24
2 Geometry of the State Space of Quantum Mechanics
2.1.7 Let T x P(H) be the tangent space of P(H) at x, elements of which can
be represented in the usual way (see e.g. [1, 74]) by (classes of mutually tangent)
differentiable curves at x. If c is such a curve (i.e. c : J → P(H) for an open interval
J in R containing 0 ∈ R, c(0) = x and t → y (c(t)) is differentiable for x ∈ N y )
denote by ˙
c x := ˙
c(0) (or simply ˙
c if the point x is fixed) the corresponding equivalence
class, ˙
c x ∈ T x P(H). With any x ∈ x, we associate an identification of T x P(H) with
[x]
⊥ by the mapping
T x x : T x P(H) → [x]
⊥
, ˙
c → T x x ( ˙
c) :=
d
dt
t=0
x (c(t)).
(2.1.12)
In (2.1.12), we identify, in the usual way, the tangent space T v [x]
⊥ of the linear
space [x]
⊥ at any of its points v ∈ [x]
⊥ with the base space [x]
⊥ itself. The mapping
T x x is a linear isomorphism for any x ∈ x, and also T x λx = λT x x (λ ∈ C). The
derivative in (2.1.12) is taken with respect to the Hilbert space norm in [x]
⊥ .
2.1.8 Let us mention two simple examples of the representation of elements ˙
c ∈
T x P(H) by curves c and of the corresponding identification of T x P(H) with [x]
⊥ .
Each vector ˙
c ∈ T x P(H) can be represented by a curve of any of the following forms
(the expressions written by bold typeface represent the projections to P(H) of the
corresponding elements of H, i.e. z(∈ H) → z ≡ P z (∈ P(H))):
c 1 (t) := λx + t y ≡ P λx+ty (λ ∈ C, y ∈ H, x ∈ x), t ∈ R,
(2.1.13)
c 2 (t) := exp(i t B)x ≡ P exp(it B)x (B = B
∗
∈ L(H ), x ∈ x), t ∈ R. (2.1.14)
If we denote corresponding tangent vectors by ˙
c 1 and ˙
c 2 , then
T x x ( ˙
c 1 ) = λ
−1
(1 − P x )y,
(2.1.15)
T x x ( ˙
c 2 ) = i(1 − P x )Bx.
(2.1.16)
Clearly ˙
c 1 = ˙
c 2 iff the right hand sides of (2.1.15) and (2.1.16) coincide as
vectors in [x]
⊥ . This is the case if e.g. y = iλBx in (2.1.15). The representants (c 1 ,
or c 2 , or …) of a given ˙
c can be chosen in many various ways. We shall use notation:
v x := T x x (v) ∈ [x]
⊥
for v ∈ T x P(H); v λx = λv x .
2.1.9 We shall consider P(H) as a real manifold of the dimension dim P(H) =
2 dim C H − 2, (if H is finite dimensional) where dim C means the complex dimension. On this manifold, we introduce a metric Q, i.e. a real-analytic symmetric
2 Geometry of the State Space of Quantum Mechanics
2.1.7 Let T x P(H) be the tangent space of P(H) at x, elements of which can
be represented in the usual way (see e.g. [1, 74]) by (classes of mutually tangent)
differentiable curves at x. If c is such a curve (i.e. c : J → P(H) for an open interval
J in R containing 0 ∈ R, c(0) = x and t → y (c(t)) is differentiable for x ∈ N y )
denote by ˙
c x := ˙
c(0) (or simply ˙
c if the point x is fixed) the corresponding equivalence
class, ˙
c x ∈ T x P(H). With any x ∈ x, we associate an identification of T x P(H) with
[x]
⊥ by the mapping
T x x : T x P(H) → [x]
⊥
, ˙
c → T x x ( ˙
c) :=
d
dt
t=0
x (c(t)).
(2.1.12)
In (2.1.12), we identify, in the usual way, the tangent space T v [x]
⊥ of the linear
space [x]
⊥ at any of its points v ∈ [x]
⊥ with the base space [x]
⊥ itself. The mapping
T x x is a linear isomorphism for any x ∈ x, and also T x λx = λT x x (λ ∈ C). The
derivative in (2.1.12) is taken with respect to the Hilbert space norm in [x]
⊥ .
2.1.8 Let us mention two simple examples of the representation of elements ˙
c ∈
T x P(H) by curves c and of the corresponding identification of T x P(H) with [x]
⊥ .
Each vector ˙
c ∈ T x P(H) can be represented by a curve of any of the following forms
(the expressions written by bold typeface represent the projections to P(H) of the
corresponding elements of H, i.e. z(∈ H) → z ≡ P z (∈ P(H))):
c 1 (t) := λx + t y ≡ P λx+ty (λ ∈ C, y ∈ H, x ∈ x), t ∈ R,
(2.1.13)
c 2 (t) := exp(i t B)x ≡ P exp(it B)x (B = B
∗
∈ L(H ), x ∈ x), t ∈ R. (2.1.14)
If we denote corresponding tangent vectors by ˙
c 1 and ˙
c 2 , then
T x x ( ˙
c 1 ) = λ
−1
(1 − P x )y,
(2.1.15)
T x x ( ˙
c 2 ) = i(1 − P x )Bx.
(2.1.16)
Clearly ˙
c 1 = ˙
c 2 iff the right hand sides of (2.1.15) and (2.1.16) coincide as
vectors in [x]
⊥ . This is the case if e.g. y = iλBx in (2.1.15). The representants (c 1 ,
or c 2 , or …) of a given ˙
c can be chosen in many various ways. We shall use notation:
v x := T x x (v) ∈ [x]
⊥
for v ∈ T x P(H); v λx = λv x .
2.1.9 We shall consider P(H) as a real manifold of the dimension dim P(H) =
2 dim C H − 2, (if H is finite dimensional) where dim C means the complex dimension. On this manifold, we introduce a metric Q, i.e. a real-analytic symmetric