27 The Goldstone Theorem for Relativistic Local Fields
195
and therefore by a general result
177
J μ (x) = ∂ μ F(x), F(x) = F(Λx).
(27.4)
Now, current conservation implies
F(x) = 0,
so that the Fourier transform is of the form ˜
F( p) = f ( p)δ( p
2
). Finally, the symmetry
breaking condition III excludes f ( p) = 0.
The Goldstone–Salam–Weinberg (GSW) version of the Goldstone theorem does
not cover the case in which the symmetry breaking involves a polynomial of the fields
(or a composite field). For these reasons a more general version is important.
178
Such a general proof also makes clear that locality and not Lorentz covariance, as
one may be led to believe on the basis of the GSW version, is the crucial ingredient.
Actually, the non-covariance of the fields of the Coulomb gauge, rather than their
non-locality, has been taken as an explanation of the evasion of the Goldstone theorem
by Higgs
179 in his proposal of the so-called Higgs mechanism. As a matter of fact,
for the two-point function of elementary fields, Lorentz covariance and locality are
deeply related
180 and therefore it is not strange that the GSW proof, which exploits
Lorentz covariance, may hide the role of locality. On the other hand, the recognition
177 K. Hepp, Helv. Phys. Acta 36, 355 (1963). The proof of (27.4) can be reduced to an exercise
in relativistic kinematics. By Poincaré covariance the Fourier transform J μ ( p) satisfies J μ ( p) =
(Λ −1 ) ν
μ J ν (Λp) and therefore if q = Rp, R a rotation,
|p|
2 J i ( p 0 , p) − p i p k J k ( p 0 , p) = |q|
2 (R
−1 J ) i ( p 0 , q) − (R
−1 q) i q k J k ( p 0 , q).
The l.h.s. vanishes for p pointing in the i-direction and therefore, multiplying the r.h.s. by R, for any
q = 0, J i ( p 0 , q) = q i q · J( p 0 , q)/|q| 2 . Again, by using rotation covariance, (omitting the variable
p 0 ),
G(p) ≡ p · J(p) = p · R
−1 J(Rp) = Rp · J(Rp) = G(Rp),
i.e. G = G(|p|). Similarly J 0 ( p) = J 0 ( p 0 , |p|). Moreover, by using covariance under Lorentz
boosts, e.g. boosts in the 3-direction, one has J i ( p 0 , p 3 , p 1 , p 2 ) = J i (Λ( p 0 , p 3 ), p 1 , p 2 ), i = 1, 2,
i.e. they are functions of the boost invariant combination p 2
0 − p 2
3 . Then, by rotation invariance,
G( p 0 , |p|)/|p| 2 = F( p 2 ). Finally
J 3 ( p) = p 3 F( p
2 ) = ((Λ)
−1 )
0
3 J 0 (Λp) + ((Λ)
−1 )
3
3 (Λp) 3 F( p
2 )
= ((Λ)
−1 )
0
3 J 0 (Λp) + p 3 F( p
2 ) − ((Λ)
−1 )
0
3 (Λp) 0 F( p
2 ),
i.e. J 0 ( p) = p 0 F( p 2 ).
178 D. Kastler, D.W. Robinson and A. Swieca, Comm. Math. Phys. 2, 108 (1966); H. Ezawa and
J.A. Swieca, Comm. Math. Phys. 5, 330 (1967). See also the beautiful reviews: D. Kastler, Broken Symmetries and the Goldstone Theorem, in Proc. 1967 Int. Conf. on Particles and Fields
(Rochester), C.R. Hagen et al. eds., Wiley 1967; J.A. Swieca, Goldstone Theorem and Related
Topics, in Chargèse Lectures 4, D. Kastler ed., Gordon and Breach 1969.
179 P.W. Higgs, Phys. Lett. 12, 133 (1964).
180 J. Bros, H. Epstein and V. Glaser, Comm. Math. Phys. 6, 77 (1967).
195
and therefore by a general result
177
J μ (x) = ∂ μ F(x), F(x) = F(Λx).
