Chapter 4
General Properties of Solutions
of Classical Field Equations
The first basic question is to identify the possible configurations of the systems (3.1),
namely the set X of initial data for which the time evolution is well defined and which
is mapped onto itself by time evolution. In the mathematical language, one has to
find the functional space X for which the Cauchy problem is well posed. In order to
see this, one has to give conditions on U
(ϕ) and to specify the class of initial data or,
equivalently, the class of solutions one is interested in. Here one faces an apparently
technical mathematical problem, which has also deep physical connections.
In the pioneering work by Jörgens
11 and Segal,
12 the choice was made of considering those initial data (and, consequently, those solutions) for which the total
“kinetic” energy is finite
13
E kin ≡
1
2
[(∇ϕ)
2
+ ϕ
2
+ ψ
2
]d
s x < ∞, ψ = ˙
ϕ.
(4.1)
From a physical point of view, condition (4.1) is unjustified and it automatically
rules out very interesting cases, like the external field problem, the symmetry breaking solutions, the soliton-like solutions and, in general, all the solutions which do
not decrease sufficiently fast at large distances to make the above integral (4.1) convergent. Actually, there is no physical reason why E kin should be finite, since even
the splitting of energy into a kinetic and a potential part is not free of ambiguities.
11 K. Jörgens, Mat. Zeit. 77, 291 (1961).
12 I. Segal, Ann. Math. 78, 339 (1963).
13 Strictly speaking, the kinetic energy should not involve the term ϕ 2 . Our abuse of language is
based on the fact that the bilinear part of the total energy corresponds to what is usually called the
“non-interacting” theory (whose treatment is generally considered as trivial or under control by an
analysis in terms of normal modes). The remaining term in the total energy is usually considered
as the true interaction potential.
© 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_4
17
General Properties of Solutions
of Classical Field Equations
The first basic question is to identify the possible configurations of the systems (3.1),
namely the set X of initial data for which the time evolution is well defined and which
is mapped onto itself by time evolution. In the mathematical language, one has to
find the functional space X for which the Cauchy problem is well posed. In order to
see this, one has to give conditions on U
(ϕ) and to specify the class of initial data or,
equivalently, the class of solutions one is interested in. Here one faces an apparently
technical mathematical problem, which has also deep physical connections.
In the pioneering work by Jörgens
11 and Segal,
12 the choice was made of considering those initial data (and, consequently, those solutions) for which the total
“kinetic” energy is finite
13
E kin ≡
1
2
[(∇ϕ)
2
+ ϕ
2
+ ψ
2
]d
s x < ∞, ψ = ˙
ϕ.
(4.1)
From a physical point of view, condition (4.1) is unjustified and it automatically
rules out very interesting cases, like the external field problem, the symmetry breaking solutions, the soliton-like solutions and, in general, all the solutions which do
not decrease sufficiently fast at large distances to make the above integral (4.1) convergent. Actually, there is no physical reason why E kin should be finite, since even
the splitting of energy into a kinetic and a potential part is not free of ambiguities.
11 K. Jörgens, Mat. Zeit. 77, 291 (1961).
12 I. Segal, Ann. Math. 78, 339 (1963).
13 Strictly speaking, the kinetic energy should not involve the term ϕ 2 . Our abuse of language is
based on the fact that the bilinear part of the total energy corresponds to what is usually called the
“non-interacting” theory (whose treatment is generally considered as trivial or under control by an
analysis in terms of normal modes). The remaining term in the total energy is usually considered
as the true interaction potential.
© 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_4
17