206
29 Symmetry Breaking in Gauge Theories
29.2 Higgs Mechanism in Local Gauges
The evasion of the Goldstone theorem by the Higgs mechanism can be understood
by a non-perturbative argument in local (renormalizable) gauges.
For concreteness, we discuss the abelian Higgs–Kibble model in the so-called α
gauges obtained by the addition of the gauge fixing −
1
2
α(∂ μ A
μ
)
2 to the gauge invariant Lagrangian (29.1). Proceeding as before with a perturbative expansion based on
the mean field ansatz ϕ = ϕ + χ 1 + iχ 2 , and performing the change of variables
W μ = A μ − (eϕ)
−1
∂ μ χ 2 , one gets a quadratic Lagrangian of the form (29.3) plus
the gauge fixing term −
1
2
α(χ 2 )
2 .
Thus, χ 2 does not disappear from the quadratic Lagrangian and satisfies a “massless” field equation
2
χ 2 = 0; this means that there are massless modes.
The problem is their physical interpretation; an indication against their physical
relevance is the α dependence of the corresponding Lagrangian term. Moreover, the
general solution of the equation
2
χ 2 = 0 is a massless field (∼ δ(k
2
)) plus a dipole
field (∼ δ
(k
2
)). A δ
(k
2
) singularity is not a measure and therefore is not allowed
to appear in the physical spectrum because the space–time translations must be
described by unitary operator, when restricted to the space of physical vectors where
positivity holds, so that their spectral representation on physical states is given by a
measure.
195
Actually, by exploiting the Gauss law relation j μ = ∂
ν F μν , one can find a general
non-perturbative argument
196 about the unphysical nature of the massless modes
associated with the breaking of the U (1) gauge group in local (renormalizable)
gauges.
As a first step, one remarks that in local renormalizable gauges, like the Feynman
gauge, the field algebra F is generated by the local charged fields ϕ(x) and by
the local vector potential A μ (x), the four components of which are quantized as
independent fields. Locality of the field algebra together with the relativistic spectral
support of the Fourier transforms of the vacuum expectations are the basic properties
shared by such local gauges, so that most of the standard wisdom on quantum field
theory is available; these are in fact the gauges used in perturbation theory.
Thanks to the locality of the field algebra F, there is no problem for the existence of
lim R→∞ [ j 0 ( f R , α), F], ∀F ∈ F. Furthermore, by locality the limit is independent
of the smearing test function α, satisfying the normalization condition ˜
α(0) = 1, i.e.
the commutators of [ j 0 ( f R , t), F] are independent of t, in the limit R → ∞.
Hence, the U (1) global gauge group is locally generated by the conserved current
j μ , the assumption of the Goldstone theorem is fulfilled and one has to discuss its
physical consequences.
For this purpose, it is important to remark that the price to pay for locality is
that one has more degrees of freedom than the physical ones (e.g. the “longitudinal
photons”) and the Maxwell equations hold in a weak form (weak Gauss’ law),
195 For the discussion of the physical interpretation of the fields of the quadratic Lagrangian, see
T.W. Kibble, Phys. Rev. 155, 1554 (1966).
196 F. Strocchi, Comm. Math. Phys. 56, 57 (1977).
29 Symmetry Breaking in Gauge Theories
29.2 Higgs Mechanism in Local Gauges
The evasion of the Goldstone theorem by the Higgs mechanism can be understood
by a non-perturbative argument in local (renormalizable) gauges.
For concreteness, we discuss the abelian Higgs–Kibble model in the so-called α
gauges obtained by the addition of the gauge fixing −
1
2
α(∂ μ A
μ
)
2 to the gauge invariant Lagrangian (29.1). Proceeding as before with a perturbative expansion based on
the mean field ansatz ϕ = ϕ + χ 1 + iχ 2 , and performing the change of variables
W μ = A μ − (eϕ)
−1
∂ μ χ 2 , one gets a quadratic Lagrangian of the form (29.3) plus
the gauge fixing term −
1
2
α(χ 2 )
2 .
Thus, χ 2 does not disappear from the quadratic Lagrangian and satisfies a “massless” field equation
2
χ 2 = 0; this means that there are massless modes.
The problem is their physical interpretation; an indication against their physical
relevance is the α dependence of the corresponding Lagrangian term. Moreover, the
general solution of the equation
2
χ 2 = 0 is a massless field (∼ δ(k
2
)) plus a dipole
field (∼ δ
(k
2
)). A δ
(k
2
) singularity is not a measure and therefore is not allowed
to appear in the physical spectrum because the space–time translations must be
described by unitary operator, when restricted to the space of physical vectors where
positivity holds, so that their spectral representation on physical states is given by a
measure.
195
Actually, by exploiting the Gauss law relation j μ = ∂
ν F μν , one can find a general
non-perturbative argument
196 about the unphysical nature of the massless modes
associated with the breaking of the U (1) gauge group in local (renormalizable)
gauges.
As a first step, one remarks that in local renormalizable gauges, like the Feynman
gauge, the field algebra F is generated by the local charged fields ϕ(x) and by
the local vector potential A μ (x), the four components of which are quantized as
independent fields. Locality of the field algebra together with the relativistic spectral
support of the Fourier transforms of the vacuum expectations are the basic properties
shared by such local gauges, so that most of the standard wisdom on quantum field
theory is available; these are in fact the gauges used in perturbation theory.
Thanks to the locality of the field algebra F, there is no problem for the existence of
lim R→∞ [ j 0 ( f R , α), F], ∀F ∈ F. Furthermore, by locality the limit is independent
of the smearing test function α, satisfying the normalization condition ˜
α(0) = 1, i.e.
the commutators of [ j 0 ( f R , t), F] are independent of t, in the limit R → ∞.
Hence, the U (1) global gauge group is locally generated by the conserved current
j μ , the assumption of the Goldstone theorem is fulfilled and one has to discuss its
physical consequences.
For this purpose, it is important to remark that the price to pay for locality is
that one has more degrees of freedom than the physical ones (e.g. the “longitudinal
photons”) and the Maxwell equations hold in a weak form (weak Gauss’ law),
195 For the discussion of the physical interpretation of the fields of the quadratic Lagrangian, see
T.W. Kibble, Phys. Rev. 155, 1554 (1966).
196 F. Strocchi, Comm. Math. Phys. 56, 57 (1977).