22
5 Stable Structures, Hilbert Sectors, Phases
I Local structure. For a reasonable interpretation of S as a “phase” or a physical
realization of the system, any two elements of S should be related by “physically
realizable operations”, in the sense, to be made more precise below, that they
should describe configurations of the system both realizable or accessible in the
same “laboratory” or phase. This implies that the two solutions can only differ
locally, but not by their behaviour at infinity, since by physically realizable
operations one cannot change the boundary conditions of the “universe” or of
the (infinite volume) thermodynamical phase in which one is living.
II Stability of S under time evolution.
III Stability of S under space translations. One may also require that the infinite
volume integral of the (renormalized) energy–momentum density is finite for
each element of S. Of particular physical interest are those sets S which satisfy
the following further condition.
IV Energy bounded from below in S.
To be more precise we have to give a mathematical formalization of the above
requirements.
I. Local structure. In order to convert condition I into a mathematical statement,
one must formalize the intuitive idea of physically realizable operations. Since our
measuring apparatuses and our possible operations on a physical system extend over
bounded regions of space, starting from a given field configuration u, by physically
realizable operations we can modify it only locally, i.e. we can reach only those configurations which essentially differ from u only locally (quasi-local modifications).
From a mathematical point of view, it is natural to identify the concept of quasi-local
modification as a H
1
(R
s
) ⊕ L
2
(R
s
) perturbation, i.e. given a solution u 1 (t), a solution u 2 (t) is a quasi- local modification of u 1 if u 1 (t) − u 2 (t) ∈ H
1
(R
s
) ⊕ L
2
(R
s
)
continuously in t, briefly
u 1 (t) − u 2 (t) ∈ C
0
(H
1
(R
s
) ⊕ L
2
(R
s
), R).
(5.1)
We are thus led to introduce the following
Definition 5.1 Let F denote the family of solutions u(t) ∈ C
0
(X loc , R) of (4.6), a
subset S ⊂ F has a local structure if (5.1) holds ∀u 1 , u 2 ∈ S.
As it appears also in other fields, the concept of “locality” plays an important rôle
for the infinite-dimensional generalization of ideas developed for finite-dimensional
systems. The emphasis on local structures is actually the key which makes possible
(and physically meaningful) the treatment of the dynamics of infinite degrees of
freedom. A crucial property is the stability of the local structure under time evolution.
II. Stability under time evolution. Since time evolution is one of the possible realizable “operations”, the above definition of local structure is physically meaningful provided it is stable under time evolution, namely if ∀u(t) ∈ S also u τ (t) ≡ u(t +τ ) ∈ S,
∀τ ∈ R. A set S with local structure satisfying such stability under time evolution
will be called a sector. Thus, all the elements u of the sector S identified by a reference element ¯
u have the property that δ(t) ≡ u(t) − ¯
u(0) ∈ H
1
(R
s
) ⊕ L
2
(R
s
),
∀t ∈ R.
5 Stable Structures, Hilbert Sectors, Phases
I Local structure. For a reasonable interpretation of S as a “phase” or a physical
realization of the system, any two elements of S should be related by “physically
realizable operations”, in the sense, to be made more precise below, that they
should describe configurations of the system both realizable or accessible in the
same “laboratory” or phase. This implies that the two solutions can only differ
locally, but not by their behaviour at infinity, since by physically realizable
operations one cannot change the boundary conditions of the “universe” or of
the (infinite volume) thermodynamical phase in which one is living.
II Stability of S under time evolution.
III Stability of S under space translations. One may also require that the infinite
volume integral of the (renormalized) energy–momentum density is finite for
each element of S. Of particular physical interest are those sets S which satisfy
the following further condition.
IV Energy bounded from below in S.
To be more precise we have to give a mathematical formalization of the above
requirements.
I. Local structure. In order to convert condition I into a mathematical statement,
one must formalize the intuitive idea of physically realizable operations. Since our
measuring apparatuses and our possible operations on a physical system extend over
bounded regions of space, starting from a given field configuration u, by physically
realizable operations we can modify it only locally, i.e. we can reach only those configurations which essentially differ from u only locally (quasi-local modifications).
From a mathematical point of view, it is natural to identify the concept of quasi-local
modification as a H
1
(R
s
) ⊕ L
2
(R
s
) perturbation, i.e. given a solution u 1 (t), a solution u 2 (t) is a quasi- local modification of u 1 if u 1 (t) − u 2 (t) ∈ H
1
(R
s
) ⊕ L
2
(R
s
)
continuously in t, briefly
u 1 (t) − u 2 (t) ∈ C
0
(H
1
(R
s
) ⊕ L
2
(R
s
), R).
(5.1)
We are thus led to introduce the following
Definition 5.1 Let F denote the family of solutions u(t) ∈ C
0
(X loc , R) of (4.6), a
subset S ⊂ F has a local structure if (5.1) holds ∀u 1 , u 2 ∈ S.
As it appears also in other fields, the concept of “locality” plays an important rôle
for the infinite-dimensional generalization of ideas developed for finite-dimensional
systems. The emphasis on local structures is actually the key which makes possible
(and physically meaningful) the treatment of the dynamics of infinite degrees of
freedom. A crucial property is the stability of the local structure under time evolution.
II. Stability under time evolution. Since time evolution is one of the possible realizable “operations”, the above definition of local structure is physically meaningful provided it is stable under time evolution, namely if ∀u(t) ∈ S also u τ (t) ≡ u(t +τ ) ∈ S,
∀τ ∈ R. A set S with local structure satisfying such stability under time evolution
will be called a sector. Thus, all the elements u of the sector S identified by a reference element ¯
u have the property that δ(t) ≡ u(t) − ¯
u(0) ∈ H
1
(R
s
) ⊕ L
2
(R
s
),
∀t ∈ R.