212
29 Symmetry Breaking in Gauge Theories
By the support properties of F(x) =
dρ(m
2
)(k 0 )δ(k
2
− m
2
)e
−ikx d
4 x, the charge
density is integrable and, in all correlation functions of the Coulomb field algebra,
lim
R→∞
[ j 0 ( f R , x 0 ), ϕ(y)] = −e
dρ(m
2
) cos(m(x 0 − y 0 ))ϕ(y).
(29.18)
The r.h.s is independent of time if and only if dρ(m
2
) = λδ(m
2
), i.e. if F μν is a free
field.
205
The same conclusions hold if instead of (29.13) one uses the regularized version of
Buchholz et al., since (29.15), (29.17), (29.18) get changed only by a convolution
with a test function h(y 0 ) ∈ D(R).
The same conclusion about the time dependence of the commutators of the charge
density is obtained by the general estimate of the space-like large distance behaviour
of the commutator (29.12) obtained by using (29.15), i.e.
[ j i (x), ϕ(y)] ∼ i(e/4π)
d
3 z∂
i
z |z − y|
−1
∂
2
0 F(x − z, x 0 − y 0 )ϕ(y),
lim
R→∞
[ ˙
Q R (x 0 ), ϕ(y)] = lim
R→∞
[divj( f R , x 0 ), ϕ(y)] = 0.
The time dependence of the charge density commutators is at the roots of the appearance of an infinite renormalization constant in equal time commutators
[ j 0 (x, t), ϕ(y, t)] = −e(Z 3 )
−1
δ(x − y)ϕ(y, t).
For such a phenomenon the vacuum polarization due to loops of charged fields
plays a crucial role, so that the semi-classical approximation does not provide relevant information about the time dependence of the charge commutators. In fact, the
phenomenon does not appear in the classical theory, where there are finite energy
localized solutions with non-zero charge and localized current j μ , only the electric
field being a Coulomb delocalized function of j 0 .
206
The above Proposition shows that the heuristic argument by which if the symmetry commutes with time translation, equivalently if the current continuity equation
holds, then the generating charge commutes with the Hamiltonian and is therefore
independent of time is not correct.
Even if the equal time commutators, in particular [j, ϕ], have a sufficient localization, the time evolution may induce a delocalization leading to a failure of (29.12).
For these reasons, no reliable information can be inferred from the equal time commutators and the check of the basic assumptions of the Goldstone theorem becomes
interlaced with the dynamical problem, as it happens for non-relativistic systems.
The failure of locality, rather than the lack of manifest covariance, is the crucial
structural property which explains the evasion of the Goldstone theorem by exactly
205 G. Morchio and F. Strocchi, J. Math,. Phys. 44, 5569 (2003), Appendix.
206 D. Buchholz, S. Doplicher, G. Morchio, J.E. Roberts and F. Strocchi, Ann. Phys. 290, 53 (2001).
29 Symmetry Breaking in Gauge Theories
By the support properties of F(x) =
dρ(m
2
)(k 0 )δ(k
2
− m
2
)e
−ikx d
4 x, the charge
density is integrable and, in all correlation functions of the Coulomb field algebra,
lim
R→∞
[ j 0 ( f R , x 0 ), ϕ(y)] = −e
dρ(m
2
) cos(m(x 0 − y 0 ))ϕ(y).
(29.18)
The r.h.s is independent of time if and only if dρ(m
2
) = λδ(m
2
), i.e. if F μν is a free
field.
205
The same conclusions hold if instead of (29.13) one uses the regularized version of
Buchholz et al., since (29.15), (29.17), (29.18) get changed only by a convolution
with a test function h(y 0 ) ∈ D(R).
The same conclusion about the time dependence of the commutators of the charge
density is obtained by the general estimate of the space-like large distance behaviour
of the commutator (29.12) obtained by using (29.15), i.e.
[ j i (x), ϕ(y)] ∼ i(e/4π)
d
3 z∂
i
z |z − y|
−1
∂
2
0 F(x − z, x 0 − y 0 )ϕ(y),
lim
R→∞
[ ˙
Q R (x 0 ), ϕ(y)] = lim
R→∞
[divj( f R , x 0 ), ϕ(y)] = 0.
The time dependence of the charge density commutators is at the roots of the appearance of an infinite renormalization constant in equal time commutators
[ j 0 (x, t), ϕ(y, t)] = −e(Z 3 )
−1
δ(x − y)ϕ(y, t).
For such a phenomenon the vacuum polarization due to loops of charged fields
plays a crucial role, so that the semi-classical approximation does not provide relevant information about the time dependence of the charge commutators. In fact, the
phenomenon does not appear in the classical theory, where there are finite energy
localized solutions with non-zero charge and localized current j μ , only the electric
field being a Coulomb delocalized function of j 0 .
206
The above Proposition shows that the heuristic argument by which if the symmetry commutes with time translation, equivalently if the current continuity equation
holds, then the generating charge commutes with the Hamiltonian and is therefore
independent of time is not correct.
Even if the equal time commutators, in particular [j, ϕ], have a sufficient localization, the time evolution may induce a delocalization leading to a failure of (29.12).
For these reasons, no reliable information can be inferred from the equal time commutators and the check of the basic assumptions of the Goldstone theorem becomes
interlaced with the dynamical problem, as it happens for non-relativistic systems.
The failure of locality, rather than the lack of manifest covariance, is the crucial
structural property which explains the evasion of the Goldstone theorem by exactly
205 G. Morchio and F. Strocchi, J. Math,. Phys. 44, 5569 (2003), Appendix.
206 D. Buchholz, S. Doplicher, G. Morchio, J.E. Roberts and F. Strocchi, Ann. Phys. 290, 53 (2001).