Chapter 20
Constructive Symmetry Breaking
Apart from simple models, like those discussed in the previous section, the existence
of a symmetry breaking order parameter is a non-trivial problem which in principle requires control on the correlation functions. In this section, we briefly discuss
constructive criteria for symmetry breaking.
A. Goldstone (Perturbative) Criterion
In a pioneering paper on symmetry breaking, Goldstone discussed a quantum (scalar)
field theory model exhibiting a symmetry breaking order parameter.
110 Since the
standard perturbative expansion based on the standard Fock representation predicts
the vanishing of the field expectation value, Goldstone suggested a strategy which
combines a perturbative expansion and a semi-classical approximation (Goldstone
criterion). Since this has become the standard approach to symmetry breaking within
the perturbative approach, it is worthwhile to discuss briefly the Goldstone criterion.
The Goldstone model is described by the following Lagrangian
L =
1
2
∂ μ ϕ ∂
μ
ϕ − U (ϕ), U (ϕ) = λ(ϕ
2
− a
2
)
2
,
(20.1)
110 J. Goldstone, Nuovo Cim. 10, 154 (1961). For a general critical discussion and for an outline
of the constructive approach to symmetry breaking in quantum field theory, see A.S. Wightman,
Constructive Field Theory, in Fundamental Interactions in Physics and Astrophysics, (Coral Gables
1972), G. Iverson et al. eds., Plenum 1973.
© 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_20
133
Constructive Symmetry Breaking
Apart from simple models, like those discussed in the previous section, the existence
of a symmetry breaking order parameter is a non-trivial problem which in principle requires control on the correlation functions. In this section, we briefly discuss
constructive criteria for symmetry breaking.
A. Goldstone (Perturbative) Criterion
In a pioneering paper on symmetry breaking, Goldstone discussed a quantum (scalar)
field theory model exhibiting a symmetry breaking order parameter.
110 Since the
standard perturbative expansion based on the standard Fock representation predicts
the vanishing of the field expectation value, Goldstone suggested a strategy which
combines a perturbative expansion and a semi-classical approximation (Goldstone
criterion). Since this has become the standard approach to symmetry breaking within
the perturbative approach, it is worthwhile to discuss briefly the Goldstone criterion.
The Goldstone model is described by the following Lagrangian
L =
1
2
∂ μ ϕ ∂
μ
ϕ − U (ϕ), U (ϕ) = λ(ϕ
2
− a
2
)
2
,
(20.1)
110 J. Goldstone, Nuovo Cim. 10, 154 (1961). For a general critical discussion and for an outline
of the constructive approach to symmetry breaking in quantum field theory, see A.S. Wightman,
Constructive Field Theory, in Fundamental Interactions in Physics and Astrophysics, (Coral Gables
1972), G. Iverson et al. eds., Plenum 1973.
© 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_20
133