22.1 Gibbs States and KMS Condition
149
temperature. Clearly, the framework discussed in Chap. 14 for the zero temperature
case requires substantial changes.
For this purpose, we recall a few basic mathematical properties of the Gibbs states.
First, we recall that for a system of free particles in a box exp(−β H 0 ), where H 0 is
the free Hamiltonian, is of trace class
123 , i.e. Tr| exp(−β H 0 )| < ∞. Under general
conditions on the interaction potential
124 also exp(−β H ) is of trace class, for all
positive β’s.
Since the product of a bounded operator and a trace class operator is an operator
of trace class
125 , also exp(−β(H − μN )) is of trace class for all β, for μ in a suitable
range, so that H − μN > 0. Thus, under such general conditions (22.1), (22.2) are
well defined.
When dealing with systems in a finite volume, we shall always assume that ρ β
and/or ρ β,μ is of trace class.
Since the thermodynamical limit is a convenient extrapolation for the description
of very large systems, it is physically reasonable to try to extract those structural
properties of the finite dimensional case which are expected to be stable in the limit.
Theorem 22.1
126 Under the above general conditions a Gibbs state, given by (22.1)
or (22.2), satisfies the KMS condition,
127 namely ∀A, B ∈ B(H)
123 For the properties of trace class operators, see, e.g. M. Reed and B. Simon, Methods of Modern
Mathematical Physics, Vol. I, Academic Press 1972, Sect. VI.6.
124 D. Ruelle, Helv. Phys. Acta 36, 789 (1963); J. Lebowitz and E. Lieb, Adv. Math. 9, 316 (1972),
Appendix by B. Simon. A sufficient condition is that the potential U is a small perturbation, i.e.
that for any a < 1 there is a b ≥ 0 such that |(Ψ, U Ψ )| < a (Ψ, H 0 Ψ ) + b (Ψ, Ψ ), for all Ψ in the
domain of the free Hamiltonian H 0 . In fact, the above inequality implies H ≥ (1 − a)H 0 − b1 and
e −β H0 of trace class implies e −β H of trace class.
125 See, e.g. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I, Academic
Press, Theorem VI.19. The point is that the operators of trace class form a vector space. In fact, for
any partial isometry S, by using Schwarz’ inequality one has
|Tr (S|A|)| ≤
n
|| |A|
1/2 S
∗ Ψ n || || |A|
1/2 Ψ n ||
≤ (
n
|| |A|
1/2 S
∗ Ψ n ||
2 )
1/2 (
n
|| |A|
1/2 Ψ n ||
2 )
1/2 = (Tr(S|A|S
∗ ))
1/2 (Tr|A|)
1/2
and Tr (S|A|S ∗ ) ≤ Tr |A|. Then, if U , U A , U B denote the partial isometries occurring in the polar
decomposition of A + B, A, B, respectively,
Tr |A + B| ≤ |Tr (U
∗ U A |A|)| + |Tr (U
∗ U B |B|)| ≤ Tr |A| + Tr |B|.
Hence, in order to prove that A B is of trace class if B is so and A is bounded, it suffices to consider
the case in which A is self-adjoint and of norm less than one; in this case A is a linear combination
of the unitary operators U ± ≡ A ± i(1 − A 2 ) 1/2 , and U ± B is clearly of trace class, if B is so, since
|U ± B| = |B|.
126 R. Haag, N.M. Hugenholtz and M. Winnink, Comm. Math. Phys. 5, 215 (1967), hereafter referred
as [HHW].
127 R. Kubo, J. Phys. Soc. Jap. 12, 570 (1957); P.C. Martin and J. Schwinger, Phys. Rev. 115, 1342
(1959).
149
temperature. Clearly, the framework discussed in Chap. 14 for the zero temperature
case requires substantial changes.
For this purpose, we recall a few basic mathematical properties of the Gibbs states.
First, we recall that for a system of free particles in a box exp(−β H 0 ), where H 0 is
the free Hamiltonian, is of trace class
123 , i.e. Tr| exp(−β H 0 )| < ∞. Under general
conditions on the interaction potential
124 also exp(−β H ) is of trace class, for all
positive β’s.
Since the product of a bounded operator and a trace class operator is an operator
of trace class
125 , also exp(−β(H − μN )) is of trace class for all β, for μ in a suitable
range, so that H − μN > 0. Thus, under such general conditions (22.1), (22.2) are
well defined.
When dealing with systems in a finite volume, we shall always assume that ρ β
and/or ρ β,μ is of trace class.
Since the thermodynamical limit is a convenient extrapolation for the description
of very large systems, it is physically reasonable to try to extract those structural
properties of the finite dimensional case which are expected to be stable in the limit.
Theorem 22.1
126 Under the above general conditions a Gibbs state, given by (22.1)
or (22.2), satisfies the KMS condition,
127 namely ∀A, B ∈ B(H)
123 For the properties of trace class operators, see, e.g. M. Reed and B. Simon, Methods of Modern
Mathematical Physics, Vol. I, Academic Press 1972, Sect. VI.6.
124 D. Ruelle, Helv. Phys. Acta 36, 789 (1963); J. Lebowitz and E. Lieb, Adv. Math. 9, 316 (1972),
Appendix by B. Simon. A sufficient condition is that the potential U is a small perturbation, i.e.
that for any a < 1 there is a b ≥ 0 such that |(Ψ, U Ψ )| < a (Ψ, H 0 Ψ ) + b (Ψ, Ψ ), for all Ψ in the
domain of the free Hamiltonian H 0 . In fact, the above inequality implies H ≥ (1 − a)H 0 − b1 and
e −β H0 of trace class implies e −β H of trace class.
125 See, e.g. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I, Academic
Press, Theorem VI.19. The point is that the operators of trace class form a vector space. In fact, for
any partial isometry S, by using Schwarz’ inequality one has
|Tr (S|A|)| ≤
n
|| |A|
1/2 S
∗ Ψ n || || |A|
1/2 Ψ n ||
≤ (
n
|| |A|
1/2 S
∗ Ψ n ||
2 )
1/2 (
n
|| |A|
1/2 Ψ n ||
2 )
1/2 = (Tr(S|A|S
∗ ))
1/2 (Tr|A|)
1/2
and Tr (S|A|S ∗ ) ≤ Tr |A|. Then, if U , U A , U B denote the partial isometries occurring in the polar
decomposition of A + B, A, B, respectively,
Tr |A + B| ≤ |Tr (U
∗ U A |A|)| + |Tr (U
∗ U B |B|)| ≤ Tr |A| + Tr |B|.
Hence, in order to prove that A B is of trace class if B is so and A is bounded, it suffices to consider
the case in which A is self-adjoint and of norm less than one; in this case A is a linear combination
of the unitary operators U ± ≡ A ± i(1 − A 2 ) 1/2 , and U ± B is clearly of trace class, if B is so, since
|U ± B| = |B|.
126 R. Haag, N.M. Hugenholtz and M. Winnink, Comm. Math. Phys. 5, 215 (1967), hereafter referred
as [HHW].
127 R. Kubo, J. Phys. Soc. Jap. 12, 570 (1957); P.C. Martin and J. Schwinger, Phys. Rev. 115, 1342
(1959).