(27.4)
Now, current conservation implies
F(x) = 0,
so that the Fourier transform is of the form ˜
F( p) = f ( p)δ( p
2
). Finally, the symmetry
breaking condition III excludes f ( p) = 0.
The Goldstone–Salam–Weinberg (GSW) version of the Goldstone theorem does
not cover the case in which the symmetry breaking involves a polynomial of the fields
(or a composite field). For these reasons a more general version is important.
178
Such a general proof also makes clear that locality and not Lorentz covariance, as
one may be led to believe on the basis of the GSW version, is the crucial ingredient.
Actually, the non-covariance of the fields of the Coulomb gauge, rather than their
non-locality, has been taken as an explanation of the evasion of the Goldstone theorem
by Higgs
179 in his proposal of the so-called Higgs mechanism. As a matter of fact,
for the two-point function of elementary fields, Lorentz covariance and locality are
deeply related
180 and therefore it is not strange that the GSW proof, which exploits
Lorentz covariance, may hide the role of locality. On the other hand, the recognition
177 K. Hepp, Helv. Phys. Acta 36, 355 (1963). The proof of (27.4) can be reduced to an exercise
in relativistic kinematics. By Poincaré covariance the Fourier transform J μ ( p) satisfies J μ ( p) =
(Λ −1 ) ν
μ J ν (Λp) and therefore if q = Rp, R a rotation,
|p|
2 J i ( p 0 , p) − p i p k J k ( p 0 , p) = |q|
2 (R
−1 J ) i ( p 0 , q) − (R
−1 q) i q k J k ( p 0 , q).
The l.h.s. vanishes for p pointing in the i-direction and therefore, multiplying the r.h.s. by R, for any
q = 0, J i ( p 0 , q) = q i q · J( p 0 , q)/|q| 2 . Again, by using rotation covariance, (omitting the variable
p 0 ),
G(p) ≡ p · J(p) = p · R
−1 J(Rp) = Rp · J(Rp) = G(Rp),
i.e. G = G(|p|). Similarly J 0 ( p) = J 0 ( p 0 , |p|). Moreover, by using covariance under Lorentz
boosts, e.g. boosts in the 3-direction, one has J i ( p 0 , p 3 , p 1 , p 2 ) = J i (Λ( p 0 , p 3 ), p 1 , p 2 ), i = 1, 2,
i.e. they are functions of the boost invariant combination p 2
0 − p 2
3 . Then, by rotation invariance,
G( p 0 , |p|)/|p| 2 = F( p 2 ). Finally
J 3 ( p) = p 3 F( p
2 ) = ((Λ)
−1 )
0
3 J 0 (Λp) + ((Λ)
−1 )
3
3 (Λp) 3 F( p
2 )
= ((Λ)
−1 )
0
3 J 0 (Λp) + p 3 F( p
2 ) − ((Λ)
−1 )
0
3 (Λp) 0 F( p
2 ),
i.e. J 0 ( p) = p 0 F( p 2 ).
178 D. Kastler, D.W. Robinson and A. Swieca, Comm. Math. Phys. 2, 108 (1966); H. Ezawa and
J.A. Swieca, Comm. Math. Phys. 5, 330 (1967). See also the beautiful reviews: D. Kastler, Broken Symmetries and the Goldstone Theorem, in Proc. 1967 Int. Conf. on Particles and Fields
(Rochester), C.R. Hagen et al. eds., Wiley 1967; J.A. Swieca, Goldstone Theorem and Related
Topics, in Chargèse Lectures 4, D. Kastler ed., Gordon and Breach 1969.
179 P.W. Higgs, Phys. Lett. 12, 133 (1964).
180 J. Bros, H. Epstein and V. Glaser, Comm. Math. Phys. 6, 77 (1967).