5.1 Multiple Systems
101
Proof. Verbally the same proof as that of 5.1.17, with M G → M
G .
5.1.36 Definitions. The generalized G-macroscopic phase space is the topological space M
G
:= g
∗
∪ {m ◦ } consisting of the finite (n-)dimensional topological
vector space g
∗ with the canonical symplectic forms defined on each orbit of the
Ad
∗
(G)-action and of an isolated point m ◦ . A state on M
G is any probability
σ-additive Borel measure μ on M
G . We shall associate with any ω ∈ S(A
) the
G-macroscopic state on M
G determined by the measure
μ
ω
g (B) := ω(E
g (B\{m ◦ })) + ω(I − p G )δ m ◦ (B), for Borel B ⊂ M
G
, (5.1.137)
where on the right hand side ω means the normal extension of the state on A
to a
normal state on (A
)
∗∗ and δ m (m ∈ M
G
) means the Dirac measure concentrated
on {m}. It is clear that every normal state ω ∈ S(M
G ) can be transformed also into
a state on M
G by the formula (5.1.137) and that its image μ
ω
g uniquely determines
ω ∈ S ∗ (M
G ). It is also clear that the state on M
G corresponding to p M ω, 5.1.32,
in this way, coincides with μ
ω
g . The association ω → μ
ω
g is G-equivariant, i.e.
μ
g·ω
g = σ
∗
g μ
ω
g , with g · ω := σ
∗
g ω, g ∈ G,
(5.1.138)
and with
σ
∗
g μ(B) := μ(Ad
∗
(g
−1
)(B\{m ◦ })) + μ({m ◦ } ∩ B),
(5.1.139)
for all g ∈ G and all Borel B ⊂ M
G . We shall use also g · μ := σ
∗
g μ. This follows
from the transformation properties of X ξ ’s and from
σ g E
g (B) = E
g (Ad
∗
(g)B),
(5.1.140)
compare 5.1.15.
Let us redefine some symbols introduced in 5.1.18. Let, (5.1.130),
S
d
g := {ω ∈ S
g : ξ ∈ L
1
(M
G
, μ
ω
g ), ∀ξ ∈ g},
(5.1.141)
where ξ is considered as the linear function F → ξ(F) := F(ξ) on g
∗
( F).
Similarly, we shall define now G-macroscopically pure states to be elements
ω ∈ E g ⊂ S(A
), where
E g := {ω ∈ S
d
g : μ
ω
g (ξ
2
) = [μ
ω
g (ξ)]
2
, ∀ξ ∈ g}.
(5.1.142)
Using (5.1.91) and (5.1.92), we can see that the Proposition 5.1.19 can be replaced
by: ω ∈ E g ⇔ μ
ω
g = δ F for some F ∈ g
∗
.
5.1.37 Definitions. A Poisson manifold M is a differentiable C
∞ -manifold
endowed with a bilinear mapping ( f ; g) → { f, g} of couples of infinitely differentiable real functions f, g ∈ C
∞
(M, R) into C
∞
(M, R), the Poisson bracket,
101
Proof. Verbally the same proof as that of 5.1.17, with M G → M
G .
5.1.36 Definitions. The generalized G-macroscopic phase space is the topological space M
G
:= g
∗
∪ {m ◦ } consisting of the finite (n-)dimensional topological
vector space g
∗ with the canonical symplectic forms defined on each orbit of the
Ad
∗
(G)-action and of an isolated point m ◦ . A state on M
G is any probability
σ-additive Borel measure μ on M
G . We shall associate with any ω ∈ S(A
) the
G-macroscopic state on M
G determined by the measure
μ
ω
g (B) := ω(E
g (B\{m ◦ })) + ω(I − p G )δ m ◦ (B), for Borel B ⊂ M
G
, (5.1.137)
where on the right hand side ω means the normal extension of the state on A
to a
normal state on (A
)
∗∗ and δ m (m ∈ M
G
) means the Dirac measure concentrated
on {m}. It is clear that every normal state ω ∈ S(M
G ) can be transformed also into
a state on M
G by the formula (5.1.137) and that its image μ
ω
g uniquely determines
ω ∈ S ∗ (M
G ). It is also clear that the state on M
G corresponding to p M ω, 5.1.32,
in this way, coincides with μ
ω
g . The association ω → μ
ω
g is G-equivariant, i.e.
μ
g·ω
g = σ
∗
g μ
ω
g , with g · ω := σ
∗
g ω, g ∈ G,
(5.1.138)
and with
σ
∗
g μ(B) := μ(Ad
∗
(g
−1
)(B\{m ◦ })) + μ({m ◦ } ∩ B),
(5.1.139)
for all g ∈ G and all Borel B ⊂ M
G . We shall use also g · μ := σ
∗
g μ. This follows
from the transformation properties of X ξ ’s and from
σ g E
g (B) = E
g (Ad
∗
(g)B),
(5.1.140)
compare 5.1.15.
Let us redefine some symbols introduced in 5.1.18. Let, (5.1.130),
S
d
g := {ω ∈ S
g : ξ ∈ L
1
(M
G
, μ
ω
g ), ∀ξ ∈ g},
(5.1.141)
where ξ is considered as the linear function F → ξ(F) := F(ξ) on g
∗
( F).
Similarly, we shall define now G-macroscopically pure states to be elements
ω ∈ E g ⊂ S(A
), where
E g := {ω ∈ S
d
g : μ
ω
g (ξ
2
) = [μ
ω
g (ξ)]
2
, ∀ξ ∈ g}.
(5.1.142)
Using (5.1.91) and (5.1.92), we can see that the Proposition 5.1.19 can be replaced
by: ω ∈ E g ⇔ μ
ω
g = δ F for some F ∈ g
∗
.
5.1.37 Definitions. A Poisson manifold M is a differentiable C
∞ -manifold
endowed with a bilinear mapping ( f ; g) → { f, g} of couples of infinitely differentiable real functions f, g ∈ C
∞
(M, R) into C
∞
(M, R), the Poisson bracket,