27 The Goldstone Theorem for Relativistic Local Fields
193
Theorem 27.1 (Goldstone Theorem for relativistic local fields) Let β
λ be a oneparameter group of
∗ -automorphisms of the field algebra F, which
I. commutes with space–time translations,
II. is locally generated by a charge, in the sense that there is a local covariant
conserved current j μ such that ∀A ∈ F
δ A = i lim
R→∞
[Q R , A],
Q R ≡ j 0 ( f R , α) ≡
d
4 x f R (x)α(x 0 ) j 0 (x, x 0 ),
(27.2)
with f R as in (25.12), and
α ∈ D(R), supp α ⊆ [−δ, δ], ˜
α(0) =
dx 0 α(x 0 ) = 1,
(27.3)
III. is spontaneously broken in the sense that there exists at least one A ∈ F with
< δ A > 0 = 0.
Then, the Fourier transform of the two-point function < j 0 (x)A > contains a
δ( p
2
) singularity (Goldstone massless modes).
Remark 1 Relativistic local fields are more singular than non-relativistic fields
and therefore a smearing in time is necessary to get mathematically well defined
objects
173 ; this is the reason for the introduction of the test function α(x 0 ) and
˜
α(0) = 1 is merely a normalization condition. Indeed, even for a free Dirac field,
j μ (x, x 0 ) is a distribution in the four variables, which does not admit a restriction
at fixed time; in fact the commutator [ j 0 (x, x 0 ), j i (y, x 0 )] is a divergent Schwinger
term.
174 However, the introduction of the smearing with α does not spoil the simple
meaning of condition II and its possible control, thanks to the following Lemma.
Remark 2 For the symmetry breaking condition it is enough to consider the case
in which A is localized in a bounded space-time region, briefly A ∈ F loc , since
< δ A > 0 = 0 for all such A implies the invariance for all A ∈ F by a density
argument.
Lemma 27.2 As a consequence of locality, for any A ∈ F the limit R → ∞ in
(27.2) exists and it is independent of α (with ˜
α(0) = 1).
Proof. In fact, if A is a local field with compact support K , the commutator
[ j μ (x, α), A] vanishes by locality for |x| sufficiently large and for a general A ∈ F
the commutator decreases faster than any inverse power of |x|. Therefore, the integrability of the charge commutators is automatically satisfied (the local integrability
is not a problem as discussed in Sect. 25.2).
173 A.S. Wightman, Ann. Inst. H. Poincaré, I, 403 (1964).
174 For a simple discussion, see, e.g. F. Strocchi, Selected Topics on the General Properties of
Quantum Field Theory, World Scientific 1993, Sect. 4.5.
193
Theorem 27.1 (Goldstone Theorem for relativistic local fields) Let β
λ be a oneparameter group of
∗ -automorphisms of the field algebra F, which
I. commutes with space–time translations,
II. is locally generated by a charge, in the sense that there is a local covariant
conserved current j μ such that ∀A ∈ F
δ A = i lim
R→∞
[Q R , A],
Q R ≡ j 0 ( f R , α) ≡
d
4 x f R (x)α(x 0 ) j 0 (x, x 0 ),
(27.2)
with f R as in (25.12), and
α ∈ D(R), supp α ⊆ [−δ, δ], ˜
α(0) =
dx 0 α(x 0 ) = 1,
(27.3)
III. is spontaneously broken in the sense that there exists at least one A ∈ F with
< δ A > 0 = 0.
Then, the Fourier transform of the two-point function < j 0 (x)A > contains a
δ( p
2
) singularity (Goldstone massless modes).
Remark 1 Relativistic local fields are more singular than non-relativistic fields
and therefore a smearing in time is necessary to get mathematically well defined
objects
173 ; this is the reason for the introduction of the test function α(x 0 ) and
˜
α(0) = 1 is merely a normalization condition. Indeed, even for a free Dirac field,
j μ (x, x 0 ) is a distribution in the four variables, which does not admit a restriction
at fixed time; in fact the commutator [ j 0 (x, x 0 ), j i (y, x 0 )] is a divergent Schwinger
term.
174 However, the introduction of the smearing with α does not spoil the simple
meaning of condition II and its possible control, thanks to the following Lemma.
Remark 2 For the symmetry breaking condition it is enough to consider the case
in which A is localized in a bounded space-time region, briefly A ∈ F loc , since
< δ A > 0 = 0 for all such A implies the invariance for all A ∈ F by a density
argument.
Lemma 27.2 As a consequence of locality, for any A ∈ F the limit R → ∞ in
(27.2) exists and it is independent of α (with ˜
α(0) = 1).
Proof. In fact, if A is a local field with compact support K , the commutator
[ j μ (x, α), A] vanishes by locality for |x| sufficiently large and for a general A ∈ F
the commutator decreases faster than any inverse power of |x|. Therefore, the integrability of the charge commutators is automatically satisfied (the local integrability
is not a problem as discussed in Sect. 25.2).
173 A.S. Wightman, Ann. Inst. H. Poincaré, I, 403 (1964).
174 For a simple discussion, see, e.g. F. Strocchi, Selected Topics on the General Properties of
Quantum Field Theory, World Scientific 1993, Sect. 4.5.