156
22 ∗ Thermal States
where the vector Ψ β represents Ω β in H β ,
2) the time translations are implemented by strongly continuous unitary operators
U (t), such that
U (t) Ψ β = Ψ β , [U (t), J ] = 0,
(22.25)
3) the generator H of U (t) satisfies
e
−β H/2
π(A) Ψ β = J π(A)
∗
Ψ β , ∀A ∈ A
(22.26)
and by (22.25) H J − J H = 0.
22.4 Pure Phases. Extremal and Primary KMS States
In this subsection, we shall discuss the characterization of the pure phases in thermodynamics.
First we recall that the thermodynamical phases are defined by equilibrium states
and the above discussion indicates that the KMS states, being the thermodynamical
limit of equilibrium Gibbs states, are the natural candidates for describing equilibrium
states.
137 Thus, one has to identify the property of KMS states which corresponds to
the phase being pure.
In the zero temperature case, the pure phases were identified by irreducible representations, but now, by the previous discussion, in particular (22.24), the GNS
representation defined by a KMS state is not irreducible and actually its commutant
is as big as the weak closure of π(A). The physical interpretation of such a violation
of irreducibility is that the role of the commutant is to account for the “degrees of
freedom” of the reservoir, whose interaction with the system is needed in order to
keep the temperature constant. Thus, irreducibility cannot be used to characterize
the pure phases at non-zero temperature.
The relevant property is that the concept of pure phase is related to that of equilibrium state which is not a mixture of other equilibrium states. KMS states which cannot
be decomposed as mixture of other KMS states are called extremal and therefore the
pure thermodynamical phases can be described by extremal KMS states.
In general, since in the standard thermodynamical sense pure phases are defined
by homogeneous states, one adds the condition that the corresponding KMS states
are invariant under space translations.
In conclusion, with respect to the zero temperature case discussed in Chap. 15,
for non-zero temperature the conditions which select the physically relevant representations have to be modified as follows.
I. (Existence of energy and momentum) The space and time translations are
implemented by strongly continuous groups of unitary operators (as in Chap. 15).
137 For further arguments involving stability properties, see Haag’s book (1996), Sect. V.3.
22 ∗ Thermal States
where the vector Ψ β represents Ω β in H β ,
2) the time translations are implemented by strongly continuous unitary operators
U (t), such that
U (t) Ψ β = Ψ β , [U (t), J ] = 0,
(22.25)
3) the generator H of U (t) satisfies
e
−β H/2
π(A) Ψ β = J π(A)
∗
Ψ β , ∀A ∈ A
(22.26)
and by (22.25) H J − J H = 0.
22.4 Pure Phases. Extremal and Primary KMS States
In this subsection, we shall discuss the characterization of the pure phases in thermodynamics.
First we recall that the thermodynamical phases are defined by equilibrium states
and the above discussion indicates that the KMS states, being the thermodynamical
limit of equilibrium Gibbs states, are the natural candidates for describing equilibrium
states.
137 Thus, one has to identify the property of KMS states which corresponds to
the phase being pure.
In the zero temperature case, the pure phases were identified by irreducible representations, but now, by the previous discussion, in particular (22.24), the GNS
representation defined by a KMS state is not irreducible and actually its commutant
is as big as the weak closure of π(A). The physical interpretation of such a violation
of irreducibility is that the role of the commutant is to account for the “degrees of
freedom” of the reservoir, whose interaction with the system is needed in order to
keep the temperature constant. Thus, irreducibility cannot be used to characterize
the pure phases at non-zero temperature.
The relevant property is that the concept of pure phase is related to that of equilibrium state which is not a mixture of other equilibrium states. KMS states which cannot
be decomposed as mixture of other KMS states are called extremal and therefore the
pure thermodynamical phases can be described by extremal KMS states.
In general, since in the standard thermodynamical sense pure phases are defined
by homogeneous states, one adds the condition that the corresponding KMS states
are invariant under space translations.
In conclusion, with respect to the zero temperature case discussed in Chap. 15,
for non-zero temperature the conditions which select the physically relevant representations have to be modified as follows.
I. (Existence of energy and momentum) The space and time translations are
implemented by strongly continuous groups of unitary operators (as in Chap. 15).
137 For further arguments involving stability properties, see Haag’s book (1996), Sect. V.3.