 1 /z 2 for
each different coset element, and elements ofCP^1 correspond to points on the
complex plane. There is however one problem with this coordinate: the point