Quantum Mechanics for Mathematicians

on the plane corresponding to (
1
0

)

does not have a well-defined value: as one approaches this point one moves off
to infinity in the complex plane. In some sense the spaceCP^1 is the complex
plane, but with a “point at infinity” added.
CP^1 is better thought of not as a plane together with a point, but as a
sphere (often called the “Riemann sphere”), with the relation to the plane and
the point at infinity given by stereographic projection. Here one creates a one-
to-one mapping by considering the lines that go from a point on the sphere
to the north pole of the sphere. Such lines will intersect the plane in a point,
and give a one-to-one mapping between points on the plane and points on the
sphere, except for the north pole. Now, the north pole can be identified with
the “point at infinity”, and thus the spaceCP^1 can be identified with the space
S^2. The picture looks like this

(0, 0 ,1)

(x 1 ,x 2 ,x 3 )

z=x+iy

R iR

C

CP^1

Figure 7.2: The complex projective lineCP^1.

and the equations relating coordinates (x 1 ,x 2 ,x 3 ) on the sphere and the complex
coordinatez 1 /z 2 =z=x+iyon the plane are given by

x=

x 1

1 −x 3

, y=

x 2

1 −x 3

and

x 1 =

2 x

x^2 +y^2 + 1

, x 2 =

2 y

x^2 +y^2 + 1

, x 3 =

x^2 +y^2 − 1

x^2 +y^2 + 1