108
16 Cluster Property and Pure Phases
The pointwise invariance of the center under space translations follows from
the relativistic spectral condition,
86 and one may argue about its validity for nonrelativistic systems. The important physical property following from it is that the
existence of the weak limits imply that they coincide with the ergodic averages
w − lim
V →∞
V
−1
V
dx α x (A),
(16.6)
which describe macroscopic observables (for simplicity V denotes both the bounded
region and its volume).
A special case in which the pointwise (space translation) invariance of the center obviously holds is that of the so-called factorial representations, defined by
the condition of having a trivial center: Z = {λ1, λ ∈ C}. The class of factorial
representations includes in particular the irreducible representations (for which
π(A)
= {λ1, λ ∈ C}), but is much more general; in fact, it can be taken as the mathematical characterization of the pure phases, also at non-zero temperature (where
the representation cannot be irreducible). The physical motivation for such a choice
is that the ergodic decomposition of a representation with respect to the space translations
87 automatically leads to definite values for the macroscopic observables.
Proposition 16.3 In any representation defined by a translationally invariant state,
satisfying weak asymptotic abelianess, with the property that the center Z is pointwise invariant under translations, one has
w − lim
|x|→∞
A x B 0 = B P inv A 0 ,
(16.7)
where P inv denotes the projection on the subspace of translationally invariant vectors
and the symbols A, B denote the representatives of elements of A in the given
representation.
Proof. One can essentially use the same argument as in the derivation of (15.2), with
P 0 replaced by the projection P inv . In fact, ∀B ∈ π(A),
A x B 0 = {[A x , B] + B [A x , P inv ] + B P inv A x } 0
(16.8)
and in the limit |x| → ∞, the first term vanishes by asymptotic abelianess, and the
last term is independent of x.
Thus, one has to discuss the weak limit of the second term. To this purpose, one
notes that ||A x || = ||A||, since the space translations are automorphisms of A and
therefore norm preserving. Then, by a compactness argument there are subsequences
{A x n }, |x n | → ∞, which have weak limits z {x n } . By asymptotic abelianess such weak
86 H. Araki, Prog. Theor. Phys. 32, 884 (1964).
87 See O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol.
1. Springer 1987, Sect. 4.3.
16 Cluster Property and Pure Phases
The pointwise invariance of the center under space translations follows from
the relativistic spectral condition,
86 and one may argue about its validity for nonrelativistic systems. The important physical property following from it is that the
existence of the weak limits imply that they coincide with the ergodic averages
w − lim
V →∞
V
−1
V
dx α x (A),
(16.6)
which describe macroscopic observables (for simplicity V denotes both the bounded
region and its volume).
A special case in which the pointwise (space translation) invariance of the center obviously holds is that of the so-called factorial representations, defined by
the condition of having a trivial center: Z = {λ1, λ ∈ C}. The class of factorial
representations includes in particular the irreducible representations (for which
π(A)
= {λ1, λ ∈ C}), but is much more general; in fact, it can be taken as the mathematical characterization of the pure phases, also at non-zero temperature (where
the representation cannot be irreducible). The physical motivation for such a choice
is that the ergodic decomposition of a representation with respect to the space translations
87 automatically leads to definite values for the macroscopic observables.
Proposition 16.3 In any representation defined by a translationally invariant state,
satisfying weak asymptotic abelianess, with the property that the center Z is pointwise invariant under translations, one has
w − lim
|x|→∞
A x B 0 = B P inv A 0 ,
(16.7)
where P inv denotes the projection on the subspace of translationally invariant vectors
and the symbols A, B denote the representatives of elements of A in the given
representation.
Proof. One can essentially use the same argument as in the derivation of (15.2), with
P 0 replaced by the projection P inv . In fact, ∀B ∈ π(A),
A x B 0 = {[A x , B] + B [A x , P inv ] + B P inv A x } 0
(16.8)
and in the limit |x| → ∞, the first term vanishes by asymptotic abelianess, and the
last term is independent of x.
Thus, one has to discuss the weak limit of the second term. To this purpose, one
notes that ||A x || = ||A||, since the space translations are automorphisms of A and
therefore norm preserving. Then, by a compactness argument there are subsequences
{A x n }, |x n | → ∞, which have weak limits z {x n } . By asymptotic abelianess such weak
86 H. Araki, Prog. Theor. Phys. 32, 884 (1964).
87 See O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol.
1. Springer 1987, Sect. 4.3.