194
27 The Goldstone Theorem for Relativistic Local Fields
Moreover, if α 1 , α 2 ∈ D(R) are two normalized test functions, then
α 1 − α 2 = dβ/dx 0 , β(x 0 ) ≡
x 0
−∞
dx
0 (α 1 (x
0 ) − α 2 (x
0 )) ∈ D(R),
and by current conservation, ∂
μ j μ = 0,
[ j 0 ( f R , α 1 ) − j 0 ( f R , α 2 ), A] = −[∂ 0 j 0 ( f R , β), A] = [j(∇ f R , β), A].
Since supp ∇ f R ⊆ {R ≤ |x| ≤ R(1 + ε)}, for R large enough, the localization region
of j(∇ f R , β) becomes space-like with respect to any (bounded) compact set K and
the commutator vanishes by locality.
Remark 3 The argument of the Lemma can be adapted to the case in which A is
replaced by a local field variable, say ϕ(y, y 0 ), since [ j 0 ( f R , α), ϕ(y, y 0 )] is a well
defined operator-valued distribution in y, y 0 and by locality the limit R → ∞ exists
and it is actually reached for R large enough. By the same argument as above, such
a limit is independent of α and therefore taking α 1 (x 0 ) = α(x 0 − y 0 ), with α as in
(27.3), the limit of shrinking support δ → 0 exists and defines a regularized version of
the equal time commutator between j 0 ( f R , x 0 ) and ϕ(y, x 0 ), for R large enough.
175
Remark 4 As a consequence of the above Lemma, the delicate problems of the
non-relativistic case (discussed in Sect. 25.2) do not arise for local field algebras. By
Remark 3, the existence and identification of a (conserved) current which generates
a given (algebraic) symmetry β
λ can be inferred by using the (equal time) CCR (or
ACR) and the stability under time evolution is guaranteed by the independence of α.
The proof of the theorem is particularly simple if the order parameter is given
by a local field, say ϕ(y, y 0 ), which transforms as in (27.1), briefly called an elementary field. For a generic element A ∈ F, one can easily obtain covariance under
space–time translations by putting A x = α x (A), but then the transformation under
the Poincaré group is not given by (27.1). The Lorentz invariance of the v.e.v. requires
that the order parameter is a scalar and thus one may take ϕ a scalar. This is the case
considered in the classic work of Goldstone, Salam and Weinberg,
176 which we
reproduce below in a somewhat simplified version.
Proof for elementary fields. The Poincaré covariance implies that
J μ (x − y) ≡< j μ (x)ϕ(y) > 0 = (Λ
−1
)
ν
μ J ν (Λ(x − y))
175 The effectiveness of such a regularization in giving finite results is clearly displayed by the equal
time commutator [ j 0 ( f R , x 0 ), j i (y, x 0 )], for a free Dirac current.
176 J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962).
27 The Goldstone Theorem for Relativistic Local Fields
Moreover, if α 1 , α 2 ∈ D(R) are two normalized test functions, then
α 1 − α 2 = dβ/dx 0 , β(x 0 ) ≡
x 0
−∞
dx
0 (α 1 (x
0 ) − α 2 (x
0 )) ∈ D(R),
and by current conservation, ∂
μ j μ = 0,
[ j 0 ( f R , α 1 ) − j 0 ( f R , α 2 ), A] = −[∂ 0 j 0 ( f R , β), A] = [j(∇ f R , β), A].
Since supp ∇ f R ⊆ {R ≤ |x| ≤ R(1 + ε)}, for R large enough, the localization region
of j(∇ f R , β) becomes space-like with respect to any (bounded) compact set K and
the commutator vanishes by locality.
Remark 3 The argument of the Lemma can be adapted to the case in which A is
replaced by a local field variable, say ϕ(y, y 0 ), since [ j 0 ( f R , α), ϕ(y, y 0 )] is a well
defined operator-valued distribution in y, y 0 and by locality the limit R → ∞ exists
and it is actually reached for R large enough. By the same argument as above, such
a limit is independent of α and therefore taking α 1 (x 0 ) = α(x 0 − y 0 ), with α as in
(27.3), the limit of shrinking support δ → 0 exists and defines a regularized version of
the equal time commutator between j 0 ( f R , x 0 ) and ϕ(y, x 0 ), for R large enough.
175
Remark 4 As a consequence of the above Lemma, the delicate problems of the
non-relativistic case (discussed in Sect. 25.2) do not arise for local field algebras. By
Remark 3, the existence and identification of a (conserved) current which generates
a given (algebraic) symmetry β
λ can be inferred by using the (equal time) CCR (or
ACR) and the stability under time evolution is guaranteed by the independence of α.
The proof of the theorem is particularly simple if the order parameter is given
by a local field, say ϕ(y, y 0 ), which transforms as in (27.1), briefly called an elementary field. For a generic element A ∈ F, one can easily obtain covariance under
space–time translations by putting A x = α x (A), but then the transformation under
the Poincaré group is not given by (27.1). The Lorentz invariance of the v.e.v. requires
that the order parameter is a scalar and thus one may take ϕ a scalar. This is the case
considered in the classic work of Goldstone, Salam and Weinberg,
176 which we
reproduce below in a somewhat simplified version.
Proof for elementary fields. The Poincaré covariance implies that
J μ (x − y) ≡< j μ (x)ϕ(y) > 0 = (Λ
−1
)
ν
μ J ν (Λ(x − y))
175 The effectiveness of such a regularization in giving finite results is clearly displayed by the equal
time commutator [ j 0 ( f R , x 0 ), j i (y, x 0 )], for a free Dirac current.
176 J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962).