9 The Goldstone Theorem
49
u ± (t) = u(t) +
±∞
t
dsU 0 (t − s) f (u(s)),
(9.3)
which express u ± (t) in terms of the solution u(t) and of the propagator W. The Yang–
Feldman equations can be interpreted as a form of the integral (4.6) with initial data
given at t = ±∞, respectively.
The problem of the existence of the asymptotic limits reduces to estimating the
asymptotic time decay of the non-linear term f (u(s)) such that the integrals on the
r.h.s. of the Yang–Feldman equations exist. This can be done by using the Basic L
∞
estimates on the time decay of the free solutions (see Strauss’ book quoted above,
pp. 5–6).
For small amplitude solutions, i.e. for initial data small in some norm, e.g. of the
form εu for fixed u, the asymptotic limits are completely governed by the behaviour
of f (u) near u = 0.
We can now state a classical counterpart of the Goldstone’s theorem.
Theorem 9.1 Let G be an N -parameter continuous (Lie) group of internal symmetries of the non-linear equation (3.1) and H ϕ the Hilbert Space Sector, defined by an
absolute minimum ϕ of the potential U , where G is spontaneously broken down to
G ϕ , the stability group of ϕ.
Then, for any generator T
α , such that T
α
ϕ = 0,
i) there are scattering configurations, associated to solutions belonging to the
sector H ϕ , which are solutions of the free wave equation (Goldstone modes).
ii) for any sphere Ω R of radius R and any time T there are solutions ϕ
α
G (x, t) = ϕ,
ϕ
α
G ∈ H ϕ , whose propagation in Ω R in the time interval t ∈ [0, T ] is that of free
waves (Goldstone-like solutions).
Proof. i) For solutions ϕ ∈ H ϕ , i.e. of the form ϕ = ϕ + χ, χ ∈ H
1 , the conservation of the current j μ = (∂ μ ϕ)T
α
ϕ, associated with the generator T
α , (without
loss of generality we can take ϕ real and T
α antisymmetric), reads
0 = ∂ μ j
μ
= χ i T
α
i j ϕ j = (χ i T
α
i j ϕ) + χ i T
α
i j χ j ,
(9.4)
and by the invariance of the potential, the second term can be written as
U
(ϕ +χ) i T
α
i j ϕ j . (In the quantum case, thanks to the vacuum expectation value,
one has only the analogue of the first term and the proof gets simpler).
Now, for small amplitude solutions χ , the asymptotic limits are governed by the
behaviour of U
(χ ) ≡ U
(ϕ + χ) near χ = 0 and in this region, by the invariance
of the potential, one has
U
i (χ )(T
α
ϕ) i = U
i jk (ϕ)χ j χ k (T
α
ϕ) i + O(χ
3
).
This implies that the small amplitude mode χ
α
≡ χ i (T
α
ϕ) i satisfies a nonlinear wave equation with an effective potential which vanishes to a degree
p ≥ 3 near χ = 0. Thus, the large time decay of the non-linear term appearing
49
u ± (t) = u(t) +
±∞
t
dsU 0 (t − s) f (u(s)),
(9.3)
which express u ± (t) in terms of the solution u(t) and of the propagator W. The Yang–
Feldman equations can be interpreted as a form of the integral (4.6) with initial data
given at t = ±∞, respectively.
The problem of the existence of the asymptotic limits reduces to estimating the
asymptotic time decay of the non-linear term f (u(s)) such that the integrals on the
r.h.s. of the Yang–Feldman equations exist. This can be done by using the Basic L
∞
estimates on the time decay of the free solutions (see Strauss’ book quoted above,
pp. 5–6).
For small amplitude solutions, i.e. for initial data small in some norm, e.g. of the
form εu for fixed u, the asymptotic limits are completely governed by the behaviour
of f (u) near u = 0.
We can now state a classical counterpart of the Goldstone’s theorem.
Theorem 9.1 Let G be an N -parameter continuous (Lie) group of internal symmetries of the non-linear equation (3.1) and H ϕ the Hilbert Space Sector, defined by an
absolute minimum ϕ of the potential U , where G is spontaneously broken down to
G ϕ , the stability group of ϕ.
Then, for any generator T
α , such that T
α
ϕ = 0,
i) there are scattering configurations, associated to solutions belonging to the
sector H ϕ , which are solutions of the free wave equation (Goldstone modes).
ii) for any sphere Ω R of radius R and any time T there are solutions ϕ
α
G (x, t) = ϕ,
ϕ
α
G ∈ H ϕ , whose propagation in Ω R in the time interval t ∈ [0, T ] is that of free
waves (Goldstone-like solutions).
Proof. i) For solutions ϕ ∈ H ϕ , i.e. of the form ϕ = ϕ + χ, χ ∈ H
1 , the conservation of the current j μ = (∂ μ ϕ)T
α
ϕ, associated with the generator T
α , (without
loss of generality we can take ϕ real and T
α antisymmetric), reads
0 = ∂ μ j
μ
= χ i T
α
i j ϕ j = (χ i T
α
i j ϕ) + χ i T
α
i j χ j ,
(9.4)
and by the invariance of the potential, the second term can be written as
U
(ϕ +χ) i T
α
i j ϕ j . (In the quantum case, thanks to the vacuum expectation value,
one has only the analogue of the first term and the proof gets simpler).
Now, for small amplitude solutions χ , the asymptotic limits are governed by the
behaviour of U
(χ ) ≡ U
(ϕ + χ) near χ = 0 and in this region, by the invariance
of the potential, one has
U
i (χ )(T
α
ϕ) i = U
i jk (ϕ)χ j χ k (T
α
ϕ) i + O(χ
3
).
This implies that the small amplitude mode χ
α
≡ χ i (T
α
ϕ) i satisfies a nonlinear wave equation with an effective potential which vanishes to a degree
p ≥ 3 near χ = 0. Thus, the large time decay of the non-linear term appearing