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22 ∗ Thermal States
In fact, [T
t , A] = U (t) [T
, A −t ]U (−t) = 0, ∀A ∈ A implies T
t ∈ π(A)
and
then
(π(A) Ψ Ω , T
t π(B) Ψ Ω ) = (Ψ Ω , π(A)
∗
π(B)T
t Ψ Ω ) =
(Ψ Ω , (π(A
∗ B)) −t T
Ψ Ω ) = ω 1 ((A
∗ B) −t ) = ω 1 (A
∗ B) =
= (π(A) Ψ Ω , T
π(B) Ψ Ω ),
where the invariance of Ω and ω 1 under time translations has been used.
4) Define
T ≡ J T
J.
(22.30)
It belongs to π(A)
by (22.24) and also to π(A)
. In fact, the time translation
invariance of Ψ Ω and (22.26) gives
T Ψ Ω = J T
J Ψ Ω = J T
Ψ Ω = J T
e
−β H/2
Ψ Ω =
= J e
−β H/2 T
Ψ Ω = T
Ψ Ω ,
where in the last step we have used that T
∈ π(A)
and the strong closure of
(2.26). Therefore
ω 1 (A) = (Ψ Ω , T
AΨ Ω ) = (Ψ Ω , AT
Ψ Ω ) = (Ψ Ω , A T Ψ Ω ) = Ω(A T ).
To conclude the argument, we use the KMS condition in the following form:
ω(A B t ) = ω(B t−iβ A),
which can be easily derived in the same way as (22.5), and the above relation
Ω(A T ) = ω 1 (A); thus, we get
Ω(AT BC) = Ω(α −iβ (BC)AT ) = ω 1 (α −iβ (BC)A) =
ω 1 (α −iβ (B)α −iβ (C)A) = ω 1 (α −iβ (C)AB) =
= Ω(α −iβ (C)ABT ) = Ω(ABT C),
i.e. [T, B] = 0, ∀B ∈ A. In conclusion, since T ∈ Z, Ω is extremal iff Z =
{λ 1, λ ∈ C}, i.e. its GNS representation is factorial, briefly iff Ω is a factor
state.
The physical relevance of the concept of factor, also called primary, state is that
in the GNS representation defined by it macroscopic observables like ergodic means
or variables at infinity have a sharp (classical) value in agreement with the physical
picture of a pure phase in thermodynamics.