(0, 0 ,1) =∞
(x 1 ,x 2 ,x 3 )
(0, 0 ,0)
(0, 0 ,x 3 )
z=x+iy
R iR
C
CP^1
similar triangles:
x
1
=
x 1
1 −x 3
y
1
=
x 2
1 −x 3
Figure 7.3: Complex-valued coordinates onCP^1 −{∞}via stereographic pro-
jection.
Digression.For another point of view onCP^1 , one constructs the quotient of
C^2 by complex scalars in two steps. Multiplication by a real scalar corresponds
to a change in normalization of the state, and we will often use this freedom to
work with normalized states, those satisfying
〈ψ|ψ〉=z 1 z 1 +z 2 z 2 = 1
Such normalized states are unit-length vectors inC^2 , which are given by points
on the unit sphereS^3 ⊂R^4 =C^2.
With such normalized states, one still must quotient out the action of mul-
tiplication by a phase, identifying elements ofS^3 that differ by multiplication
byeiθ. The set of these elements forms a new geometrical space, often writ-
tenS^3 /U(1). This structure is called a “fibering” ofS^3 by circles (the action
by phase multiplication traces out non-intersecting circles) and is known as the
“Hopf fibration”. Try an internet search for various visualizations of the ge-
ometrical structure involved, a surprising decomposition of three dimensional
space (identifying points at infinity to getS^3 ) into non-intersecting curves.
Acting onH=C^2 by linear maps
(
z 1
z 2
)
→
(
α β
γ δ
)(
z 1
z 2