154
22 ∗ Thermal States
nuity of α t implies that U (t) can be chosen weakly and therefore strongly continuous.
Now, the condition
U (t) π(A) U (t)
−1
= π(α t (A))
(22.20)
implies
U (t) = π(U F (t)) V (t),
(22.21)
where U F (t) is generated by the Fock Hamiltonian H F (or by the Fock operator
H F − μN ) (for simplicity, the two possibilities will both be denoted by ˜
H F ) and
V (t) ∈ π(A)
.
The invariance of Ψ r 0 under U (t) uniquely fixes V (t). In fact, since π(A)
=
π
(A)
, V (t) is of the form V (t) = π
(V F (t)), V F (t) ∈ B(H F ) = A and therefore
Ψ r 0 = π(U F (t)) π
(V F (t)) Ψ r 0 = Ψ U F (t)r 0 V F (t) ∗ .
Since r 0 commutes with U F (t) and the representation is faithful
r 0 (1 − U F (t) V F (t)
∗
) = 0, i.e. V F (t) = U F (t).
In conclusion the generator of U (t) is not the Fock Hamiltonian H F (or H F − μ N )
but
H = π(H F ) − π
(H F )
(22.22)
where the second term on the right-hand side has the meaning of the contribution to
the energy by the reservoir.
It is instructive to work out the case of a (quantum) lattice spin system and explicitly check the properties i), ii) (in particular (22.17) is easily proven). We shall leave
this exercise to the reader.
134
As we shall see below, the occurrence of a subtraction in the definition of the
generator of the time translations allows for the existence of the generator H also in
the infinite volume limit, when the average energy becomes divergent and the Fock
Hamiltonian H F (or H F − μ N ) does not exist.
22.3 KMS States in the Thermodynamical Limit
The power of the KMS condition is that it makes sense also in the thermodynamical
limit and can be used as a characterization of the equilibrium states in such a limit,
where the Gibbs prescription becomes meaningless. It can actually be proven that
the KMS condition survives the thermodynamical limit under general conditions, as
stated in Theorem 22.3 below.
For this purpose, a few comments on the thermodynamical limit are useful. A
state Ω V describing the system in a finite volume V is a positive linear functional
on A(V ) ⊆ A, (A ≡ the quasi-local algebra, see Chap. 14). By the Hahn–Banach
134 For help see Hugenholtz’s lectures in London (1972).
22 ∗ Thermal States
nuity of α t implies that U (t) can be chosen weakly and therefore strongly continuous.
Now, the condition
U (t) π(A) U (t)
−1
= π(α t (A))
(22.20)
implies
U (t) = π(U F (t)) V (t),
(22.21)
where U F (t) is generated by the Fock Hamiltonian H F (or by the Fock operator
H F − μN ) (for simplicity, the two possibilities will both be denoted by ˜
H F ) and
V (t) ∈ π(A)
.
The invariance of Ψ r 0 under U (t) uniquely fixes V (t). In fact, since π(A)
=
π
(A)
, V (t) is of the form V (t) = π
(V F (t)), V F (t) ∈ B(H F ) = A and therefore
Ψ r 0 = π(U F (t)) π
(V F (t)) Ψ r 0 = Ψ U F (t)r 0 V F (t) ∗ .
Since r 0 commutes with U F (t) and the representation is faithful
r 0 (1 − U F (t) V F (t)
∗
) = 0, i.e. V F (t) = U F (t).
In conclusion the generator of U (t) is not the Fock Hamiltonian H F (or H F − μ N )
but
H = π(H F ) − π
(H F )
(22.22)
where the second term on the right-hand side has the meaning of the contribution to
the energy by the reservoir.
It is instructive to work out the case of a (quantum) lattice spin system and explicitly check the properties i), ii) (in particular (22.17) is easily proven). We shall leave
this exercise to the reader.
134
As we shall see below, the occurrence of a subtraction in the definition of the
generator of the time translations allows for the existence of the generator H also in
the infinite volume limit, when the average energy becomes divergent and the Fock
Hamiltonian H F (or H F − μ N ) does not exist.
22.3 KMS States in the Thermodynamical Limit
The power of the KMS condition is that it makes sense also in the thermodynamical
limit and can be used as a characterization of the equilibrium states in such a limit,
where the Gibbs prescription becomes meaningless. It can actually be proven that
the KMS condition survives the thermodynamical limit under general conditions, as
stated in Theorem 22.3 below.
For this purpose, a few comments on the thermodynamical limit are useful. A
state Ω V describing the system in a finite volume V is a positive linear functional
on A(V ) ⊆ A, (A ≡ the quasi-local algebra, see Chap. 14). By the Hahn–Banach
134 For help see Hugenholtz’s lectures in London (1972).