Chapter 9
The Goldstone Theorem
The mechanism of SSB does not only provide a general strategy for unifying the
description of apparently different systems, but it also provides information on the
energy spectrum of an infinite dimensional system, by means of the so-called Goldstone theorem,
40 according to which to each broken generator T of a continuous
symmetry there corresponds a massless mode, i.e. a free wave. The quantum version
of such a statement has been turned into a theorem,
41 whereas, as far as we know,
no analogous theorem has been proved for classical (infinite- dimensional) systems
and the standard accounts seem to rely on heuristic arguments.
The standard heuristic argument, which actually goes back to Goldstone, considers
as a prototype the non-linear equation (3.1)
ϕ + U
(ϕ) = 0,
where the multi-component real field ϕ transforms as a linear representation of a
Lie group G and the potential U is invariant under the transformations of G. This
implies that for the generator T
α one has
0 = δ
α U (ϕ) = U
j (ϕ)T
α
jk ϕ k , ∀ϕ
(9.1)
and therefore the derivative of this equation at ϕ = ϕ gives
U
jk (ϕ)(T
α
ϕ) k = 0.
(9.2)
40 J. Goldstone, Nuovo Cimento 19, 154 (1961).
41 J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962); J. Swieca, Goldstone’s
theorem and related topics, Cargèse lectures 1969.
© 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_9
47
The Goldstone Theorem
The mechanism of SSB does not only provide a general strategy for unifying the
description of apparently different systems, but it also provides information on the
energy spectrum of an infinite dimensional system, by means of the so-called Goldstone theorem,
40 according to which to each broken generator T of a continuous
symmetry there corresponds a massless mode, i.e. a free wave. The quantum version
of such a statement has been turned into a theorem,
41 whereas, as far as we know,
no analogous theorem has been proved for classical (infinite- dimensional) systems
and the standard accounts seem to rely on heuristic arguments.
The standard heuristic argument, which actually goes back to Goldstone, considers
as a prototype the non-linear equation (3.1)
ϕ + U
(ϕ) = 0,
where the multi-component real field ϕ transforms as a linear representation of a
Lie group G and the potential U is invariant under the transformations of G. This
implies that for the generator T
α one has
0 = δ
α U (ϕ) = U
j (ϕ)T
α
jk ϕ k , ∀ϕ
(9.1)
and therefore the derivative of this equation at ϕ = ϕ gives
U
jk (ϕ)(T
α
ϕ) k = 0.
(9.2)
40 J. Goldstone, Nuovo Cimento 19, 154 (1961).
41 J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962); J. Swieca, Goldstone’s
theorem and related topics, Cargèse lectures 1969.
© 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_9
47