22.2 GNS Representation Defined by a Gibbs State
153
Theorem 22.2
132 The GNS representation defined by a Gibbs state has the following
properties
i) the commutant π(A)
of π(A) is the weak closure of π
(A)
π(A)
= (π
(A))
,
(22.12)
(equivalently π(A)
= (π
(A))
)
ii) there exists an antiunitary operator J such that
J π(A) J = π
(A), ∀A ∈ A,
(22.13)
J
2
= 1,
(22.14)
J Ψ r 0 = Ψ r 0 .
(22.15)
Proof. i) By definition
π(A) π
(B) Ψ r 0 = Ψ Ar 0 B ∗ = π
(B)π(A)Ψ r 0 ,
so that π(A) ⊆ (π
(A))
. By taking the weak closure one gets
π(A)
⊆ ((π
(A))
)
= (π
(A))
.
(22.16)
On the other hand, the subalgebra A 0 ⊆ A of Hilbert–Schmidt operators is a
Hilbert algebra
133 and for such algebras
π(A 0 )
= (π
(A 0 ))
.
(22.17)
Then,
π(A)
⊇ π(A 0 )
= (π
(A 0 ))
⊇ π
(A))
.
(22.18)
Equations (22.16), (22.18) imply (22.12).
ii) The operator J is defined by J Ψ r = Ψ r ∗ . Equations (22.14), (22.15) are obvious
and (22.13) follows from
J π(A)J Ψ r = J π(A)Ψ r ∗ = J Ψ Ar ∗ = Ψ r A ∗ = π
(A)Ψ r .
(22.19)
We can now clarify the relation between the generator of the time translations and
the many-particle (Fock) Hamiltonian.
The time invariance of a Gibbs state Ω β implies that in the corresponding GNS
representation the time translations are implemented by a one-parameter group of
unitary operators U (t), t ∈ R (we omit the finite volume suffix V ). The weak conti132 [HHW]; see also the book by Haag [1996] and the London lectures by Hugenholtz (1972), quoted
in footnote 121.
133 J. Dixmier, Von Neumann algebras, North-Holland 1981, Chap. 11.