Appendix A: Long Range Dynamics and Vacuum Seizing
227
cutoff in the equations of motions (see, Eq. (25.23)), is that, for ∀ A, B, C ∈ A, with
A x the x-translated of A, the decay rate of the correlation functions be such that
|x|
−d
ω(C A x B) is absolutely integrable in x. We already applied this strategy in
Chap. 25, in the case of a two-body Coulomb potential.
In general, the rate of convergence of α
t
V (A), ∀A ∈ A L , (equivalently the infrared
regularity of the state such that the removal of the infrared cutoff is possible and
defines a one-parameter group α
t
, t ∈ R), requires some care from the mathematical
point of view, which shall not be dealt with here.
5 In the following, we shall take
for granted that the family Σ of states satisfies the required infrared regularity to
ensure that
α
t
= τ − weak lim
V →∞
α
t
V , t ∈ R,
(A.1)
defines a one-parameter group of automorphisms of the algebra (of observables or
of dynamical variables) A ≡ A L
τ .
b) Delocalization by variables at infinity; vacuum seizing
The algebra A, stable under α
t , may contain highly delocalized variables, like the
observables at infinity, which are localized outside any bounded region, e.g. the
ergodic averages, see Eq. (16.6), or the infinite volume limit of averages around the
boundary (C R a suitable normalization constant)
A ∞ = lim
R→∞
C R
R≤|x|≤R(1+ε)
d
3 x A x .
(A.2)
By asymptotic abelianess such variables belong to the centre of A (see Chap. 16), so
that in different pure phases the time evolution of A may not be the same (vacuum
seizing).
The mechanism is simply accounted for by the following considerations. In the
case of a two-body potential U , the time evolution of an element A localized in a
neighbourhood of the origin involves a convolution with U (see Eq. (25.23)), i.e.
a coupling between A and the (average) variables on the “boundary” ∂V , say the
surface S R of a sphere or radius R; then, if the coupling between A and such boundary
variables does not decrease faster than R
−2 , in general, in the limit V → ∞, one gets
an effective coupling with variables at infinity of order one. Then, the time evolution
of A involves variables at infinity, namely central variables.
5 We briefly mention that the family Σ is assumed to have the following properties: 1) Σ is closed
under linear combination; 2) Σ is norm closed and separating, i.e. ω(A) = 0, ∀ ω ∈ Σ implies
A = 0; 3) Σ is stable “under local operations”, i.e. if ω ∈ Σ, for any local A, B, also ω AB defined
by ω AB (C) ≡ ω(A C B), ∀C ∈ A L , belongs to Σ.
A condition of infrared regularity of the states of Σ is that α t
V (A), A ∈ A L converges, in the
ultra-strong topology defined by Σ, to a one-parameter group of automorphisms α t of the algebra
A ≡ A L
τ , the closure of A L with respect to the weak topology τ induced by Σ. For a detailed
discussion of such a mathematical structure, see G. Morchio and F. Strocchi [1987].
227
cutoff in the equations of motions (see, Eq. (25.23)), is that, for ∀ A, B, C ∈ A, with
A x the x-translated of A, the decay rate of the correlation functions be such that
|x|
−d
ω(C A x B) is absolutely integrable in x. We already applied this strategy in
Chap. 25, in the case of a two-body Coulomb potential.
In general, the rate of convergence of α
t
V (A), ∀A ∈ A L , (equivalently the infrared
regularity of the state such that the removal of the infrared cutoff is possible and
defines a one-parameter group α
t
, t ∈ R), requires some care from the mathematical
point of view, which shall not be dealt with here.
5 In the following, we shall take
for granted that the family Σ of states satisfies the required infrared regularity to
ensure that
α
t
= τ − weak lim
V →∞
α
t
V , t ∈ R,
(A.1)
defines a one-parameter group of automorphisms of the algebra (of observables or
of dynamical variables) A ≡ A L
τ .
b) Delocalization by variables at infinity; vacuum seizing
The algebra A, stable under α
t , may contain highly delocalized variables, like the
observables at infinity, which are localized outside any bounded region, e.g. the
ergodic averages, see Eq. (16.6), or the infinite volume limit of averages around the
boundary (C R a suitable normalization constant)
A ∞ = lim
R→∞
C R
R≤|x|≤R(1+ε)
d
3 x A x .
(A.2)
By asymptotic abelianess such variables belong to the centre of A (see Chap. 16), so
that in different pure phases the time evolution of A may not be the same (vacuum
seizing).
The mechanism is simply accounted for by the following considerations. In the
case of a two-body potential U , the time evolution of an element A localized in a
neighbourhood of the origin involves a convolution with U (see Eq. (25.23)), i.e.
a coupling between A and the (average) variables on the “boundary” ∂V , say the
surface S R of a sphere or radius R; then, if the coupling between A and such boundary
variables does not decrease faster than R
−2 , in general, in the limit V → ∞, one gets
an effective coupling with variables at infinity of order one. Then, the time evolution
of A involves variables at infinity, namely central variables.
5 We briefly mention that the family Σ is assumed to have the following properties: 1) Σ is closed
under linear combination; 2) Σ is norm closed and separating, i.e. ω(A) = 0, ∀ ω ∈ Σ implies
A = 0; 3) Σ is stable “under local operations”, i.e. if ω ∈ Σ, for any local A, B, also ω AB defined
by ω AB (C) ≡ ω(A C B), ∀C ∈ A L , belongs to Σ.
A condition of infrared regularity of the states of Σ is that α t
V (A), A ∈ A L converges, in the
ultra-strong topology defined by Σ, to a one-parameter group of automorphisms α t of the algebra
A ≡ A L
τ , the closure of A L with respect to the weak topology τ induced by Σ. For a detailed
discussion of such a mathematical structure, see G. Morchio and F. Strocchi [1987].