172
25 Breaking of Continuous Symmetries: Goldstone’s Theorem
This is incompatible with an energy gap at k → 0.
148
The above standard (heuristic) argument would completely settle the statement of
Goldstone’s theorem (apart from somewhat pedantic mathematical polishing) were
it not for the existence of physically interesting models which seem to evade the
conclusions of the theorem. The attention on these examples arose especially in the
early 1960s in connection with attempts to interpret the SU (3) eightfold way as a
spontaneously broken symmetry, notwithstanding the absence of the corresponding
Goldstone bosons. Among such examples we mention the BCS model of superconductivity, where the U (1) internal symmetry is spontaneously broken with energy
gap, the breaking of the Galilei symmetry in Coulomb systems, which is accompanied by the plasmon energy gap, the Higgs mechanism in the Coulomb gauge and
the breaking of the axial U (1) symmetry in quantum chromodynamics (QCD) (the
so-called U (1) problem), both with no corresponding massless Goldstone bosons,
etc.
Clearly, in such examples some of the assumptions of the theorem must fail,
but the long discussions on the possible mechanisms for evading the conclusions
of the theorem seem to have led more to a series of catchwords or perturbative
prescriptions, rather than to a sharp and clear identification of the crucial points. For
non-relativistic systems, the standard explanation for the presence of an energy gap
is that the Coulomb potential leads to a shift of energy (at k → 0), by a mechanism
advocated on the basis of clever ad hoc approximations, rather than in terms of
a general non-perturbative mechanism. The problem with such an explanation is
that long range correlations and interactions, which always occur when there are
massless particles, do not invalidate the applicability of the theorem in relativistic
local quantum field theory. The standard explanation of the Higgs mechanism relies
on the perturbative expansion, and for the U (1) problem the standard explanation,
in terms of the chiral anomaly and instanton calculations, does not seem to provide
a general clear-cut solution and some questions remain open.
149
The above considerations justify a critical analysis of the hypotheses of the theorem and their verification. As we shall see, the standard explanations of the “evasion”
of the theorem are somewhat incomplete, if not misleading, since they seem to overlook the basic delicate points and miss the general mechanism.
148 In fact, if |k, ω(k) l , l > denote the improper eigenstates of momentum and energy, with l the
additional quantum numbers needed to remove possible degeneracies, then lim k→0 ω l (k) ≥ μ > 0
implies that
lim
k→0
˜
J (k, ω) = lim
k→0
4π
2 2 Im
l
< AΨ 0 |k, ω l , l >< k, ω l , l| j 0 (0, 0)Ψ 0 >
cannot satisfy (25.9).
149 For a critical discussion, see F. Strocchi, Selected Topics on the General Properties of Quantum
Field theory, World Scientific (1993), Sect. 7.4 iv. See also Appendices E, F, below.
25 Breaking of Continuous Symmetries: Goldstone’s Theorem
This is incompatible with an energy gap at k → 0.
148
The above standard (heuristic) argument would completely settle the statement of
Goldstone’s theorem (apart from somewhat pedantic mathematical polishing) were
it not for the existence of physically interesting models which seem to evade the
conclusions of the theorem. The attention on these examples arose especially in the
early 1960s in connection with attempts to interpret the SU (3) eightfold way as a
spontaneously broken symmetry, notwithstanding the absence of the corresponding
Goldstone bosons. Among such examples we mention the BCS model of superconductivity, where the U (1) internal symmetry is spontaneously broken with energy
gap, the breaking of the Galilei symmetry in Coulomb systems, which is accompanied by the plasmon energy gap, the Higgs mechanism in the Coulomb gauge and
the breaking of the axial U (1) symmetry in quantum chromodynamics (QCD) (the
so-called U (1) problem), both with no corresponding massless Goldstone bosons,
etc.
Clearly, in such examples some of the assumptions of the theorem must fail,
but the long discussions on the possible mechanisms for evading the conclusions
of the theorem seem to have led more to a series of catchwords or perturbative
prescriptions, rather than to a sharp and clear identification of the crucial points. For
non-relativistic systems, the standard explanation for the presence of an energy gap
is that the Coulomb potential leads to a shift of energy (at k → 0), by a mechanism
advocated on the basis of clever ad hoc approximations, rather than in terms of
a general non-perturbative mechanism. The problem with such an explanation is
that long range correlations and interactions, which always occur when there are
massless particles, do not invalidate the applicability of the theorem in relativistic
local quantum field theory. The standard explanation of the Higgs mechanism relies
on the perturbative expansion, and for the U (1) problem the standard explanation,
in terms of the chiral anomaly and instanton calculations, does not seem to provide
a general clear-cut solution and some questions remain open.
149
The above considerations justify a critical analysis of the hypotheses of the theorem and their verification. As we shall see, the standard explanations of the “evasion”
of the theorem are somewhat incomplete, if not misleading, since they seem to overlook the basic delicate points and miss the general mechanism.
148 In fact, if |k, ω(k) l , l > denote the improper eigenstates of momentum and energy, with l the
additional quantum numbers needed to remove possible degeneracies, then lim k→0 ω l (k) ≥ μ > 0
implies that
lim
k→0
˜
J (k, ω) = lim
k→0
4π
2 2 Im
l
< AΨ 0 |k, ω l , l >< k, ω l , l| j 0 (0, 0)Ψ 0 >
cannot satisfy (25.9).
149 For a critical discussion, see F. Strocchi, Selected Topics on the General Properties of Quantum
Field theory, World Scientific (1993), Sect. 7.4 iv. See also Appendices E, F, below.