152
22 ∗ Thermal States
product.
130 Then, the Gibbs state defined by ρ β (or by ρ β,μ ) can be written as
Ω β (A) = Tr (r 0 A r 0 ) = (Ψ r 0 , Ψ A r 0 ) ≡ (Ψ r 0 , π(A) Ψ r 0 ).
(22.9)
The above equation is well defined since A r 0 = 0 implies A = 0: in fact, as an
operator in the GNS representation space H β , r
−1
0 = Z
1/2
β e
β H/2 has a dense domain
D and therefore 0 = A r 0 (r
−1
0 D) = A D implies A = 0. This shows that Ψ r 0 is a
separating vector for the algebra A V , in this subsection simply denoted by A.
Furthermore, Ψ r 0 is a cyclic vector for A, since
(Ψ r , π(A) Ψ r 0 ) = Tr (r
∗
A r 0 ) = 0
implies Tr (r
∗ rr 0 r 0 ) = Tr (r 0 r
∗ rr 0 ) = 0, i.e. r r 0 = 0 and therefore r = 0.
In conclusion, (22.9) displays the explicit GNS representation π of A as operators
in the GNS representation space H β = D 0 , with a cyclic and separating vector Ψ r 0 .
The so constructed GNS representation space D 0 is also the carrier of a conjugate,
i.e. antilinear, representation π
of A given by
π
(A)Ψ r 0 = Ψ r 0 A ∗ .
(22.10)
Clearly, π
(λ A) = λ π
(A), ∀λ ∈ C, where λ denotes the complex conjugate of λ.
Furthermore,
||π(A)|| = ||π
(A)|| = ||A||,
(22.11)
so that the representation is faithful.
131
In order to characterize the infinite volume limit of the KMS states, we need to
derive other general properties of the GNS representation defined by a Gibbs state
(which we shall show to be stable under the thermodynamical limit). Such additional
information is provided by the following theorem.
130 In fact, τ convergence implies operator norm convergence, since
||B||
2 = ||B
∗ B|| = sup
||x||=1
(x, B
∗ Bx) ≤ Tr (B
∗ B).
Furthermore, the convergence of Tr (B ∗
n B n ) implies that the sequence is uniformly bounded, so
that
N
k=1 ||B n x k || 2 ≤ C uniformly in n, N and
N
k=1 ||B x k || 2 ≤ C, uniformly in N ; hence, the
limit operator B has a finite Hilbert–Schmidt norm.
131 In fact,
||π(A)||
2 = sup
r
Tr (r
∗ A
∗ Ar )/Tr (r
∗ r ) = sup
r
Tr (A rr
∗ A
∗ )/Tr (r
∗ r )
= sup
r
Tr (Ar
∗ r A
∗ )/Tr(rr
∗ ) = ||π
(A)||
2 .
The equality ||π(A)|| 2 = ||A|| 2 follows since one can choose r as the projection on a state with
spectral support relative to A ∗ A as close as one likes to ||A|| 2 .
22 ∗ Thermal States
product.
130 Then, the Gibbs state defined by ρ β (or by ρ β,μ ) can be written as
Ω β (A) = Tr (r 0 A r 0 ) = (Ψ r 0 , Ψ A r 0 ) ≡ (Ψ r 0 , π(A) Ψ r 0 ).
(22.9)
The above equation is well defined since A r 0 = 0 implies A = 0: in fact, as an
operator in the GNS representation space H β , r
−1
0 = Z
1/2
β e
β H/2 has a dense domain
D and therefore 0 = A r 0 (r
−1
0 D) = A D implies A = 0. This shows that Ψ r 0 is a
separating vector for the algebra A V , in this subsection simply denoted by A.
Furthermore, Ψ r 0 is a cyclic vector for A, since
(Ψ r , π(A) Ψ r 0 ) = Tr (r
∗
A r 0 ) = 0
implies Tr (r
∗ rr 0 r 0 ) = Tr (r 0 r
∗ rr 0 ) = 0, i.e. r r 0 = 0 and therefore r = 0.
In conclusion, (22.9) displays the explicit GNS representation π of A as operators
in the GNS representation space H β = D 0 , with a cyclic and separating vector Ψ r 0 .
The so constructed GNS representation space D 0 is also the carrier of a conjugate,
i.e. antilinear, representation π
of A given by
π
(A)Ψ r 0 = Ψ r 0 A ∗ .
(22.10)
Clearly, π
(λ A) = λ π
(A), ∀λ ∈ C, where λ denotes the complex conjugate of λ.
Furthermore,
||π(A)|| = ||π
(A)|| = ||A||,
(22.11)
so that the representation is faithful.
131
In order to characterize the infinite volume limit of the KMS states, we need to
derive other general properties of the GNS representation defined by a Gibbs state
(which we shall show to be stable under the thermodynamical limit). Such additional
information is provided by the following theorem.
130 In fact, τ convergence implies operator norm convergence, since
||B||
2 = ||B
∗ B|| = sup
||x||=1
(x, B
∗ Bx) ≤ Tr (B
∗ B).
Furthermore, the convergence of Tr (B ∗
n B n ) implies that the sequence is uniformly bounded, so
that
N
k=1 ||B n x k || 2 ≤ C uniformly in n, N and
N
k=1 ||B x k || 2 ≤ C, uniformly in N ; hence, the
limit operator B has a finite Hilbert–Schmidt norm.
131 In fact,
||π(A)||
2 = sup
r
Tr (r
∗ A
∗ Ar )/Tr (r
∗ r ) = sup
r
Tr (A rr
∗ A
∗ )/Tr (r
∗ r )
= sup
r
Tr (Ar
∗ r A
∗ )/Tr(rr
∗ ) = ||π
(A)||
2 .
The equality ||π(A)|| 2 = ||A|| 2 follows since one can choose r as the projection on a state with
spectral support relative to A ∗ A as close as one likes to ||A|| 2 .