16 Cluster Property and Pure Phases
109
limits belong to the center Z and (by hypothesis) commute with the spectral projections of U (x), in particular with P inv . Thus, for all convergent subsequences
w − lim
|x n |→∞
[A x n , P inv ] = 0.
(16.9)
This implies that the second term in (16.8) converges to zero, since otherwise
there is an ε > 0, a pair , ∈ H and a sequence {y n }, |y n | → ∞, such that
((, [A y n , P inv ] )| > ε for all |y n | sufficiently large; by the compactness argument
the sequence {A y n } has a convergent subsequence which would therefore not satisfy
(16.9). In conclusion, the weak limit |x| → ∞ of the left-hand side of (16.8) exists
and (16.7) holds.
Proposition 16.4 In a representation defined by a translationally invariant state,
satisfying weak asymptotic abelianess, with the center pointwise invariant under
space translations, the cluster property (16.1) is equivalent to the uniqueness of the
translationally invariant state.
In a factorial representation, the translationally invariant state is unique.
Proof. It follows easily from (16.7) that the cluster property holds iff P inv is onedimensional. For factorial representation (16.7) holds with P inv replaced by the
projection P 0 on the translationally invariant vector state 0 , which defines the
representation, since the (trivial) center obviously commutes with P 0 and therefore
the cluster property holds.
It is worthwhile to remark that irreducibility is a much too strong condition for
non-isolated systems, like those in thermodynamical equilibrium at non-zero temperature, which requires a heat exchange with the reservoir (or thermal bath). The
GNS representation defined by a translationally invariant equilibrium state has the
property that the equilibrium vector state is cyclic with respect to the observable
algebra, but there are operators (e.g. those describing the “dynamical variables” of
the reservoir), which commute with the observables of the system and therefore irreducibility fails. However, if the representation is factorial, by Proposition 16.4 the
translationally invariant equilibrium state cannot be decomposed as a convex combination of other translationally invariant states and in this sense describes a pure
phase. We shall return to non-zero temperature states later.
The physically motivated factorization of the correlation functions of infinitely
separated observables does not require irreducibility, but rather the uniqueness of the
translationally invariant state, which holds if the representation is factorial.
As it is clear from the above discussion, in a factorial representation defined
by a translationally invariant equilibrium state, the ergodic averages of observables
are c-numbers and coincide with the expectation values of the observables on the
equilibrium state. Such a state therefore encodes the information on the macroscopic
observables, as well as the large distance behaviour of the observables.
In the next section we shall confront the general framework discussed above with
some concrete examples.
109
limits belong to the center Z and (by hypothesis) commute with the spectral projections of U (x), in particular with P inv . Thus, for all convergent subsequences
w − lim
|x n |→∞
[A x n , P inv ] = 0.
(16.9)
This implies that the second term in (16.8) converges to zero, since otherwise
there is an ε > 0, a pair , ∈ H and a sequence {y n }, |y n | → ∞, such that
((, [A y n , P inv ] )| > ε for all |y n | sufficiently large; by the compactness argument
the sequence {A y n } has a convergent subsequence which would therefore not satisfy
(16.9). In conclusion, the weak limit |x| → ∞ of the left-hand side of (16.8) exists
and (16.7) holds.
Proposition 16.4 In a representation defined by a translationally invariant state,
satisfying weak asymptotic abelianess, with the center pointwise invariant under
space translations, the cluster property (16.1) is equivalent to the uniqueness of the
translationally invariant state.
In a factorial representation, the translationally invariant state is unique.
Proof. It follows easily from (16.7) that the cluster property holds iff P inv is onedimensional. For factorial representation (16.7) holds with P inv replaced by the
projection P 0 on the translationally invariant vector state 0 , which defines the
representation, since the (trivial) center obviously commutes with P 0 and therefore
the cluster property holds.
It is worthwhile to remark that irreducibility is a much too strong condition for
non-isolated systems, like those in thermodynamical equilibrium at non-zero temperature, which requires a heat exchange with the reservoir (or thermal bath). The
GNS representation defined by a translationally invariant equilibrium state has the
property that the equilibrium vector state is cyclic with respect to the observable
algebra, but there are operators (e.g. those describing the “dynamical variables” of
the reservoir), which commute with the observables of the system and therefore irreducibility fails. However, if the representation is factorial, by Proposition 16.4 the
translationally invariant equilibrium state cannot be decomposed as a convex combination of other translationally invariant states and in this sense describes a pure
phase. We shall return to non-zero temperature states later.
The physically motivated factorization of the correlation functions of infinitely
separated observables does not require irreducibility, but rather the uniqueness of the
translationally invariant state, which holds if the representation is factorial.
As it is clear from the above discussion, in a factorial representation defined
by a translationally invariant equilibrium state, the ergodic averages of observables
are c-numbers and coincide with the expectation values of the observables on the
equilibrium state. Such a state therefore encodes the information on the macroscopic
observables, as well as the large distance behaviour of the observables.
In the next section we shall confront the general framework discussed above with
some concrete examples.