Chapter 2
Geometry of the State Space of Quantum
Mechanics
2.1 Manifold Structure of P(H)
2.1.1 Let H be a complex separable Hilbert space with the scalar product (x, y) ∈
C (x, y, ∈ H), which is linear in the second factor y. Let P(H) := H/C
∗ be the
factor-space of H by the multiplicative group C
∗ of nonzero complex numbers acting
on H by multiplications by scalars. Any element x ∈ P(H) has the form
x := {y ∈ H : y = λx, λ ∈ C
∗
}, 0 = x ∈ H.
(2.1.1)
The natural topology on P(H) is the factor-topology coming from the normtopology in H. This topological space P(H) is the projective Hilbert space of
H. The space P(H) can be considered as the set of all one-dimensional complex
subspaces of H, or the set of all one-dimensional projectors P x ∈ L(H ) (0 = x ∈
H), P
∗
x = P x = P
2
x , P x x = x, with the natural bijective correspondence P x ↔ x.
It is known that there is a natural Kähler structure on complex projective spaces.
We shall describe it in some details in the case of P(H).
1
2.1.2 Let us define two natural (mutually equivalent) metrices (i.e. distance functions) d 1 , d 2 on P(H) (as usual: x
2
:= (x, x), x ∈ H):
d 1 (x, y) :=
√
2 inf
x
x
− e
iλ y
y
: λ ∈ R
,
(2.1.2)
d 2 (x, y) :=
√
2 P x − P y .
(2.1.3)
1 Another, more intuitive and more detailed approach to the structure of quantum state space can be
found in [16]. For geometry and dynamics (also nonlinear) of general—not only pure—states see
also [37, Sect. 2.1].
© Springer Nature Switzerland AG 2020, corrected publication 2020
P. Bóna, Classical Systems in Quantum Mechanics,
https://doi.org/10.1007/978-3-030-45070-0_2
21
Geometry of the State Space of Quantum
Mechanics
2.1 Manifold Structure of P(H)
2.1.1 Let H be a complex separable Hilbert space with the scalar product (x, y) ∈
C (x, y, ∈ H), which is linear in the second factor y. Let P(H) := H/C
∗ be the
factor-space of H by the multiplicative group C
∗ of nonzero complex numbers acting
on H by multiplications by scalars. Any element x ∈ P(H) has the form
x := {y ∈ H : y = λx, λ ∈ C
∗
}, 0 = x ∈ H.
(2.1.1)
The natural topology on P(H) is the factor-topology coming from the normtopology in H. This topological space P(H) is the projective Hilbert space of
H. The space P(H) can be considered as the set of all one-dimensional complex
subspaces of H, or the set of all one-dimensional projectors P x ∈ L(H ) (0 = x ∈
H), P
∗
x = P x = P
2
x , P x x = x, with the natural bijective correspondence P x ↔ x.
It is known that there is a natural Kähler structure on complex projective spaces.
We shall describe it in some details in the case of P(H).
1
2.1.2 Let us define two natural (mutually equivalent) metrices (i.e. distance functions) d 1 , d 2 on P(H) (as usual: x
2
:= (x, x), x ∈ H):
d 1 (x, y) :=
√
2 inf
x
x
− e
iλ y
y
: λ ∈ R
,
(2.1.2)
d 2 (x, y) :=
√
2 P x − P y .
(2.1.3)
1 Another, more intuitive and more detailed approach to the structure of quantum state space can be
found in [16]. For geometry and dynamics (also nonlinear) of general—not only pure—states see
also [37, Sect. 2.1].
© Springer Nature Switzerland AG 2020, corrected publication 2020
P. Bóna, Classical Systems in Quantum Mechanics,
https://doi.org/10.1007/978-3-030-45070-0_2
21