Chapter 16
Cluster Property and Pure Phases
The irreducible (physically relevant) representations selected in the previous section
have a further important property, called cluster property.
Proposition 16.1 Under the same conditions of Proposition 15.2, the ground state
correlation of two quasi-local operators factorize, when one is translated at space
infinity
lim
|x|→∞
[< A B x > 0 − < A > 0 < B > 0 ] = 0
(16.1)
The proof follows easily from (15.2).
The reasons for stressing this property are many. First, the cluster property plays
a crucial role for the foundations of the S-matrix theory in quantum field theory.
82
In fact, the possibility itself of defining a scattering process requires such a factorization of the amplitude relative to clusters of fields which are infinitely separated in
space. Otherwise, a scattering process localized in a space–time region O would be
influenced by a scattering taking place at very large distances.
The physical meaning of the cluster property is that the ground state reacts locally
to local operations, and it cannot support non-trivial correlations between far separated observables. In a certain sense, this condition neutralizes the non-local content
of the ground state to the effect that the latter does not spoil the local structure of
the physically realizable operations, at the level of the correlation functions, and it
is essentially confined to the property of accounting for the large distance limits of
the observables.
For representations satisfying conditions I, II, III i), one can show that the cluster
property implies irreducibility and therefore it is equivalent to it, but, from a constructive point of view, the cluster property is much better controlled since it can
be directly read off from the knowledge of the correlation functions. Thus, for zero
82 R. Haag, Phys. Rev. 112, 669 (1958); Local Quantum Physics, Springer 1996, esp. Sect. II.4; D.
Ruelle, Helv. Phys. Acta 35, 147 (1962). For a systematic account of the Haag–Ruelle theory, see
R. Jost, The General Theory of Quantized Fields, Am. Math. Soc. 1965.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_16
105
Cluster Property and Pure Phases
The irreducible (physically relevant) representations selected in the previous section
have a further important property, called cluster property.
Proposition 16.1 Under the same conditions of Proposition 15.2, the ground state
correlation of two quasi-local operators factorize, when one is translated at space
infinity
lim
|x|→∞
[< A B x > 0 − < A > 0 < B > 0 ] = 0
(16.1)
The proof follows easily from (15.2).
The reasons for stressing this property are many. First, the cluster property plays
a crucial role for the foundations of the S-matrix theory in quantum field theory.
82
In fact, the possibility itself of defining a scattering process requires such a factorization of the amplitude relative to clusters of fields which are infinitely separated in
space. Otherwise, a scattering process localized in a space–time region O would be
influenced by a scattering taking place at very large distances.
The physical meaning of the cluster property is that the ground state reacts locally
to local operations, and it cannot support non-trivial correlations between far separated observables. In a certain sense, this condition neutralizes the non-local content
of the ground state to the effect that the latter does not spoil the local structure of
the physically realizable operations, at the level of the correlation functions, and it
is essentially confined to the property of accounting for the large distance limits of
the observables.
For representations satisfying conditions I, II, III i), one can show that the cluster
property implies irreducibility and therefore it is equivalent to it, but, from a constructive point of view, the cluster property is much better controlled since it can
be directly read off from the knowledge of the correlation functions. Thus, for zero
82 R. Haag, Phys. Rev. 112, 669 (1958); Local Quantum Physics, Springer 1996, esp. Sect. II.4; D.
Ruelle, Helv. Phys. Acta 35, 147 (1962). For a systematic account of the Haag–Ruelle theory, see
R. Jost, The General Theory of Quantized Fields, Am. Math. Soc. 1965.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_16
105