9 The Goldstone Theorem
51
where δ denotes the Dirac delta function, one has
H r (k) = μ(I k )
dλh r (k(1 − λ))h r (λk),
where μ is the Lebesgue measure and
I k ≡ {q; q ∈ supp h r ∩ {kq = 0, k
2
= 0, q
2
= 0}}
For s ≥ 2 this appears to exclude that h ∈ H
1
(R
s
).
The above argument indicates that the solutions with the properties of ii) can be
constructed, e.g. as
ϕ
α
G (x, t) = e
h k (x,t) f R+2T (x)T
α ϕ,
with f R (x) = 1 for |x| ≤ R and = 0 for |x| ≥ R(1 + ε).
The above discussion also shows that in one space dimension s = 1 one may
find solutions of (9.5) belonging to H
1 and therefore one proves the existence
of genuine Goldstone modes all over the space. In fact, any function h(x − t) or
h(x + t), h ∈ H
1
(R), is a solution of (9.5).