192
27 The Goldstone Theorem for Relativistic Local Fields
U (a, Λ(A))ϕ j (x)U (a, Λ(A))
−1
= S jk (A
−1
)ϕ k (Λ(A)x + a), A ∈ SL(2, C),
(27.1)
where S jk is a finite dimensional representation of SL(2, C) (the universal covering
of the restricted Lorentz group L
↑
+ ).
For example, for a scalar field ϕ, S jk = 1 and for a vector field j μ , S μν (Λ
−1
) =
(Λ
−1
)
ν
μ , etc.
The construction of a C
∗ -algebra A F ⊃ A obs , in terms of the polynomial algebra
F generated by the smeared fields {ϕ j ( f ), f ∈ S(R
4
)}, requires self-adjoint conditions on the smeared fields, which are not easy to control (in contrast with the finite
dimensional case). Therefore, following Wightman,
169 and also in analogy with the
perturbative approach to quantum field theory, one usually works directly with the
(polynomial) field algebra F.
In general, it is not automatic that the field algebra F, needed for the formulation
and solution of the dynamical problem, is a local algebra, i.e. it satisfies (14.2) or its
extension for anticommuting fields.
For example, this is not the case of the Coulomb gauge field algebra of quantum
electrodynamics (QED), where the electron field ψ and the electromagnetic field F μν
do not commute at space-like separations; moreover, ψ and the vector potential do
not transform as in (27.1).
On the other hand, a local covariant field algebra F is at the basis of the so-called
renormalizable gauges of gauge quantum field theory (e.g. the Feynman gauge in
QED), at the price that the vacuum state is not a positive functional on F.
170
Even in this more general case without positivity,
171 one can introduce the
concept of symmetry breaking of a one-parameter group of automorphisms β
λ of
F, when the vacuum expectation values (v.e.v.) of the fields, briefly denoted by
< A > 0 , ∀A ∈ F, are not invariant under β
λ , i.e. < δ A > 0 = 0, for some A ∈ F.
One may then investigate the consequences of such a breaking for the spectral support of the Fourier transforms of the v.e.v.
We shall now discuss a version of the Goldstone theorem, which applies to local
field algebras with v.e.v. which satisfy space-time translation invariance, relativistic
spectral support, but not necessarily positivity.
172
169 R.F. Streater and A.S. Wightman, PCT, Spin and Statistics and All That Benjamin-Cummings
1980.
170 For a general discussion of the interplay between locality and positivity in gauge quantum field
theory, see F. Strocchi, Selected Topics on the General Properties of Quantum Field Theory, World
Scientific 1993 and references therein.
171 For the discussion of this more general formulation of quantum field theories, which is particularly relevant for two-dimensional models involving a massless scalar field and for gauge quantum
field theories, see F. Strocchi and A.S. Wightman, J. Math. Phys. 15, 2198 (1974); G. Morchio and
F. Strocchi, Ann. Inst. H. Poincaré, A 33, 251 (1980) and for a general review F. Strocchi, Selected
Topics on the General Properties of Quantum Field Theory, World Scientific 1993, Chap. VI.
172 F. Strocchi, Comm. Math. Phys. 56, 57 (1977).
27 The Goldstone Theorem for Relativistic Local Fields
U (a, Λ(A))ϕ j (x)U (a, Λ(A))
−1
= S jk (A
−1
)ϕ k (Λ(A)x + a), A ∈ SL(2, C),
(27.1)
where S jk is a finite dimensional representation of SL(2, C) (the universal covering
of the restricted Lorentz group L
↑
+ ).
For example, for a scalar field ϕ, S jk = 1 and for a vector field j μ , S μν (Λ
−1
) =
(Λ
−1
)
ν
μ , etc.
The construction of a C
∗ -algebra A F ⊃ A obs , in terms of the polynomial algebra
F generated by the smeared fields {ϕ j ( f ), f ∈ S(R
4
)}, requires self-adjoint conditions on the smeared fields, which are not easy to control (in contrast with the finite
dimensional case). Therefore, following Wightman,
169 and also in analogy with the
perturbative approach to quantum field theory, one usually works directly with the
(polynomial) field algebra F.
In general, it is not automatic that the field algebra F, needed for the formulation
and solution of the dynamical problem, is a local algebra, i.e. it satisfies (14.2) or its
extension for anticommuting fields.
For example, this is not the case of the Coulomb gauge field algebra of quantum
electrodynamics (QED), where the electron field ψ and the electromagnetic field F μν
do not commute at space-like separations; moreover, ψ and the vector potential do
not transform as in (27.1).
On the other hand, a local covariant field algebra F is at the basis of the so-called
renormalizable gauges of gauge quantum field theory (e.g. the Feynman gauge in
QED), at the price that the vacuum state is not a positive functional on F.
170
Even in this more general case without positivity,
171 one can introduce the
concept of symmetry breaking of a one-parameter group of automorphisms β
λ of
F, when the vacuum expectation values (v.e.v.) of the fields, briefly denoted by
< A > 0 , ∀A ∈ F, are not invariant under β
λ , i.e. < δ A > 0 = 0, for some A ∈ F.
One may then investigate the consequences of such a breaking for the spectral support of the Fourier transforms of the v.e.v.
We shall now discuss a version of the Goldstone theorem, which applies to local
field algebras with v.e.v. which satisfy space-time translation invariance, relativistic
spectral support, but not necessarily positivity.
172
169 R.F. Streater and A.S. Wightman, PCT, Spin and Statistics and All That Benjamin-Cummings
1980.
170 For a general discussion of the interplay between locality and positivity in gauge quantum field
theory, see F. Strocchi, Selected Topics on the General Properties of Quantum Field Theory, World
Scientific 1993 and references therein.
171 For the discussion of this more general formulation of quantum field theories, which is particularly relevant for two-dimensional models involving a massless scalar field and for gauge quantum
field theories, see F. Strocchi and A.S. Wightman, J. Math. Phys. 15, 2198 (1974); G. Morchio and
F. Strocchi, Ann. Inst. H. Poincaré, A 33, 251 (1980) and for a general review F. Strocchi, Selected
Topics on the General Properties of Quantum Field Theory, World Scientific 1993, Chap. VI.
172 F. Strocchi, Comm. Math. Phys. 56, 57 (1977).