48
9 The Goldstone Theorem
Thus, in an expansion of the potential around ϕ, the quadratic term, which has the
meaning of a mass term, has a zero eigenvalue in the direction T
α
ϕ. This is taken as
evidence that there is a massless mode.
In our opinion, the argument is not conclusive since it involves an expansion and
one should in some way control the effect of higher order terms; moreover, it is not
clear that there are (physically meaningful) solutions in the direction of T
α
ϕ for all
times, so that for them the quadratic term disappears. In any case, the argument does
not show that there are massless solutions as in the quantum case.
Another heuristic argument appeals to the finite-dimensional analogy, where the
motion of a particle along the bottom of the potential, i.e. along the orbit {g
α
(λ)ϕ},
where g
α
(λ) , λ ∈ R, is the one-parameter subgroup generated by T
α , does not feel
the potential, since U
(g
α
ϕ) = 0, and therefore the motion is like a free motion. This
is considered as evidence that, correspondingly, in the infinite-dimensional case there
are massless modes. Again the argument does not appear complete, since it is not at
all clear that there are physically meaningful solutions, i.e. belonging to the physical
sector of ϕ and therefore of the form ϕ = ϕ + χ, χ ∈ H
1
(R
s
), s = space dimension,
of zero mass.
We propose a version
42 of the Goldstone theorem for classical fields as a mathematically acceptable substitute and correction of the above heuristic arguments.
We consider the case of space dimension s = 3, unless otherwise stated and for
simplicity the case of compact semi-simple Lie group G of internal symmetries. The
potential is assumed to be of class C
3 .
The argument relies on some basic fact on the asymptotic solutions of (4.6) which
we briefly recall for the convenience of the reader.
Given a solution u(t) of the integral equation (4.6), its asymptotic time (t → ±∞)
behaviour defines the so-called scattering configurations or asymptotic states u ± (t)
associated with u(t).
The behaviour of f (u) near u = 0 plays a crucial role for such asymptotic limits
and if f (u) − f
(0)u vanishes to a sufficiently high degree, e.g. as O(u
3
), i) such
limits u ± (t) exist and ii) their time evolution is that corresponding to the differential
operator + f
(0), i.e.
u ± (t
) = W(t
− t)u ± (t),
where W(t) denotes the propagator corresponding to the differential operator
+ f
(0), (if f
(0) = 0, W(t) is the free wave equation propagator W (t) defined
in Chap. 4).
The mathematical theory of scattering for the non-linear wave equation is well
developed and it is beautifully reviewed by W. Strauss, Non-linear Wave Equations,
Am. Math. Soc. 1989.
The mathematical problem of the existence of the scattering configurations (the
so-called scattering theory) is to guarantee the well definiteness of the Yang–Feldman
equations
42 F. Strocchi, Phys. Lett. A267, 40 (2000).
9 The Goldstone Theorem
Thus, in an expansion of the potential around ϕ, the quadratic term, which has the
meaning of a mass term, has a zero eigenvalue in the direction T
α
ϕ. This is taken as
evidence that there is a massless mode.
In our opinion, the argument is not conclusive since it involves an expansion and
one should in some way control the effect of higher order terms; moreover, it is not
clear that there are (physically meaningful) solutions in the direction of T
α
ϕ for all
times, so that for them the quadratic term disappears. In any case, the argument does
not show that there are massless solutions as in the quantum case.
Another heuristic argument appeals to the finite-dimensional analogy, where the
motion of a particle along the bottom of the potential, i.e. along the orbit {g
α
(λ)ϕ},
where g
α
(λ) , λ ∈ R, is the one-parameter subgroup generated by T
α , does not feel
the potential, since U
(g
α
ϕ) = 0, and therefore the motion is like a free motion. This
is considered as evidence that, correspondingly, in the infinite-dimensional case there
are massless modes. Again the argument does not appear complete, since it is not at
all clear that there are physically meaningful solutions, i.e. belonging to the physical
sector of ϕ and therefore of the form ϕ = ϕ + χ, χ ∈ H
1
(R
s
), s = space dimension,
of zero mass.
We propose a version
42 of the Goldstone theorem for classical fields as a mathematically acceptable substitute and correction of the above heuristic arguments.
We consider the case of space dimension s = 3, unless otherwise stated and for
simplicity the case of compact semi-simple Lie group G of internal symmetries. The
potential is assumed to be of class C
3 .
The argument relies on some basic fact on the asymptotic solutions of (4.6) which
we briefly recall for the convenience of the reader.
Given a solution u(t) of the integral equation (4.6), its asymptotic time (t → ±∞)
behaviour defines the so-called scattering configurations or asymptotic states u ± (t)
associated with u(t).
The behaviour of f (u) near u = 0 plays a crucial role for such asymptotic limits
and if f (u) − f
(0)u vanishes to a sufficiently high degree, e.g. as O(u
3
), i) such
limits u ± (t) exist and ii) their time evolution is that corresponding to the differential
operator + f
(0), i.e.
u ± (t
) = W(t
− t)u ± (t),
where W(t) denotes the propagator corresponding to the differential operator
+ f
(0), (if f
(0) = 0, W(t) is the free wave equation propagator W (t) defined
in Chap. 4).
The mathematical theory of scattering for the non-linear wave equation is well
developed and it is beautifully reviewed by W. Strauss, Non-linear Wave Equations,
Am. Math. Soc. 1989.
The mathematical problem of the existence of the scattering configurations (the
so-called scattering theory) is to guarantee the well definiteness of the Yang–Feldman
equations
42 F. Strocchi, Phys. Lett. A267, 40 (2000).