32
6 Stability Under Space Translations. Positive Energy
δϕ 0 =
1
0
dσa i ∇ i ϕ 0 (x + σa)
defines a functional which satisfies
|δϕ 0 (χ)| ≤ |a| ||∇ϕ 0 || L 2 ||χ|| L 2
as a consequence of (6.1). Thus, it has a continuous extension to L
2
(R
s
) and by the
Riesz representation theorem it defines an element of L
2
(R
s
).
For the proof of the second statement, since ψ may be an arbitrary element of L
2
(R
s
),
P is integrable provided ∇ϕ ∈ L
2
(R
s
), i.e. ∇ϕ 0 ∈ L
2
(R
s
), since χ = ϕ − ϕ 0 ∈
H
1
(R
s
).
The curly brackets in (6.2) defines a linear operator ˜
P which is well defined in H ϕ
and generates the space translations iff (6.1) holds.
Clearly, the possibility of using solutions of non-linear field equations for the
description of physical systems requires that such solutions have finite energy–
momentum, and the localization properties of the physical measurements require
the existence of an energy–momentum density.
The conventional expression of the energy density for the theory described by
(4.6) is
E(ϕ, ψ) =
1
2
[(∇ϕ)
2
+ ψ
2
] + U (ϕ).
(6.3)
However, if one adds any function of x, the (Hamilton) equations of motion will
remain unchanged and the new expression of the total energy is still formally conserved.
This ambiguity is related to the fact that only energy differences have a physical meaning, so that the concept of finite energy solutions must necessarily make
reference to some chosen reference solution. Such a fixing of the energy scale will
generally depend on the sector, since E(ϕ, ψ) is locally but in general not globally
integrable. The fixing of the energy scale corresponds to the so-called infinite volume
renormalization which occurs in the treatment of infinitely extended systems.
Thus, given an Hilbert space sector H ϕ 0 , one is led to define a renormalized energy
density (without loss of generality we can take ψ 0 = 0)
E ren (ϕ, ψ) ≡ E(ϕ, ψ) − E(ϕ 0 , 0)
=
1
2
[(∇χ)
2
+ ψ
2
] + ∇χ∇ϕ 0 + G(χ) + U
(ϕ 0 )χ,
(6.4)
where χ = ϕ − ϕ 0 and G(χ) is defined by (5.9).
The background subtraction is, however, not enough for assuring that the renormalized density is globally integrable. The most which can be said, without additional
assumptions, is that E ren is integrable if χ is of compact support and that it identifies
6 Stability Under Space Translations. Positive Energy
δϕ 0 =
1
0
dσa i ∇ i ϕ 0 (x + σa)
defines a functional which satisfies
|δϕ 0 (χ)| ≤ |a| ||∇ϕ 0 || L 2 ||χ|| L 2
as a consequence of (6.1). Thus, it has a continuous extension to L
2
(R
s
) and by the
Riesz representation theorem it defines an element of L
2
(R
s
).
For the proof of the second statement, since ψ may be an arbitrary element of L
2
(R
s
),
P is integrable provided ∇ϕ ∈ L
2
(R
s
), i.e. ∇ϕ 0 ∈ L
2
(R
s
), since χ = ϕ − ϕ 0 ∈
H
1
(R
s
).
The curly brackets in (6.2) defines a linear operator ˜
P which is well defined in H ϕ
and generates the space translations iff (6.1) holds.
Clearly, the possibility of using solutions of non-linear field equations for the
description of physical systems requires that such solutions have finite energy–
momentum, and the localization properties of the physical measurements require
the existence of an energy–momentum density.
The conventional expression of the energy density for the theory described by
(4.6) is
E(ϕ, ψ) =
1
2
[(∇ϕ)
2
+ ψ
2
] + U (ϕ).
(6.3)
However, if one adds any function of x, the (Hamilton) equations of motion will
remain unchanged and the new expression of the total energy is still formally conserved.
This ambiguity is related to the fact that only energy differences have a physical meaning, so that the concept of finite energy solutions must necessarily make
reference to some chosen reference solution. Such a fixing of the energy scale will
generally depend on the sector, since E(ϕ, ψ) is locally but in general not globally
integrable. The fixing of the energy scale corresponds to the so-called infinite volume
renormalization which occurs in the treatment of infinitely extended systems.
Thus, given an Hilbert space sector H ϕ 0 , one is led to define a renormalized energy
density (without loss of generality we can take ψ 0 = 0)
E ren (ϕ, ψ) ≡ E(ϕ, ψ) − E(ϕ 0 , 0)
=
1
2
[(∇χ)
2
+ ψ
2
] + ∇χ∇ϕ 0 + G(χ) + U
(ϕ 0 )χ,
(6.4)
where χ = ϕ − ϕ 0 and G(χ) is defined by (5.9).
The background subtraction is, however, not enough for assuring that the renormalized density is globally integrable. The most which can be said, without additional
assumptions, is that E ren is integrable if χ is of compact support and that it identifies