Chapter 27
The Goldstone Theorem for Relativistic
Local Fields
Relativistic systems, like elementary particles, are described by an algebra of observables A obs which satisfies the causality condition, (14.2), and is stable under the
automorphisms α(a, Λ) which describe space–time translations and Lorentz transformations (with parameters a, Λ, respectively).
The physically relevant representations of A obs have to satisfy the relativistic
version of conditions I–III (Chap. 15):
I. (Poincaré Covariance) The automorphisms α(a, Λ) are implemented by a
strongly continuous group of unitary operators U (a, Λ).
II. (Relativistic spectral condition)
H ≥ 0, H
2
− P
2
≥ 0,
equivalently, the Fourier transform of the matrix elements of U (a) have support
in the closed forward cone V + = {p
2
≥ 0, p 0 ≥ 0}.
III. (Vacuum state) There is a unique space-time translationally invariant state Ψ 0
(vacuum state) cyclic for the algebra A obs .
As we have also seen in the case of non-relativistic systems (e.g. the free Bose gas),
it is convenient (if not necessary) for the formulation and solution of the dynamical
problem to work with an extension of the algebra of observables. This amounts
to introducing a field algebra F, which plays the role of the algebra of canonical
variables of the non-relativistic systems.
The algebra F is generated by a set of fields {ϕ j (x), x = (x, x 0 ), j ∈ I =
finite index set}, which are operator-valued (tempered) distributions and in general
transform covariantly under the Poincaré group
© 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_27
191
The Goldstone Theorem for Relativistic
Local Fields
Relativistic systems, like elementary particles, are described by an algebra of observables A obs which satisfies the causality condition, (14.2), and is stable under the
automorphisms α(a, Λ) which describe space–time translations and Lorentz transformations (with parameters a, Λ, respectively).
The physically relevant representations of A obs have to satisfy the relativistic
version of conditions I–III (Chap. 15):
I. (Poincaré Covariance) The automorphisms α(a, Λ) are implemented by a
strongly continuous group of unitary operators U (a, Λ).
II. (Relativistic spectral condition)
H ≥ 0, H
2
− P
2
≥ 0,
equivalently, the Fourier transform of the matrix elements of U (a) have support
in the closed forward cone V + = {p
2
≥ 0, p 0 ≥ 0}.
III. (Vacuum state) There is a unique space-time translationally invariant state Ψ 0
(vacuum state) cyclic for the algebra A obs .
As we have also seen in the case of non-relativistic systems (e.g. the free Bose gas),
it is convenient (if not necessary) for the formulation and solution of the dynamical
problem to work with an extension of the algebra of observables. This amounts
to introducing a field algebra F, which plays the role of the algebra of canonical
variables of the non-relativistic systems.
The algebra F is generated by a set of fields {ϕ j (x), x = (x, x 0 ), j ∈ I =
finite index set}, which are operator-valued (tempered) distributions and in general
transform covariantly under the Poincaré group
© 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_27
191