6.3 Time Evolution in Generalized Mean-Field Theories
141
is valid too. We shall verify completness of C bs in this norm. For any Cauchy sequence
{ f n ; n ∈ Z + } in C bs , the sequence { f n (F), n ∈ Z + } is Cauchy in A for any F ∈ K .
The completness of A gives the existence of pointwise limits
f (F) := n- lim
k
f k (F) ∈ A, F ∈ K .
(6.3.11)
By defining the norm of any function f : K → A ( f could be infinite in
general) by (6.3.4), we have the norm-convergence of f k to f from (6.3.11): If
f n − f m < δ for all n, m > n δ , then f n − f < δ for all n > n δ , for any positive δ, since lim m f n (F) − f m (F) = = f n (F) − f (F) for all F ∈ K . Considering the cyclic representation (π ω , H ω , , ω ) corresponding to any ω ∈ p G S(A) as
a subrepresentation of the universal representation π u in p G π u (A), we have with the
identification of A with p G π u (A) (cf. (6.2.19)):
ˆ
p ω ( f (F) − f (F 0 )) = =( f (F) − f (F 0 ))) ω
≤ 2 f m − f + +( f m (F) − f m (F 0 ))) ω , (6.3.12)
and the s-continuity of f m ’s gives the s-continuity of f . A use of the norm-continuity
of the
∗ -operation gives us the s
∗ -continuity of f , i.e. f ∈ C bs . The remaining assertions of the proposition follow now easily.
6.3.4 Lemma. The functions f 0 from (6.3.7) belong to C bs . Hence, C
G
bs and C
G J
bs are
C
∗ -subalgebras of C bs .
Proof. Since f ∈ C b can be considered as an element of C bs , it suffices to prove
f 0 ∈ C bs for f 0 given by (6.3.7) with f := constant function. This will be proved
by proving the s
∗ -continuity of σ(G). For any x ∈ A and any ω ∈ p G S(A), we have
ˆ
p ω
σ g (x) − σ g 0 (x)
2 = ω
(σ g (x ∗ ) − σ g 0 (x ∗ ))(σ g (x) − σ g 0 (x))
=
= ω
σ g (x ∗ x) − σ g 0 (x ∗ x)
+
ω
(σ g 0 (x ∗ ) − σ g (x ∗ )) σ g 0 (x)
+ ω
σ g 0 (x ∗ )(σ g 0 (x) − σ g (x))
, (6.3.13)
and the s-continuity follows from the assumption (6.3.1) by repeated use of the
polarization identity (expressing nondiagonal matrix elements of bounded operators in a Hilbert space by a finite linear combination of the diagonal ones).
The s
∗ -continuity is then obtained by the replacement of x by x
∗ in the above
considerations.
6.3.5 The quasilocal C
∗ -algebra A of quantum (microscopic) observables is naturally embedded into C bs as a C
∗ -subalgebra by the identification of any x ∈ A with
a constant function f ∈ C
G
bs :
f (F) := x = 1(F) σ e (x), F ∈ K ,
(6.3.14)
141
is valid too. We shall verify completness of C bs in this norm. For any Cauchy sequence
{ f n ; n ∈ Z + } in C bs , the sequence { f n (F), n ∈ Z + } is Cauchy in A for any F ∈ K .
The completness of A gives the existence of pointwise limits
f (F) := n- lim
k
f k (F) ∈ A, F ∈ K .
(6.3.11)
By defining the norm of any function f : K → A ( f could be infinite in
general) by (6.3.4), we have the norm-convergence of f k to f from (6.3.11): If
f n − f m < δ for all n, m > n δ , then f n − f < δ for all n > n δ , for any positive δ, since lim m f n (F) − f m (F) = = f n (F) − f (F) for all F ∈ K . Considering the cyclic representation (π ω , H ω , , ω ) corresponding to any ω ∈ p G S(A) as
a subrepresentation of the universal representation π u in p G π u (A), we have with the
identification of A with p G π u (A) (cf. (6.2.19)):
ˆ
p ω ( f (F) − f (F 0 )) = =( f (F) − f (F 0 ))) ω
≤ 2 f m − f + +( f m (F) − f m (F 0 ))) ω , (6.3.12)
and the s-continuity of f m ’s gives the s-continuity of f . A use of the norm-continuity
of the
∗ -operation gives us the s
∗ -continuity of f , i.e. f ∈ C bs . The remaining assertions of the proposition follow now easily.
6.3.4 Lemma. The functions f 0 from (6.3.7) belong to C bs . Hence, C
G
bs and C
G J
bs are
C
∗ -subalgebras of C bs .
Proof. Since f ∈ C b can be considered as an element of C bs , it suffices to prove
f 0 ∈ C bs for f 0 given by (6.3.7) with f := constant function. This will be proved
by proving the s
∗ -continuity of σ(G). For any x ∈ A and any ω ∈ p G S(A), we have
ˆ
p ω
σ g (x) − σ g 0 (x)
2 = ω
(σ g (x ∗ ) − σ g 0 (x ∗ ))(σ g (x) − σ g 0 (x))
=
= ω
σ g (x ∗ x) − σ g 0 (x ∗ x)
+
ω
(σ g 0 (x ∗ ) − σ g (x ∗ )) σ g 0 (x)
+ ω
σ g 0 (x ∗ )(σ g 0 (x) − σ g (x))
, (6.3.13)
and the s-continuity follows from the assumption (6.3.1) by repeated use of the
polarization identity (expressing nondiagonal matrix elements of bounded operators in a Hilbert space by a finite linear combination of the diagonal ones).
The s
∗ -continuity is then obtained by the replacement of x by x
∗ in the above
considerations.
6.3.5 The quasilocal C
∗ -algebra A of quantum (microscopic) observables is naturally embedded into C bs as a C
∗ -subalgebra by the identification of any x ∈ A with
a constant function f ∈ C
G
bs :
f (F) := x = 1(F) σ e (x), F ∈ K ,
(6.3.14)