28
5 Stable Structures, Hilbert Sectors, Phases
1
2
|k|
−1
| ˜
h(k)| ≤ (|k|
−2 sin |k| − |k|
−1
)| ˜
h|,
for |k| ≥ 2, imply | ˜
h|(1 + |k|
2
)
−1/2
∈ L
2 , by (5.19), (5.20).
Remarks The conclusions of the above theorem hold in the more general case in
which the condition ϕ 0 ∈ L
∞
(R
s
) is replaced by that of ϕ 0 being such that i) and ii),
i.e. (5.10) and (5.11), hold; in this case ϕ 0 is said to be a regular point (or admissible)
with respect to U,(see Ref. II).
The conditions (5.2), (5.3) characterize those initial data for which the time derivative preserves some sort of localization; in particular (5.3) says that the time derivative
of the second component is H
−1 localized.
The HSSs defined by the absolute minima of the potential are the analogs of the
vacuum sectors of quantum field theory; those defined by relative minima of U are
the analogs of the false vacuum sectors
22 and are classically stable (no tunnelling).
The solutions of (4.6) which correspond to initial data u 0 satisfying (5.2), (5.3) will
be briefly called generalized stationary solutions. In general, a sector H u 0 identified
by a generalized stationary solution does not contain static solutions; a necessary
and sufficient condition is that the elliptic equation
Δχ − G
ϕ 0
(x, χ) = −h(ϕ 0 )
with h(ϕ 0 ) ≡ Δϕ 0 − U
(ϕ 0 ) ∈ H
−1
(R
s
), has solutions χ ∈ H
1
(R
s
).
The occurrence of disjoint Hilbert structures, stable under time evolution, associated with generalized stationary solutions is a rather remarkable feature in a fully
non-linear problem without any approximation or linearization being involved. In a
certain sense the generalized stationary solutions play a hierarchical role and exhibit
some sort of stability property since they keep their H
1
⊕ L
2 perturbations steadily
trapped around them. A non-linear structure characterizes the labelling of the sectors
by the generalized stationary solutions, since the corresponding initial data do not
have a linear structure; however, within a given sector H ϕ 0 all the initial data are
described by the affine space generated by ϕ 0 through H
1
⊕ L
2 . In general, the time
evolution is not described by a linear operator on H ϕ 0 .
It is worthwhile to remark that the emergence of disjoint stable structures in the set
of solutions of the non-linear equation (4.6) has been made possible by the framework
adopted in Chap. 4, in which the Cauchy data were not restricted to be in H
1
⊕ L
2 . In
that case one would have only gotten the sector corresponding to the trivial vacuum
ϕ 0 = 0, ψ 0 = 0.
23
The occurrence of Hilbert space sectors in the set solutions of non-linear field
equations allows to establish strong connections with quantum mechanical structures
and to recover the analog of quantum mechanical phenomena like linear representa22 S. Coleman, Phys. Rev. D15, 2929 (1977).
23 See Chap. 4, footnotes 11, 12.
5 Stable Structures, Hilbert Sectors, Phases
1
2
|k|
−1
| ˜
h(k)| ≤ (|k|
−2 sin |k| − |k|
−1
)| ˜
h|,
for |k| ≥ 2, imply | ˜
h|(1 + |k|
2
)
−1/2
∈ L
2 , by (5.19), (5.20).
Remarks The conclusions of the above theorem hold in the more general case in
which the condition ϕ 0 ∈ L
∞
(R
s
) is replaced by that of ϕ 0 being such that i) and ii),
i.e. (5.10) and (5.11), hold; in this case ϕ 0 is said to be a regular point (or admissible)
with respect to U,(see Ref. II).
The conditions (5.2), (5.3) characterize those initial data for which the time derivative preserves some sort of localization; in particular (5.3) says that the time derivative
of the second component is H
−1 localized.
The HSSs defined by the absolute minima of the potential are the analogs of the
vacuum sectors of quantum field theory; those defined by relative minima of U are
the analogs of the false vacuum sectors
22 and are classically stable (no tunnelling).
The solutions of (4.6) which correspond to initial data u 0 satisfying (5.2), (5.3) will
be briefly called generalized stationary solutions. In general, a sector H u 0 identified
by a generalized stationary solution does not contain static solutions; a necessary
and sufficient condition is that the elliptic equation
Δχ − G
ϕ 0
(x, χ) = −h(ϕ 0 )
with h(ϕ 0 ) ≡ Δϕ 0 − U
(ϕ 0 ) ∈ H
−1
(R
s
), has solutions χ ∈ H
1
(R
s
).
The occurrence of disjoint Hilbert structures, stable under time evolution, associated with generalized stationary solutions is a rather remarkable feature in a fully
non-linear problem without any approximation or linearization being involved. In a
certain sense the generalized stationary solutions play a hierarchical role and exhibit
some sort of stability property since they keep their H
1
⊕ L
2 perturbations steadily
trapped around them. A non-linear structure characterizes the labelling of the sectors
by the generalized stationary solutions, since the corresponding initial data do not
have a linear structure; however, within a given sector H ϕ 0 all the initial data are
described by the affine space generated by ϕ 0 through H
1
⊕ L
2 . In general, the time
evolution is not described by a linear operator on H ϕ 0 .
It is worthwhile to remark that the emergence of disjoint stable structures in the set
of solutions of the non-linear equation (4.6) has been made possible by the framework
adopted in Chap. 4, in which the Cauchy data were not restricted to be in H
1
⊕ L
2 . In
that case one would have only gotten the sector corresponding to the trivial vacuum
ϕ 0 = 0, ψ 0 = 0.
23
The occurrence of Hilbert space sectors in the set solutions of non-linear field
equations allows to establish strong connections with quantum mechanical structures
and to recover the analog of quantum mechanical phenomena like linear representa22 S. Coleman, Phys. Rev. D15, 2929 (1977).
23 See Chap. 4, footnotes 11, 12.