Quantum Mechanics for Mathematicians

(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

)