22.1 Gibbs States and KMS Condition
151
An equivalent form of the KMS condition, which will turn useful in the applications is the following.
128 Let ˜
F
β
(w), ˜
G
β
(w) be the (distributional) Fourier transforms
of F
β
(t), G
β
(t), (defined in (22.3)), respectively. Then
˜
F
β
(w) = e
−β w ˜
G
β
(w).
(22.6)
This follows from the fact that F
β
(t), G
β
(t) and F
β
(t + iβ) are all bounded continuous functions of t, hence tempered distributions, and the Fourier transform of
F
β
(t + iβ) is e
β w ˜
F
β
(w). By a similar argument, one shows that the KMS condition
is equivalent to the following one:
f (t − iβ) F
β
(t) dt =
f (t) G
β
(t) dt, ∀ ˜
f ∈ D(R).
(22.7)
22.2 GNS Representation Defined by a Gibbs State
As we shall see below, the main virtue of the KMS condition with respect to the Gibbs
formula ((22.1) or (22.2)) is that the former one survives the thermodynamical limit
and can therefore be used to characterize the equilibrium states in this limit, whereas
the Gibbs formula becomes meaningless.
The physical reason is that in the infinite volume limit the average energy diverges.
On the other hand, by Theorem 22.1, quite generally the KMS condition implies
the invariance of the state under time translations and therefore the existence of a
one-parameter group of (strongly continuous) unitary operators U (t), implementing
the time translations, in the GNS representation defined by such a KMS state. The
apparent conflict between the existence of the generator of U (t) and the divergence
of the average energy requires a better understanding of the structure of the GNS
representation defined by a KMS state. Again, we shall start from the case of finite
volume.
From the definition of a Gibbs state, we have that Z
−1
β ρ β is a positive operator and
we denote by r 0 its square root; it is a Hilbert–Schmidt operator, i.e. such that r
∗
0 r 0 is
of trace class. The vector space D 0 of Hilbert–Schmidt operators is invariant under
right and left multiplication by bounded operators
129 and it is naturally equipped by
a Hilbert scalar product
(Ψ r , Ψ r ) ≡ Tr (r
∗ r ),
(22.8)
where Ψ r denotes the vector identified by the Hilbert–Schmidt operator r . Actually,
D 0 is a Hilbert space, i.e. it is closed under the topology τ defined by the above scalar
128 [HHW].
129 See, e.g. M. Reed and B. Simon, loc. cit., Theorem VI.22.
151
An equivalent form of the KMS condition, which will turn useful in the applications is the following.
128 Let ˜
F
β
(w), ˜
G
β
(w) be the (distributional) Fourier transforms
of F
β
(t), G
β
(t), (defined in (22.3)), respectively. Then
˜
F
β
(w) = e
−β w ˜
G
β
(w).
(22.6)
This follows from the fact that F
β
(t), G
β
(t) and F
β
(t + iβ) are all bounded continuous functions of t, hence tempered distributions, and the Fourier transform of
F
β
(t + iβ) is e
β w ˜
F
β
(w). By a similar argument, one shows that the KMS condition
is equivalent to the following one:
f (t − iβ) F
β
(t) dt =
f (t) G
β
(t) dt, ∀ ˜
f ∈ D(R).
(22.7)
22.2 GNS Representation Defined by a Gibbs State
As we shall see below, the main virtue of the KMS condition with respect to the Gibbs
formula ((22.1) or (22.2)) is that the former one survives the thermodynamical limit
and can therefore be used to characterize the equilibrium states in this limit, whereas
the Gibbs formula becomes meaningless.
The physical reason is that in the infinite volume limit the average energy diverges.
On the other hand, by Theorem 22.1, quite generally the KMS condition implies
the invariance of the state under time translations and therefore the existence of a
one-parameter group of (strongly continuous) unitary operators U (t), implementing
the time translations, in the GNS representation defined by such a KMS state. The
apparent conflict between the existence of the generator of U (t) and the divergence
of the average energy requires a better understanding of the structure of the GNS
representation defined by a KMS state. Again, we shall start from the case of finite
volume.
From the definition of a Gibbs state, we have that Z
−1
β ρ β is a positive operator and
we denote by r 0 its square root; it is a Hilbert–Schmidt operator, i.e. such that r
∗
0 r 0 is
of trace class. The vector space D 0 of Hilbert–Schmidt operators is invariant under
right and left multiplication by bounded operators
129 and it is naturally equipped by
a Hilbert scalar product
(Ψ r , Ψ r ) ≡ Tr (r
∗ r ),
(22.8)
where Ψ r denotes the vector identified by the Hilbert–Schmidt operator r . Actually,
D 0 is a Hilbert space, i.e. it is closed under the topology τ defined by the above scalar
128 [HHW].
129 See, e.g. M. Reed and B. Simon, loc. cit., Theorem VI.22.