34
6 Stability Under Space Translations. Positive Energy
energy be bounded from below. Now, even if the potential is bounded from below,
in general the renormalized energy may not be so.
Proposition 6.3 The renormalized energy is bounded from below in the HSSs defined
by absolute minima of the potential (vacuum sectors)
Proof. In fact, in (6.4) ∇ϕ 0 = 0 and, since ϕ 0 is an absolute minimum
G(χ) + U
(ϕ 0 )χ = U (ϕ 0 + χ) − U (ϕ 0 ) ≥ 0.
The energy is not bounded from below in the sectors defined by relative minima of
the potential (false vacuum sectors) and one expects instability against external field
perturbations.
In conclusion, the set of solutions of the non-linear field equation (4.6) which have
a reasonable physical interpretation are those belonging to Hilbert space sectors with
energy–momentum density (physical sectors), and a distinguished role is played by
the vacuum sectors. (For time-independent solutions defining physical sectors, see
Appendix 10.5). The analogy with the corresponding structures in quantum field
theory
29 is rather remarkable.
29 See references in Chap. 5, footnotes 16 and 17.
6 Stability Under Space Translations. Positive Energy
energy be bounded from below. Now, even if the potential is bounded from below,
in general the renormalized energy may not be so.
Proposition 6.3 The renormalized energy is bounded from below in the HSSs defined
by absolute minima of the potential (vacuum sectors)
Proof. In fact, in (6.4) ∇ϕ 0 = 0 and, since ϕ 0 is an absolute minimum
G(χ) + U
(ϕ 0 )χ = U (ϕ 0 + χ) − U (ϕ 0 ) ≥ 0.
The energy is not bounded from below in the sectors defined by relative minima of
the potential (false vacuum sectors) and one expects instability against external field
perturbations.
In conclusion, the set of solutions of the non-linear field equation (4.6) which have
a reasonable physical interpretation are those belonging to Hilbert space sectors with
energy–momentum density (physical sectors), and a distinguished role is played by
the vacuum sectors. (For time-independent solutions defining physical sectors, see
Appendix 10.5). The analogy with the corresponding structures in quantum field
theory
29 is rather remarkable.
29 See references in Chap. 5, footnotes 16 and 17.