228
Appendix A: Long Range Dynamics and Vacuum Seizing
The next step is to investigate whether some generalized localization survives
allowing the local generation of continuous symmetries, a crucial issue for the consequences of their breaking.
A subalgebra A l of A is essentially localized if it is generated by elements of the
form
d
s x f (x) A x , A ∈ A L with test functions f (x) decreasing at space infinity,
e.g. f (x) of fast decrease or square integrable, rather than of compact support. In
view of the discussion of symmetry breaking, more generally an algebra A l may be
considered as essentially localized if the continuous symmetries of the dynamics are
generated by local charges on it, in the sense of Eq. (25.1), with Q R satisfying the
charge integrability condition.
The dynamics is said to be essentially local if there exists an essentially localized
subalgebra A l of A, larger than A L , with the properties:
i) A l is weakly dense in A;
ii) A l is faithfully represented in each factorial representation of A by states of Σ;
iii) A l is stable under the time evolution α
t
= w − lim V α
t
V .
Since A l is weakly dense in A, in a factorial representation of A, the centre of
π(A l ) is trivial.
6
This case is very close to the quasi-local case, with the technical change which
takes into account the infinite propagation speed of the non-relativistic interactions;
see, e.g. the discussion of the free Bose gas in Sect. 17.2. Clearly, for strictly local
dynamics A l = A L .
The situation drastically changes if the time evolution is not essentially local.
However, some effective localization survives if the algebra A (stable under α
t )
contains an essentially localized algebra A l with properties i)–ii), whose instability
under the time evolution α
t is only due to the occurrence of variables at infinity. In this
case, the time evolution shall be briefly said to give rise to a central delocalization.
This means that A is generated by A l and by variables at infinity.
The interest of such a structure is its being realized in physically interesting models
like the Curie–Weiss model (discussed in Chap. 21, Sect. 5), the Anderson spin model
of superconductivity, the Coulomb Fermi gas in uniform background, (Appendices
B, C below), the bosonized Schwinger model, and the Stückelberg–Kibble model of
the Higgs phenomenon.
7
The resulting framework is not drastically different from the Haag–Kastler local
structure, since the infinite delocalization induced by the time evolution does not
affect the essentially local structure of the commutators; with respect to the essentially
local case, the time evolution involves observables with “support at infinity”.
Clearly, if the dynamical instability of an essentially localized subalgebra A l is
only due to the involvement of variables at infinity, a factorial representation π of A,
6 In fact, if z belongs to the centre Z l of π(A l ) it also belongs to the centre of π(A), since it commutes
with the dense subalgebra π(A l ), and in a factorial representation the centre is represented by
multiples of the identity.
7 G. Morchio and F. Strocchi, Comm. Math. Phys. 99, 153 (1985); ibid, 111, 593 (1987); Ann. Phys.
170, 310 (1986); J. Math. Phys. 28, 1912 (1987).
Appendix A: Long Range Dynamics and Vacuum Seizing
The next step is to investigate whether some generalized localization survives
allowing the local generation of continuous symmetries, a crucial issue for the consequences of their breaking.
A subalgebra A l of A is essentially localized if it is generated by elements of the
form
d
s x f (x) A x , A ∈ A L with test functions f (x) decreasing at space infinity,
e.g. f (x) of fast decrease or square integrable, rather than of compact support. In
view of the discussion of symmetry breaking, more generally an algebra A l may be
considered as essentially localized if the continuous symmetries of the dynamics are
generated by local charges on it, in the sense of Eq. (25.1), with Q R satisfying the
charge integrability condition.
The dynamics is said to be essentially local if there exists an essentially localized
subalgebra A l of A, larger than A L , with the properties:
i) A l is weakly dense in A;
ii) A l is faithfully represented in each factorial representation of A by states of Σ;
iii) A l is stable under the time evolution α
t
= w − lim V α
t
V .
Since A l is weakly dense in A, in a factorial representation of A, the centre of
π(A l ) is trivial.
6
This case is very close to the quasi-local case, with the technical change which
takes into account the infinite propagation speed of the non-relativistic interactions;
see, e.g. the discussion of the free Bose gas in Sect. 17.2. Clearly, for strictly local
dynamics A l = A L .
The situation drastically changes if the time evolution is not essentially local.
However, some effective localization survives if the algebra A (stable under α
t )
contains an essentially localized algebra A l with properties i)–ii), whose instability
under the time evolution α
t is only due to the occurrence of variables at infinity. In this
case, the time evolution shall be briefly said to give rise to a central delocalization.
This means that A is generated by A l and by variables at infinity.
The interest of such a structure is its being realized in physically interesting models
like the Curie–Weiss model (discussed in Chap. 21, Sect. 5), the Anderson spin model
of superconductivity, the Coulomb Fermi gas in uniform background, (Appendices
B, C below), the bosonized Schwinger model, and the Stückelberg–Kibble model of
the Higgs phenomenon.
7
The resulting framework is not drastically different from the Haag–Kastler local
structure, since the infinite delocalization induced by the time evolution does not
affect the essentially local structure of the commutators; with respect to the essentially
local case, the time evolution involves observables with “support at infinity”.
Clearly, if the dynamical instability of an essentially localized subalgebra A l is
only due to the involvement of variables at infinity, a factorial representation π of A,
6 In fact, if z belongs to the centre Z l of π(A l ) it also belongs to the centre of π(A), since it commutes
with the dense subalgebra π(A l ), and in a factorial representation the centre is represented by
multiples of the identity.
7 G. Morchio and F. Strocchi, Comm. Math. Phys. 99, 153 (1985); ibid, 111, 593 (1987); Ann. Phys.
170, 310 (1986); J. Math. Phys. 28, 1912 (1987).