3.2 Classical Phase Spaces from the Quantal State Space
39
the space g
∗ dual to the Lie algebra g endowed with the natural Kirillov-Kostant
symplectic form.
3.2.2 Let
◦ denotes the restriction of the form onto the immersed submanifold
O z (z ∈ PA G ) of P(H). Since the vector fields σ ξ (ξ ∈ g) span T x O z at each point
x ∈ O z , the form
◦ is uniquely defined by its values on vectors σ ξ (x) (ξ ∈ g, x ∈
O z ):
◦
x (σ ξ , σ η ) := x (σ ξ , σ η ) = i T r(P x [X ξ , X η ]),
(3.2.1)
where we used formula (2.3.10) and the restrictions of the fields σ ξ onto O z are
equally denoted as the unrestricted fields. According to the definition (3.1.5) and
with the use (2.3.21), we can write
◦
x (σ ξ , σ η ) = −F x ([ξ, η]).
(3.2.2)
If we denote by
u ◦ : O z = u(G/K
◦
) → P(H)
the inclusion of the orbit into P(H), then the form
◦ is simply the pull-back of
by u ◦ :
◦
= u
∗
◦ .
(3.2.3)
Since exterior derivative commutes with any pull-back, e.g. [74, p.204], we see
that the two-form
◦ on O z is closed. It is clear from (3.2.2), that
◦ is degenerate
iff for some η = 0 and for all ξ ∈ g the term F x ([ξ, η]) = 0 for some x in the orbit.
This is, however, the situation discussed in 3.1.7.
3.2.3 The mapping F x : g → R, ξ → F x (ξ ) is linear because of linearity of ξ →
X ξ , hence F x ∈ g
∗ for any x ∈ O z . Define the action of G on the functionals F x (x ∈
O z ) by
g · F x := F g·x , for all g ∈ G.
(3.2.4)
Then analogous computations to those in 3.1.5 lead to:
F g·x (ξ ) = F x (Ad(g
−1
)ξ ), what means: g · F x = Ad
∗
(g)F x .
(3.2.5)
Let now K x be, as above, the stability subgroup of G of the coadjoint action at
the point F x ∈ g
∗
. Since Ad
∗ is continuous, K x is closed. Let k x be the Lie algebra
of K x . Then it is clear, that:
3.2.4 Lemma. Let x ∈ PA G , y := g · x. Then K y = gK x g
−1
, k y = Ad(g)k x , and
K
◦
x ⊂ K x for all x and all g ∈ G. It is ξ ∈ k x iff
39
the space g
∗ dual to the Lie algebra g endowed with the natural Kirillov-Kostant
symplectic form.
3.2.2 Let
◦ denotes the restriction of the form onto the immersed submanifold
O z (z ∈ PA G ) of P(H). Since the vector fields σ ξ (ξ ∈ g) span T x O z at each point
x ∈ O z , the form
◦ is uniquely defined by its values on vectors σ ξ (x) (ξ ∈ g, x ∈
O z ):
◦
x (σ ξ , σ η ) := x (σ ξ , σ η ) = i T r(P x [X ξ , X η ]),
(3.2.1)
where we used formula (2.3.10) and the restrictions of the fields σ ξ onto O z are
equally denoted as the unrestricted fields. According to the definition (3.1.5) and
with the use (2.3.21), we can write
◦
x (σ ξ , σ η ) = −F x ([ξ, η]).
(3.2.2)
If we denote by
u ◦ : O z = u(G/K
◦
) → P(H)
the inclusion of the orbit into P(H), then the form
◦ is simply the pull-back of
by u ◦ :
◦
= u
∗
◦ .
(3.2.3)
Since exterior derivative commutes with any pull-back, e.g. [74, p.204], we see
that the two-form
◦ on O z is closed. It is clear from (3.2.2), that
◦ is degenerate
iff for some η = 0 and for all ξ ∈ g the term F x ([ξ, η]) = 0 for some x in the orbit.
This is, however, the situation discussed in 3.1.7.
3.2.3 The mapping F x : g → R, ξ → F x (ξ ) is linear because of linearity of ξ →
X ξ , hence F x ∈ g
∗ for any x ∈ O z . Define the action of G on the functionals F x (x ∈
O z ) by
g · F x := F g·x , for all g ∈ G.
(3.2.4)
Then analogous computations to those in 3.1.5 lead to:
F g·x (ξ ) = F x (Ad(g
−1
)ξ ), what means: g · F x = Ad
∗
(g)F x .
(3.2.5)
Let now K x be, as above, the stability subgroup of G of the coadjoint action at
the point F x ∈ g
∗
. Since Ad
∗ is continuous, K x is closed. Let k x be the Lie algebra
of K x . Then it is clear, that:
3.2.4 Lemma. Let x ∈ PA G , y := g · x. Then K y = gK x g
−1
, k y = Ad(g)k x , and
K
◦
x ⊂ K x for all x and all g ∈ G. It is ξ ∈ k x iff