106
16 Cluster Property and Pure Phases
temperature states the cluster property can be used to identify the pure phases. Actually, as we shall see later, the cluster property characterizes the pure phases also at
non-zero temperature, where irreducibility cannot hold.
The important property which makes the cluster property so relevant (also at nonzero temperature) is that of being equivalent to the uniqueness of the translationally
invariant state. In particular, this condition (assumed in III) appears justified also on
the basis of the motivations for the validity of the cluster property discussed above
and, in fact, can be replaced by the latter.
For the proof of such an equivalence, we remark that the cluster property in the
form of (16.1) states both the existence of the limit of the first term and the property
of being equal to the second. In order to make such a relation more transparent and
to point out its basic physical content, we shall first discuss a weaker form of the
cluster property in which the limit in (16.1) is taken in the Cesaro sense (weak cluster
property), and we shall prove the equivalence between such a weak form and the
uniqueness of the translationally invariant state.
We recall that such a weaker form of the limit (for brevity also called mean-limit or
mean ergodic limit) is defined in the following way, for locally measurable functions
(for simplicity we consider the case of one variable):
mean − lim
|x|→∞
f (x) ≡ lim
L→∞
L
−1
L
0
dx f (x).
(16.2)
The limit can easily be proved to exist for a large class of functions, e.g. if the Fourier
transform of f is a finite measure. It is clear that the values taken by f in any bounded
interval [0, L 0 ] do not affect the right-hand side since the latter is also equal to
lim
L→∞
L
−1
L
L 0
dx f (x).
The only thing which matters for the limit is the behaviour of f at infinity and clearly,
if f (x) has a limit in the ordinary sense, the mean-limit coincides with it.
Theorem 16.2 In any representation π defined by a translationally invariant state
and satisfying weak asymptotic abelianess, the weak cluster property,
lim
|V |→∞
|V |
−1
V
dx [< AB x > − < A >< B >] = 0,
(16.3)
where V is a bounded (regular) region centered at the origin, e.g. a sphere or a cube,
|V | denotes the volume of V and the limit is taken by expanding it equally in all
directions, is equivalent to the uniqueness of the translationally invariant state.
16 Cluster Property and Pure Phases
temperature states the cluster property can be used to identify the pure phases. Actually, as we shall see later, the cluster property characterizes the pure phases also at
non-zero temperature, where irreducibility cannot hold.
The important property which makes the cluster property so relevant (also at nonzero temperature) is that of being equivalent to the uniqueness of the translationally
invariant state. In particular, this condition (assumed in III) appears justified also on
the basis of the motivations for the validity of the cluster property discussed above
and, in fact, can be replaced by the latter.
For the proof of such an equivalence, we remark that the cluster property in the
form of (16.1) states both the existence of the limit of the first term and the property
of being equal to the second. In order to make such a relation more transparent and
to point out its basic physical content, we shall first discuss a weaker form of the
cluster property in which the limit in (16.1) is taken in the Cesaro sense (weak cluster
property), and we shall prove the equivalence between such a weak form and the
uniqueness of the translationally invariant state.
We recall that such a weaker form of the limit (for brevity also called mean-limit or
mean ergodic limit) is defined in the following way, for locally measurable functions
(for simplicity we consider the case of one variable):
mean − lim
|x|→∞
f (x) ≡ lim
L→∞
L
−1
L
0
dx f (x).
(16.2)
The limit can easily be proved to exist for a large class of functions, e.g. if the Fourier
transform of f is a finite measure. It is clear that the values taken by f in any bounded
interval [0, L 0 ] do not affect the right-hand side since the latter is also equal to
lim
L→∞
L
−1
L
L 0
dx f (x).
The only thing which matters for the limit is the behaviour of f at infinity and clearly,
if f (x) has a limit in the ordinary sense, the mean-limit coincides with it.
Theorem 16.2 In any representation π defined by a translationally invariant state
and satisfying weak asymptotic abelianess, the weak cluster property,
lim
|V |→∞
|V |
−1
V
dx [< AB x > − < A >< B >] = 0,
(16.3)
where V is a bounded (regular) region centered at the origin, e.g. a sphere or a cube,
|V | denotes the volume of V and the limit is taken by expanding it equally in all
directions, is equivalent to the uniqueness of the translationally invariant state.