3.2 Classical Phase Spaces from the Quantal State Space
41
3.2.9 Lemma. All the vector fields σ ξ (ξ ∈ g) on O z are projected onto unambiguously defined (analytic, if z ∈ PA G ) vector fields σ
M
ξ on M z :
σ
M
ξ ([x]) := T p M σ ξ (h · x)
(3.2.12)
for all h ∈ K x .
3.2.10 Proposition. There is a unique symplectic form
M on M z satisfying
M
[x] (σ
M
ξ , σ
M
η ) =
◦
x (σ ξ , σ η ) = ( p
∗
M
M
) x (σ ξ , σ η )
(3.2.13)
for all ξ, η ∈ g and all x ∈ O z . p
∗
M in (3.2.13) is the pull-back corresponding to the
projector p M (compare [1], resp. also [37, A.3.11] for the definition).
Proof. The first equality can be considered as a definition of a two-form
M , which
is correct due to two preceding lemmas and the fact, that vectors σ
M
ξ ( p M x) (ξ ∈ g)
contain a basis of T [x] M z : σ
M
η ( p M x) = 0 implies η ∈ k x and M z is diffeomorphic to
G/K x . This ensures also the uniqueness of
M . The second equality is a consequence
of the definition (3.2.12) of σ
M
ξ and it shows, how
◦ can be reconstructed from
M .
The bilinearity of
M follows from linearity of the mapping T x p M and the bilinearity of
◦ , antisymmetry is trivial and closedness holds due to commutativity of
the exterior derivative with the pull-bacs: d p
∗
M = p
∗
M d, and due to closedness of
◦
.
Nondegeneracy follows from (3.2.1) and 3.2.4, which completes the proof.
3.2.11 As it was pointed out, the manifold M := M z (z ∈ PA G ) is diffeomorphic
to G/K z , where K z is the stability group of the point F z ∈ g
∗ with respect to the
coadjoint representation of G. On the other hand, the form
M on M has the expression
M
[z] (σ
M
ξ , σ
M
η ) = −F z ([ξ, η]),
(3.2.14)
which follows from (3.2.2). This is, up to the sign, the canonical symplectic form
on the orbit of Ad
∗
(G) passing through F z and diffeomorphic to G/K z . Hence the
symplectic manifold (M;
M
) is symplectomorphic to a Kirillov-Kostant symplectic
orbit, compare [174]. This manifold is here interpreted as a classical phase space
obtained by the above described canonical procedure from a given quantal system,
in which interpretation of observables is (at least partly) determined by a Lie group
action U (G). This action is projected on the coadjoint action Ad
∗
(G) on M, see
(3.2.5). Almost obvious is the following
3.2.12 Proposition. The vector fields σ
M
ξ (ξ ∈ g) are globally Hamiltonian vector
fields on the symplectic manifold (M;
M
) corresponding to Hamiltonian functions
f ξ : [x] → f ξ ([x]) := F x (ξ ).
(3.2.15)
They generate Hamiltonian flows F
ξ
t on M:
41
3.2.9 Lemma. All the vector fields σ ξ (ξ ∈ g) on O z are projected onto unambiguously defined (analytic, if z ∈ PA G ) vector fields σ
M
ξ on M z :
σ
M
ξ ([x]) := T p M σ ξ (h · x)
(3.2.12)
for all h ∈ K x .
3.2.10 Proposition. There is a unique symplectic form
M on M z satisfying
M
[x] (σ
M
ξ , σ
M
η ) =
◦
x (σ ξ , σ η ) = ( p
∗
M
M
) x (σ ξ , σ η )
(3.2.13)
for all ξ, η ∈ g and all x ∈ O z . p
∗
M in (3.2.13) is the pull-back corresponding to the
projector p M (compare [1], resp. also [37, A.3.11] for the definition).
Proof. The first equality can be considered as a definition of a two-form
M , which
is correct due to two preceding lemmas and the fact, that vectors σ
M
ξ ( p M x) (ξ ∈ g)
contain a basis of T [x] M z : σ
M
η ( p M x) = 0 implies η ∈ k x and M z is diffeomorphic to
G/K x . This ensures also the uniqueness of
M . The second equality is a consequence
of the definition (3.2.12) of σ
M
ξ and it shows, how
◦ can be reconstructed from
M .
The bilinearity of
M follows from linearity of the mapping T x p M and the bilinearity of
◦ , antisymmetry is trivial and closedness holds due to commutativity of
the exterior derivative with the pull-bacs: d p
∗
M = p
∗
M d, and due to closedness of
◦
.
Nondegeneracy follows from (3.2.1) and 3.2.4, which completes the proof.
3.2.11 As it was pointed out, the manifold M := M z (z ∈ PA G ) is diffeomorphic
to G/K z , where K z is the stability group of the point F z ∈ g
∗ with respect to the
coadjoint representation of G. On the other hand, the form
M on M has the expression
M
[z] (σ
M
ξ , σ
M
η ) = −F z ([ξ, η]),
(3.2.14)
which follows from (3.2.2). This is, up to the sign, the canonical symplectic form
on the orbit of Ad
∗
(G) passing through F z and diffeomorphic to G/K z . Hence the
symplectic manifold (M;
M
) is symplectomorphic to a Kirillov-Kostant symplectic
orbit, compare [174]. This manifold is here interpreted as a classical phase space
obtained by the above described canonical procedure from a given quantal system,
in which interpretation of observables is (at least partly) determined by a Lie group
action U (G). This action is projected on the coadjoint action Ad
∗
(G) on M, see
(3.2.5). Almost obvious is the following
3.2.12 Proposition. The vector fields σ
M
ξ (ξ ∈ g) are globally Hamiltonian vector
fields on the symplectic manifold (M;
M
) corresponding to Hamiltonian functions
f ξ : [x] → f ξ ([x]) := F x (ξ ).
(3.2.15)
They generate Hamiltonian flows F
ξ
t on M: