Chapter 6
Stability Under Space Translations.
Positive Energy
In this chapter we discuss the requirements III and IV stated in Chap. 5.
In the sequel we shall denote by H ϕ 0 the Hilbert space sector (HSS) defined by a
ϕ 0 ∈ L
∞
(R
s
) satisfying (5.3), taking always for granted that ψ 0 ∈ L
2
(R
s
).
Stability under space translations means that if u(x, t) ∈ H ϕ 0 so does u a (x, t) ≡
u(x + a, t), ∀a ∈ R
s . Clearly, by the above characterization of Hilbert sectors, such
a condition is equivalent to the condition that H ϕ 0 a = H ϕ 0 , ∀a ∈ R
s
, ϕ 0 a (x) ≡
ϕ 0 (x + a).
Proposition 6.1 The Hilbert sector H ϕ 0 is stable under space translations if
∇ϕ 0 ∈ L
2
(R
s
).
(6.1)
Furthermore, the momentum density
P i (ϕ, ψ) = ψ∇ i ϕ, ψ ∈ L
2
(R
s
), ϕ − ϕ 0 ∈ H
1
(R
s
)
is integrable for any element of H ϕ 0 , equivalently its integral P defines a linear
operator ˜
P in H ϕ 0 , which acts as the generator of translations
˜
P i u = {u, P i } = ∇ i u,
(6.2)
({, } denotes the Poisson bracket), if and only if (6.1) holds.
Proof. The proof of the first statement, namely that δϕ 0 ≡ ϕ 0 (x + a) − ϕ(x) ∈
H
1
(R
s
), reduces to the proof that δϕ 0 ∈ L
2
(R
s
) if (6.1) holds. In fact, as a distribution
on C
∞
0 (R
s
),
© 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_6
31
Stability Under Space Translations.
Positive Energy
In this chapter we discuss the requirements III and IV stated in Chap. 5.
In the sequel we shall denote by H ϕ 0 the Hilbert space sector (HSS) defined by a
ϕ 0 ∈ L
∞
(R
s
) satisfying (5.3), taking always for granted that ψ 0 ∈ L
2
(R
s
).
Stability under space translations means that if u(x, t) ∈ H ϕ 0 so does u a (x, t) ≡
u(x + a, t), ∀a ∈ R
s . Clearly, by the above characterization of Hilbert sectors, such
a condition is equivalent to the condition that H ϕ 0 a = H ϕ 0 , ∀a ∈ R
s
, ϕ 0 a (x) ≡
ϕ 0 (x + a).
Proposition 6.1 The Hilbert sector H ϕ 0 is stable under space translations if
∇ϕ 0 ∈ L
2
(R
s
).
(6.1)
Furthermore, the momentum density
P i (ϕ, ψ) = ψ∇ i ϕ, ψ ∈ L
2
(R
s
), ϕ − ϕ 0 ∈ H
1
(R
s
)
is integrable for any element of H ϕ 0 , equivalently its integral P defines a linear
operator ˜
P in H ϕ 0 , which acts as the generator of translations
˜
P i u = {u, P i } = ∇ i u,
(6.2)
({, } denotes the Poisson bracket), if and only if (6.1) holds.
Proof. The proof of the first statement, namely that δϕ 0 ≡ ϕ 0 (x + a) − ϕ(x) ∈
H
1
(R
s
), reduces to the proof that δϕ 0 ∈ L
2
(R
s
) if (6.1) holds. In fact, as a distribution
on C
∞
0 (R
s
),
© 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_6
31