6 Stability Under Space Translations. Positive Energy
33
an energy functional defined on the whole HSS by a suitable extension.
26 In general, however, the so extended functional will not be the integral over a density and
therefore the concept of local energy is problematic. Such a difficulty does not arise
if the HSS is defined by a ϕ 0 ∈ L
∞
(R
s
) with ∇ϕ 0 ∈ L
2
(R
s
).
Proposition 6.2
27 Given a Hilbert space sector defined by a ϕ 0 ∈ L
∞
(R
s
), a (renormalized) energy density can be defined on it with a convergent infinite volume integral
if
∇ϕ 0 ∈ L
2
(R
s
).
(6.5)
Proof. By Lemma 5.3, G
(χ) is globally Lipschitz continuous and therefore G
(χ) ∈
L
2
(R
s
), ∀χ ∈ H
1
(R
s
). Now, from the identity
G(χ 1 ) − G(χ 2 ) =
1
0
dσ
d
dσ
G(χ 1 + σ(χ 2 − χ 1 )) =
=
1
0
dσ(χ 2 − χ 1 )G
(χ 1 + σ(χ 2 − χ 1 )),
one has
d
s x|G(χ 1 ) − G(χ 2 )| ≤ sup
0≤σ≤1
G
(χ 1 + σ(χ 2 − χ 1 )) L 2 χ 2 − χ 1 L 2
and, since G(0) = 0, G(χ) ∈ L
1
(R
s
).
On the other hand,
∇χ∇ϕ 0 + U
(ϕ 0 )χ = ∇(χ∇ϕ 0 ) − h(ϕ 0 )χ
and the second term on the r.h.s. is integrable since h ∈ H
−1
(R
s
), χ ∈ H
1
(R
s
). By
(6.5), χ∇ϕ 0 ∈ L
1
(R
s
) and therefore the infinite volume limit of the integral of the
first term vanishes. The other terms in (6.4) are clearly integrable.
No renormalization is needed for the momentum, since without loss of generality
we can take ψ 0 = 0 and the background momentum subtraction vanishes.
It is not difficult to show
28 that the infinite volume integrals of the renormalized
energy–momentum densities define conserved quantities and that the corresponding
functionals are continuous in the Hilbert space topology of the HSS, if (6.5) holds.
A HSS defined by a ϕ 0 ∈ L
∞
(R
s
) with ∇ϕ 0 ∈ L
2
(R
s
) will be called a Hilbert
space sector with energy–momentum density or briefly a physical sector.
A related question is the stability of a sector under external perturbations and an
important role is played by condition IV of Chap. 5, namely that the (renormalized)
26 Ref. II quoted in Chap.3, footnote 5.
27 See Ref. II.
28 See Ref. II.
33
an energy functional defined on the whole HSS by a suitable extension.
26 In general, however, the so extended functional will not be the integral over a density and
therefore the concept of local energy is problematic. Such a difficulty does not arise
if the HSS is defined by a ϕ 0 ∈ L
∞
(R
s
) with ∇ϕ 0 ∈ L
2
(R
s
).
Proposition 6.2
27 Given a Hilbert space sector defined by a ϕ 0 ∈ L
∞
(R
s
), a (renormalized) energy density can be defined on it with a convergent infinite volume integral
if
∇ϕ 0 ∈ L
2
(R
s
).
(6.5)
Proof. By Lemma 5.3, G
(χ) is globally Lipschitz continuous and therefore G
(χ) ∈
L
2
(R
s
), ∀χ ∈ H
1
(R
s
). Now, from the identity
G(χ 1 ) − G(χ 2 ) =
1
0
dσ
d
dσ
G(χ 1 + σ(χ 2 − χ 1 )) =
=
1
0
dσ(χ 2 − χ 1 )G
(χ 1 + σ(χ 2 − χ 1 )),
one has
d
s x|G(χ 1 ) − G(χ 2 )| ≤ sup
0≤σ≤1
G
(χ 1 + σ(χ 2 − χ 1 )) L 2 χ 2 − χ 1 L 2
and, since G(0) = 0, G(χ) ∈ L
1
(R
s
).
On the other hand,
∇χ∇ϕ 0 + U
(ϕ 0 )χ = ∇(χ∇ϕ 0 ) − h(ϕ 0 )χ
and the second term on the r.h.s. is integrable since h ∈ H
−1
(R
s
), χ ∈ H
1
(R
s
). By
(6.5), χ∇ϕ 0 ∈ L
1
(R
s
) and therefore the infinite volume limit of the integral of the
first term vanishes. The other terms in (6.4) are clearly integrable.
No renormalization is needed for the momentum, since without loss of generality
we can take ψ 0 = 0 and the background momentum subtraction vanishes.
It is not difficult to show
28 that the infinite volume integrals of the renormalized
energy–momentum densities define conserved quantities and that the corresponding
functionals are continuous in the Hilbert space topology of the HSS, if (6.5) holds.
A HSS defined by a ϕ 0 ∈ L
∞
(R
s
) with ∇ϕ 0 ∈ L
2
(R
s
) will be called a Hilbert
space sector with energy–momentum density or briefly a physical sector.
A related question is the stability of a sector under external perturbations and an
important role is played by condition IV of Chap. 5, namely that the (renormalized)
26 Ref. II quoted in Chap.3, footnote 5.
27 See Ref. II.
28 See Ref. II.