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5 Macroscopic Limits
5.2.13 A scheme of ‘macroscopic quantization’.
Having once a classical limit in the form of the couple {(g
∗
, λ; ϕ G ), (M G ; σ G )},
where σ G ⊂
∗ - Aut M G , we are interested in the question: Can the original algebra
A be reconstructed from this classical limit? Keeping in mind the model of Sect. 5.1
we propose the following scheme for obtaining the algebra A of a system (A; σ G ),
the macroscopic limit of which is (M G ; σ G ; E g ) (here the measure E g symbolizes
the connection with the classical system (g
∗
, λ; ϕ G )), (let us denote by MQ the following scheme):
(MQ) Find a faithful representation ρ of M G in a Hilbert space H ρ (necessarily
nonseparable) with the properties:
(i) There is a simple C
∗ -subalgebra A of L(H ρ ) such that the center of its commutant A’ contains ρ(M G ); σ G extends to an automorphism group of A.
(ii) A is expressible as the norm-closure of union of a net of von Neumann subalgebras A j ( j ∈ J := a directed set): j ≺ k ⇒ A j ⊂ A k .
(iii) Each A j is a σ G -invariant subset of A and the restriction of σ G to any A j ( j ∈ J )
is unitarily implementable (i.e. it exists a strongly continuous unitary representation U
j of G in H ρ such that σ g (x) = U
j
(g)xU
j
(g
−1
) for all x ∈ A j , g ∈ G
and j ∈ J ).
(iv) Each A k (k ∈ J ) is generated by all A j with j ≺ k ( j = k) as well as by the
bounded Borel functions of the selfadjoint generators X
k
ξ (ξ ∈ g) of the one
parameter groups t → U
k
(exp(tξ)).
Hence the proposed ‘quantization procedure’ of the classical system (g
∗
, λ; ϕ G )
consists in finding an ‘imprimitivity system’ (M G , σ G ) (cf. [321]) determined by
a choice of a G-measure E g (in some commutative C
∗ -algebra M G , where σ G ⊂
∗ - Aut M G is determined by σ g E g (B) := E g (ϕ g B), g ∈ G, B = Borel subsets in
g
∗ ), and afterwards applying the scheme (MQ) of ‘macroscopic quantization’ to
(M G ; σ G ). We shall not investigate here conditions of existence and a ‘degree of
uniqueness’ of this recipe. The scheme is nonempty, since it is fulfilled e.g. by the
models considered in Sect. 5.1 if U (G) is irreducible, 5.1.3.
The question of obtaining a microscopic quantum dynamics of this ‘quantized
macroscopic system’ corresponding to its given classical time evolution is posed and
solved in the next Chap. 6.
5 Macroscopic Limits
5.2.13 A scheme of ‘macroscopic quantization’.
Having once a classical limit in the form of the couple {(g
∗
, λ; ϕ G ), (M G ; σ G )},
where σ G ⊂
∗ - Aut M G , we are interested in the question: Can the original algebra
A be reconstructed from this classical limit? Keeping in mind the model of Sect. 5.1
we propose the following scheme for obtaining the algebra A of a system (A; σ G ),
the macroscopic limit of which is (M G ; σ G ; E g ) (here the measure E g symbolizes
the connection with the classical system (g
∗
, λ; ϕ G )), (let us denote by MQ the following scheme):
(MQ) Find a faithful representation ρ of M G in a Hilbert space H ρ (necessarily
nonseparable) with the properties:
(i) There is a simple C
∗ -subalgebra A of L(H ρ ) such that the center of its commutant A’ contains ρ(M G ); σ G extends to an automorphism group of A.
(ii) A is expressible as the norm-closure of union of a net of von Neumann subalgebras A j ( j ∈ J := a directed set): j ≺ k ⇒ A j ⊂ A k .
(iii) Each A j is a σ G -invariant subset of A and the restriction of σ G to any A j ( j ∈ J )
is unitarily implementable (i.e. it exists a strongly continuous unitary representation U
j of G in H ρ such that σ g (x) = U
j
(g)xU
j
(g
−1
) for all x ∈ A j , g ∈ G
and j ∈ J ).
(iv) Each A k (k ∈ J ) is generated by all A j with j ≺ k ( j = k) as well as by the
bounded Borel functions of the selfadjoint generators X
k
ξ (ξ ∈ g) of the one
parameter groups t → U
k
(exp(tξ)).
Hence the proposed ‘quantization procedure’ of the classical system (g
∗
, λ; ϕ G )
consists in finding an ‘imprimitivity system’ (M G , σ G ) (cf. [321]) determined by
a choice of a G-measure E g (in some commutative C
∗ -algebra M G , where σ G ⊂
∗ - Aut M G is determined by σ g E g (B) := E g (ϕ g B), g ∈ G, B = Borel subsets in
g
∗ ), and afterwards applying the scheme (MQ) of ‘macroscopic quantization’ to
(M G ; σ G ). We shall not investigate here conditions of existence and a ‘degree of
uniqueness’ of this recipe. The scheme is nonempty, since it is fulfilled e.g. by the
models considered in Sect. 5.1 if U (G) is irreducible, 5.1.3.
The question of obtaining a microscopic quantum dynamics of this ‘quantized
macroscopic system’ corresponding to its given classical time evolution is posed and
solved in the next Chap. 6.