120
6 Dynamics of Quantum Mechanical Macroscopic Systems
6.2 Spin Systems with Polynomial Local Hamiltonians Q N
6.2.1 Let us consider the system described in Sect. 5.1: The C
∗ -algebra of quasilocal observables A is the C
∗ -inductive limit of the sequence of the von Neumann
algebras A
N
:= L(H N ), A := A
. A compact Lie group G acts on A by the subgroup σ(G) := σ G of
∗ -automorphisms of A introduced in 5.1.5. It is assumed in
this section that the generators X ξ (ξ ∈ g) of the representation U (G) in H introduced in 5.1.3 are bounded operators. We shall use the notation of the subsections
from 5.1.2 to 5.1.28; we shall write for the set of all positive integers. It will be
convenient for definiteness and for some technical reasons to work in the subrepresentation s G π u of the universal representation π u of the algebra A in the Hilbert
space H u . The bidual A
∗∗ is canonically identified with the bicommutant π u (A)
of π u (A) in L(H u ), and s G ∈ Z := π u (A)
∩ π u (A)
⊂ L(H u ) is defined in 5.1.11.
The following considerations could be extended to the larger representation p G π u ,
where p G ∈ Z is introduced in 5.1.29. Hence we shall work in the framework of
the von Neumann algebra s G A
∗∗ which is isomorphic with the subalgebra P G B
# of
L(H ) via the mapping ρ G , cf. 5.1.11. The quasilocal algebra A will be identified
with its representation s G π u (A) in the Hilbert space s G H u or, equivalently, with the
corresponding C
∗ -subalgebra of the abstract W
∗ -algebra s G A
∗∗ . Remember that A
is simple, hence any of its nonzero representations as a C
∗ -algebra is faithful.
Let us introduce notation for various elements and subsets of s G A
∗∗ :
6.2.2 Notation. Let us denote:
(i) E g denotes the projection measure (G-measure, 5.2.3) on the linear space g
∗
generated by E g (F) (F ∈ g
∗
) from 5.1.13; in the notation of 5.1.16 E g (B) =
c(B) for any subset B ≡ B of g
∗ .
(ii) E g ( f ) :=
f (F) E g (dF) for any complex valued function f ∈ L
1
(g
∗
, μ
ω
g ) for
all ω ∈ S ∗ (s G A
∗∗
) := the normal states on s G A
∗∗ , i.e. the integral E g ( f ) is
assumed to converge in the w
∗ -sense.
(iii) B
N
0 := A
N
∪ {X ξ K : ξ ∈ g, K ∈ } ⊂ s G A
∗∗ , if the generators X ξ ∈ L(H)
are bounded, 5.1.3.
(iv) Let N
c be the C
∗ -subalgebra of s G A
∗∗ generated by all the elements E g ( f )
with uniformly bounded continuous f ∈ C b (g
∗
, C).
(v) C
N := the C
∗ - algebra generated by A
N and N
c ; C
N is isomorphic to the
C
∗ -tensor product A
N
⊗ N
c , the isomorphism being: x ⊗ z → xz ∈ C
N
(x ∈
A
N
, z ∈ N
c
), cf [274, 1.22] and [306, IV.4.7].
(vi) C will denote the C
∗ - algebra generated by {C
N
: N ∈ }; C is isomorphic
to the tensor product A ⊗ N
c , cf. 6.2.13.
6.2.3 Notation. Let {ξ j : j = 1, 2, . . . n} be a fixed basis of g. Let
X
N
j := |N | X j N :=
|N |
k=1
π k (X (ξ j )), X (ξ) := X ξ ,
(6.2.1)
6 Dynamics of Quantum Mechanical Macroscopic Systems
6.2 Spin Systems with Polynomial Local Hamiltonians Q N
6.2.1 Let us consider the system described in Sect. 5.1: The C
∗ -algebra of quasilocal observables A is the C
∗ -inductive limit of the sequence of the von Neumann
algebras A
N
:= L(H N ), A := A
. A compact Lie group G acts on A by the subgroup σ(G) := σ G of
∗ -automorphisms of A introduced in 5.1.5. It is assumed in
this section that the generators X ξ (ξ ∈ g) of the representation U (G) in H introduced in 5.1.3 are bounded operators. We shall use the notation of the subsections
from 5.1.2 to 5.1.28; we shall write for the set of all positive integers. It will be
convenient for definiteness and for some technical reasons to work in the subrepresentation s G π u of the universal representation π u of the algebra A in the Hilbert
space H u . The bidual A
∗∗ is canonically identified with the bicommutant π u (A)
of π u (A) in L(H u ), and s G ∈ Z := π u (A)
∩ π u (A)
⊂ L(H u ) is defined in 5.1.11.
The following considerations could be extended to the larger representation p G π u ,
where p G ∈ Z is introduced in 5.1.29. Hence we shall work in the framework of
the von Neumann algebra s G A
∗∗ which is isomorphic with the subalgebra P G B
# of
L(H ) via the mapping ρ G , cf. 5.1.11. The quasilocal algebra A will be identified
with its representation s G π u (A) in the Hilbert space s G H u or, equivalently, with the
corresponding C
∗ -subalgebra of the abstract W
∗ -algebra s G A
∗∗ . Remember that A
is simple, hence any of its nonzero representations as a C
∗ -algebra is faithful.
Let us introduce notation for various elements and subsets of s G A
∗∗ :
6.2.2 Notation. Let us denote:
(i) E g denotes the projection measure (G-measure, 5.2.3) on the linear space g
∗
generated by E g (F) (F ∈ g
∗
) from 5.1.13; in the notation of 5.1.16 E g (B) =
c(B) for any subset B ≡ B of g
∗ .
(ii) E g ( f ) :=
f (F) E g (dF) for any complex valued function f ∈ L
1
(g
∗
, μ
ω
g ) for
all ω ∈ S ∗ (s G A
∗∗
) := the normal states on s G A
∗∗ , i.e. the integral E g ( f ) is
assumed to converge in the w
∗ -sense.
(iii) B
N
0 := A
N
∪ {X ξ K : ξ ∈ g, K ∈ } ⊂ s G A
∗∗ , if the generators X ξ ∈ L(H)
are bounded, 5.1.3.
(iv) Let N
c be the C
∗ -subalgebra of s G A
∗∗ generated by all the elements E g ( f )
with uniformly bounded continuous f ∈ C b (g
∗
, C).
(v) C
N := the C
∗ - algebra generated by A
N and N
c ; C
N is isomorphic to the
C
∗ -tensor product A
N
⊗ N
c , the isomorphism being: x ⊗ z → xz ∈ C
N
(x ∈
A
N
, z ∈ N
c
), cf [274, 1.22] and [306, IV.4.7].
(vi) C will denote the C
∗ - algebra generated by {C
N
: N ∈ }; C is isomorphic
to the tensor product A ⊗ N
c , cf. 6.2.13.
6.2.3 Notation. Let {ξ j : j = 1, 2, . . . n} be a fixed basis of g. Let
X
N
j := |N | X j N :=
|N |
k=1
π k (X (ξ j )), X (ξ) := X ξ ,
(6.2.1)