5.2 Generalized Macroscopic Limits
107
where a θ is a unique differentiable mapping from G to g
∗ with the properties,
[212]:
(i) a θ (gh) = Ad
∗
(g)(a θ (h)) + a θ (g), ∀g, h, ∈ G,
(ii) T e a θ (ξ)(η) = θ(ξ, η), ∀ξ, η ∈ g, where T e a θ : g → g
∗ is the tangent map of
a θ at e ∈ G.
The system (A; σ G ) represents a quantal system and (g
∗
, λ; ϕ G ) is a (generalized) classical system which will play the role of a macroscopic limit of the system
(A; σ G ). Let us introduce candidates for this micro-macro connection:
5.2.3 Definitions. Let B(g
∗
) be the set of all complex-valued uniformly bounded
Borel functions on g
∗ and let G be the Borel σ-algebra of subsets of g
∗ . Let the
G-measure E (of the system (g
∗
, λ; ϕ G ), resp. of (A; σ G )) be any projection-valued
measure on g
∗ with values in Z, which is G-equivariant, i.e.
E : G → Z, B → E(B) = E(B)
∗
= E(B)
2
∈ Z (B ∈ G ),
(5.2.5a)
B j ∩ B k = ∅ ( j = k, j, k ∈ Z + ) ⇒ E(∪ j B j ) =
j
E(B j ),
(5.2.5b)
E(ϕ g B) = σ g E(B), for all B ∈ G , and for all g ∈ G.
(5.2.5c)
Denote by E( f ) ∈ Z the integral of f ∈ B(g
∗
) over E.
Let p E := E(g
∗
), I := the unit of A
∗∗ .
Denote by N(E) the W
∗ -subalgebra of Z generated by E( f ), f ∈ B(g
∗
).
Let B(E) denote the Borel
∗ -algebra [235, 4.5.5] in Z generated by all the
E( f ), f ∈ B(g
∗
); this means that B(E) is the smallest C
∗ -subalgebra of Z containing all the E(B) (B ∈ G ) and with each monotone (increasing or decreasing)
sequence x j ∈ B(E) s it is also s- lim x j ∈ B(E). Clearly B(E) ⊂ N(E). Here M s
is the set of all selfadjoint elements of a C
∗ -algebra M. The projector p E is the
common unit of B(E) and N(E). Any projector q ∈ B(E) is of the form q = E(B)
for some B ∈ G , what need not be the case for N(E). Projections in N(E) separate
various kinds of spectra of E (resp. of operators E( f ) etc.) what need not be the
case of B(E).
Let supp E ⊂ g
∗ be the minimal closed B = B ∈ G such that E(B) = p E . On
the other hand, supp E(B) := {m ∈ Z = ES(Z) : m(E(B)) = 1} is a clopen subset
of Z. Let
dim(F) := dimension of the orbit ϕ G F ⊂ g
∗
, dim(F) = 2k ≤ dim g
∗
,
and dim(E) := max{dim(F) : F ∈ supp E}. The G-measure E is trivial iff
dim(E) = 0. The quantal system (A; σ G ) has a nontrivial macroscopic limit in
107
where a θ is a unique differentiable mapping from G to g
∗ with the properties,
[212]:
(i) a θ (gh) = Ad
∗
(g)(a θ (h)) + a θ (g), ∀g, h, ∈ G,
(ii) T e a θ (ξ)(η) = θ(ξ, η), ∀ξ, η ∈ g, where T e a θ : g → g
∗ is the tangent map of
a θ at e ∈ G.
The system (A; σ G ) represents a quantal system and (g
∗
, λ; ϕ G ) is a (generalized) classical system which will play the role of a macroscopic limit of the system
(A; σ G ). Let us introduce candidates for this micro-macro connection:
5.2.3 Definitions. Let B(g
∗
) be the set of all complex-valued uniformly bounded
Borel functions on g
∗ and let G be the Borel σ-algebra of subsets of g
∗ . Let the
G-measure E (of the system (g
∗
, λ; ϕ G ), resp. of (A; σ G )) be any projection-valued
measure on g
∗ with values in Z, which is G-equivariant, i.e.
E : G → Z, B → E(B) = E(B)
∗
= E(B)
2
∈ Z (B ∈ G ),
(5.2.5a)
B j ∩ B k = ∅ ( j = k, j, k ∈ Z + ) ⇒ E(∪ j B j ) =
j
E(B j ),
(5.2.5b)
E(ϕ g B) = σ g E(B), for all B ∈ G , and for all g ∈ G.
(5.2.5c)
Denote by E( f ) ∈ Z the integral of f ∈ B(g
∗
) over E.
Let p E := E(g
∗
), I := the unit of A
∗∗ .
Denote by N(E) the W
∗ -subalgebra of Z generated by E( f ), f ∈ B(g
∗
).
Let B(E) denote the Borel
∗ -algebra [235, 4.5.5] in Z generated by all the
E( f ), f ∈ B(g
∗
); this means that B(E) is the smallest C
∗ -subalgebra of Z containing all the E(B) (B ∈ G ) and with each monotone (increasing or decreasing)
sequence x j ∈ B(E) s it is also s- lim x j ∈ B(E). Clearly B(E) ⊂ N(E). Here M s
is the set of all selfadjoint elements of a C
∗ -algebra M. The projector p E is the
common unit of B(E) and N(E). Any projector q ∈ B(E) is of the form q = E(B)
for some B ∈ G , what need not be the case for N(E). Projections in N(E) separate
various kinds of spectra of E (resp. of operators E( f ) etc.) what need not be the
case of B(E).
Let supp E ⊂ g
∗ be the minimal closed B = B ∈ G such that E(B) = p E . On
the other hand, supp E(B) := {m ∈ Z = ES(Z) : m(E(B)) = 1} is a clopen subset
of Z. Let
dim(F) := dimension of the orbit ϕ G F ⊂ g
∗
, dim(F) = 2k ≤ dim g
∗
,
and dim(E) := max{dim(F) : F ∈ supp E}. The G-measure E is trivial iff
dim(E) = 0. The quantal system (A; σ G ) has a nontrivial macroscopic limit in