50
9 The Goldstone Theorem
in the corresponding Yang–Feldman equation is not worse than in the case of
a wave equation with potential vanishing with degree p ≥ 3 near the origin
(other massive modes occurring in U
have faster decay properties). Then, one
can appeal to standard results
43 to obtain the existence of the asymptotic limits
χ
α
± (t) satisfying the free wave equation.
ii) The existence of free waves ϕ(x, t) = ϕ + χ(x, t) within a given region Ω R
in the time interval [0, T ] is equivalent to U
(ϕ + χ(x, t)) = 0, ∀x ∈ Ω R ,
t ∈ [0, T ], so that if the absolute minima of the potential consist of a single
orbit ϕ + χ(x, t) = exp(h
α
(x, t)T
α
)ϕ, h
α
(x, t) real ∈ H
1 and for solutions
associated to a given generator T
α , with T
α
ϕ = 0, one has solutions of the form
ϕ
α
(x, t) = e
h(x,t)T
α ϕ.
Now, the wave equation ϕ(x, t) = 0 requires
h(x, t) = 0, (∂ μ h∂
μ h)(x, t) = 0,
(9.5)
(since T
α and (T
α
)
2 have different symmetry properties).
This implies that any C
2 function of h also satisfies (9.5) and in particular
χ
(α)
≡ ϕ
(α)
− ϕ also does.
Equations (9.5) have solutions of the form h k (x, t) = h(k 0 t − k · x), with h an
arbitrary C
2 function and k = (k 0 , k) a light-like four vector, but they are not
in H
1
(R
s
) for s ≥ 2. One can argue more generally that the above equations
do not have solutions h ∈ H
1
(R
s
) for s ≥ 2. In fact, the wave equation requires
that the support of the s + 1-dimensional Fourier transform ˆ
h(k), k ∈ R
s+1 is
contained in {k
2
= 0}, and the second equation becomes
k
2
d
s+1 q ˆ
h(q − k) ˆ
h(q) = 0,
since kq − q
2
= k
2
− (k − q)
2
, (k − q)
2 ˆ
h(k − q) = 0. Thus,
H (k) ≡
d
s+1 q ˆ
h(q − k) ˆ
h(q)
must have support in k
2
= 0. Now, the sum of two light-like four vectors k −
q, q may be a light-like vector k only if k and q are parallel or antiparallel,
corresponding to sign k 0 q 0 =+1 or =−1, respectively, i.e. only if q = λk, λ ∈ R.
Hence, if k ∈ supp H and q and q − k belong to the support of ˆ
h, q must lie
in the intersection of the light cone q
2
= 0 and the hyperplane kq = 0; thus,
writing
ˆ
h(q) = δ(q
2
)h r (q), H (k) = δ(k
2
)H r (k),
43 H. Pecher. Math. Zeit. 185, 261 (1984); 198, 277 (1988).
9 The Goldstone Theorem
in the corresponding Yang–Feldman equation is not worse than in the case of
a wave equation with potential vanishing with degree p ≥ 3 near the origin
(other massive modes occurring in U
have faster decay properties). Then, one
can appeal to standard results
43 to obtain the existence of the asymptotic limits
χ
α
± (t) satisfying the free wave equation.
ii) The existence of free waves ϕ(x, t) = ϕ + χ(x, t) within a given region Ω R
in the time interval [0, T ] is equivalent to U
(ϕ + χ(x, t)) = 0, ∀x ∈ Ω R ,
t ∈ [0, T ], so that if the absolute minima of the potential consist of a single
orbit ϕ + χ(x, t) = exp(h
α
(x, t)T
α
)ϕ, h
α
(x, t) real ∈ H
1 and for solutions
associated to a given generator T
α , with T
α
ϕ = 0, one has solutions of the form
ϕ
α
(x, t) = e
h(x,t)T
α ϕ.
Now, the wave equation ϕ(x, t) = 0 requires
h(x, t) = 0, (∂ μ h∂
μ h)(x, t) = 0,
(9.5)
(since T
α and (T
α
)
2 have different symmetry properties).
This implies that any C
2 function of h also satisfies (9.5) and in particular
χ
(α)
≡ ϕ
(α)
− ϕ also does.
Equations (9.5) have solutions of the form h k (x, t) = h(k 0 t − k · x), with h an
arbitrary C
2 function and k = (k 0 , k) a light-like four vector, but they are not
in H
1
(R
s
) for s ≥ 2. One can argue more generally that the above equations
do not have solutions h ∈ H
1
(R
s
) for s ≥ 2. In fact, the wave equation requires
that the support of the s + 1-dimensional Fourier transform ˆ
h(k), k ∈ R
s+1 is
contained in {k
2
= 0}, and the second equation becomes
k
2
d
s+1 q ˆ
h(q − k) ˆ
h(q) = 0,
since kq − q
2
= k
2
− (k − q)
2
, (k − q)
2 ˆ
h(k − q) = 0. Thus,
H (k) ≡
d
s+1 q ˆ
h(q − k) ˆ
h(q)
must have support in k
2
= 0. Now, the sum of two light-like four vectors k −
q, q may be a light-like vector k only if k and q are parallel or antiparallel,
corresponding to sign k 0 q 0 =+1 or =−1, respectively, i.e. only if q = λk, λ ∈ R.
Hence, if k ∈ supp H and q and q − k belong to the support of ˆ
h, q must lie
in the intersection of the light cone q
2
= 0 and the hyperplane kq = 0; thus,
writing
ˆ
h(q) = δ(q
2
)h r (q), H (k) = δ(k
2
)H r (k),
43 H. Pecher. Math. Zeit. 185, 261 (1984); 198, 277 (1988).