Appendix A
Long Range Dynamics and Vacuum Seizing
As stressed in Chap. 25, the delocalization induced by the time evolution crucially
enters into the explanation of important physical phenomena; in particular, its effect
on the field commutators has been pointed out in connection with the evasion of
the Goldstone theorem. In this Appendix, we shall directly investigate the relation
between the range of the interaction and the delocalization of the observables (or
more generally of the relevant dynamical variables). The emerging phenomena are
the role of the boundary conditions for the time evolution, leading to volume effects,
the occurrence of variables at infinity and the so-called seizing of the vacuum; they
represent a radical departure from the standard local case.
a) Existence of infinite volume dynamics
As emphasized in Part II, the local structure of the observables is at the basis of
the standard algebraic approach to the theory of infinitely extended systems.
1 One
of the main consequences is that different pure phases are not distinguished by
different dynamics, as the mean field approximation might lead to think, but rather
(and exclusively) by the properties of the ground/equilibrium states which define
them though the corresponding correlation functions. Such a property underlies the
algebraic approach to thermodynamics and statistical mechanics and is one of the
pillars of the modern treatment of phase transitions.
As discussed in Chaps. 17, 19, 22, 25, for infinitely extended systems the time
evolution α
t of observables is defined as the infinite volume limit of the finite volume
dynamics α
t
V . Hence, a crucial physical condition is that such an infinite volume
limit exists and defines a one-parameter group of automorphisms of the algebra of
observables.
We recall that, if the time evolution α
t
V is characterized by a finite propagation
speed, briefly is strictly local, (see Chap. 17, Sect. 17.3 and the case of relativistic local
1 R. Haag, Local Quantum Physics, Springer 1996.
© 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
225
Long Range Dynamics and Vacuum Seizing
As stressed in Chap. 25, the delocalization induced by the time evolution crucially
enters into the explanation of important physical phenomena; in particular, its effect
on the field commutators has been pointed out in connection with the evasion of
the Goldstone theorem. In this Appendix, we shall directly investigate the relation
between the range of the interaction and the delocalization of the observables (or
more generally of the relevant dynamical variables). The emerging phenomena are
the role of the boundary conditions for the time evolution, leading to volume effects,
the occurrence of variables at infinity and the so-called seizing of the vacuum; they
represent a radical departure from the standard local case.
a) Existence of infinite volume dynamics
As emphasized in Part II, the local structure of the observables is at the basis of
the standard algebraic approach to the theory of infinitely extended systems.
1 One
of the main consequences is that different pure phases are not distinguished by
different dynamics, as the mean field approximation might lead to think, but rather
(and exclusively) by the properties of the ground/equilibrium states which define
them though the corresponding correlation functions. Such a property underlies the
algebraic approach to thermodynamics and statistical mechanics and is one of the
pillars of the modern treatment of phase transitions.
As discussed in Chaps. 17, 19, 22, 25, for infinitely extended systems the time
evolution α
t of observables is defined as the infinite volume limit of the finite volume
dynamics α
t
V . Hence, a crucial physical condition is that such an infinite volume
limit exists and defines a one-parameter group of automorphisms of the algebra of
observables.
We recall that, if the time evolution α
t
V is characterized by a finite propagation
speed, briefly is strictly local, (see Chap. 17, Sect. 17.3 and the case of relativistic local
1 R. Haag, Local Quantum Physics, Springer 1996.
© 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
225