5.2 Generalized Macroscopic Limits
111
requirement of ‘maximal sensitivity’ of the corresponding macroscopic description
of the system (A; σ G ).
We shall not proceed further in an analysis of the set [E] G and we shall not
try to specify some ‘most convenient’ element E ∈ [E] G as a representative of the
macroscopic limit. Let us choose any fixed E g ∈ [E] G .
5.2.10 Definitions. The projection-valued measure E g ∈ [E] G on the Poisson manifold (g
∗
, λ; ϕ G ), the G-action on which is ’maximal’ (i.e. orbits ϕ G F are maximal
symplectic immersed submanifolds of g
∗ the Poisson bracket on which is given by λ,
for any F ∈ g
∗ ), with values in Z (:= the center of A
∗∗ ) is called the G-macroscopic
limit of the system (A; σ G ) in the classical system (g
∗
, λ; ϕ G ). The projector
p G := E g (g
∗
) is the support projector of the macroscopic limit. The dimension n G
will be called also the dimension of E g . The Borel*- (resp. the W*-) algebra [235,
4.5.5] generated by E g (resp. by E g and I ∈ Z) will be called the B*- (resp. W*-)
macroscopic algebra of G-definiteness (resp. the G-macroscopic algebra) of the
system (A; σ G ) and will be denoted (in the W*-cases) by N G (resp. by M G ).
Denote by p M : S(A) → S ∗ (M G ), ω → p M ω := r M ◦ e ∗ (ω), where r M is the
restriction of S(A
∗∗
) to S(M G ) and e ∗ is the natural extension from S(A) to
S ∗ (A
∗∗
). Let μ
ω
g be the probability measure on M
G (:= g
∗
∪ {m ◦ }, m ◦ is an isolated
point) given by
μ
ω
g (B) := ω(E g (B\{m ◦ })) + ω(I − p G )δ m ◦ (B), any Borel B ⊂ M
G
, (5.2.17)
compare (5.1.137). Let us introduce the set
E g := {ω ∈ S(A) : e ∗ ω( p G ) = 1, μ
ω
g (ξ
2
) = [μ
ω
g (ξ)]
2
< ∞, ∀ξ ∈ g}, (5.2.18)
compare (5.1.142), where ξ ∈ g is considered as a linear function on g
∗ , since g ⊂
g
∗∗ .
We can introduce also unbounded operators X ξ := E g (ξ) on the Hilbert space H u
of the universal representation of A. Then we have
5.2.11 Theorem. The Theorem 5.1.38 as well as its proof are valid also after the
omission of the index everywhere in its formulation and exchange of Ad
∗
(G) by
ϕ G , with the interpretation of symbols according to 5.2.10.
5.2.12 Note. We could now, after the recognizing of the Theorem, to continue in
the choices of E g ∈ [E] G according to the following idea: Choose E g such that
the sets S F of states with sharp values of the macroscopic observables (cf. 5.1.38
(iii)) are in a certain sense ‘maximal’. We shall not make this idea precise here. We
believe, however, that continuing in this direction we could obtain E g ‘essentially
uniquely’—up to natural coordinate transformations in the g
∗ .
111
requirement of ‘maximal sensitivity’ of the corresponding macroscopic description
of the system (A; σ G ).
We shall not proceed further in an analysis of the set [E] G and we shall not
try to specify some ‘most convenient’ element E ∈ [E] G as a representative of the
macroscopic limit. Let us choose any fixed E g ∈ [E] G .
5.2.10 Definitions. The projection-valued measure E g ∈ [E] G on the Poisson manifold (g
∗
, λ; ϕ G ), the G-action on which is ’maximal’ (i.e. orbits ϕ G F are maximal
symplectic immersed submanifolds of g
∗ the Poisson bracket on which is given by λ,
for any F ∈ g
∗ ), with values in Z (:= the center of A
∗∗ ) is called the G-macroscopic
limit of the system (A; σ G ) in the classical system (g
∗
, λ; ϕ G ). The projector
p G := E g (g
∗
) is the support projector of the macroscopic limit. The dimension n G
will be called also the dimension of E g . The Borel*- (resp. the W*-) algebra [235,
4.5.5] generated by E g (resp. by E g and I ∈ Z) will be called the B*- (resp. W*-)
macroscopic algebra of G-definiteness (resp. the G-macroscopic algebra) of the
system (A; σ G ) and will be denoted (in the W*-cases) by N G (resp. by M G ).
Denote by p M : S(A) → S ∗ (M G ), ω → p M ω := r M ◦ e ∗ (ω), where r M is the
restriction of S(A
∗∗
) to S(M G ) and e ∗ is the natural extension from S(A) to
S ∗ (A
∗∗
). Let μ
ω
g be the probability measure on M
G (:= g
∗
∪ {m ◦ }, m ◦ is an isolated
point) given by
μ
ω
g (B) := ω(E g (B\{m ◦ })) + ω(I − p G )δ m ◦ (B), any Borel B ⊂ M
G
, (5.2.17)
compare (5.1.137). Let us introduce the set
E g := {ω ∈ S(A) : e ∗ ω( p G ) = 1, μ
ω
g (ξ
2
) = [μ
ω
g (ξ)]
2
< ∞, ∀ξ ∈ g}, (5.2.18)
compare (5.1.142), where ξ ∈ g is considered as a linear function on g
∗ , since g ⊂
g
∗∗ .
We can introduce also unbounded operators X ξ := E g (ξ) on the Hilbert space H u
of the universal representation of A. Then we have
5.2.11 Theorem. The Theorem 5.1.38 as well as its proof are valid also after the
omission of the index everywhere in its formulation and exchange of Ad
∗
(G) by
ϕ G , with the interpretation of symbols according to 5.2.10.
5.2.12 Note. We could now, after the recognizing of the Theorem, to continue in
the choices of E g ∈ [E] G according to the following idea: Choose E g such that
the sets S F of states with sharp values of the macroscopic observables (cf. 5.1.38
(iii)) are in a certain sense ‘maximal’. We shall not make this idea precise here. We
believe, however, that continuing in this direction we could obtain E g ‘essentially
uniquely’—up to natural coordinate transformations in the g
∗ .