Part II: SYMMETRY BREAKING IN QUANTUM SYSTEMS
71
General criteria and non-perturbative constructive approaches to spontaneous
symmetry breaking are briefly discussed in Chaps. 18–20 and applied to simple
examples in Chaps. 21 and 23. In particular the Ising model displays the discrepancy
between the non-perturbative (Ruelle and Bogoliubov) approaches and the perturbative (Goldstone) criterion.
The modification of the general structure for systems at non-zero temperature and
the basic role of the KMS condition is reviewed in Chap. 22 and applied to simple
examples of many-body systems and of quantum fields.
The spontaneous breaking of continuous symmetries and the implication on the
energy spectrum are discussed in detail in Chap. 25. The Goldstone theorem is carefully discussed with a critical analysis of its hypotheses. In particular, the integrability
of the charge density commutators and the localization properties of the dynamics
are argued to be the relevant ingredients for a clear and mathematical control of the
Goldstone theorem for non-relativistic systems. The relation between the range of the
potential and the critical delocalization of the dynamics leading to an evasion of the
Goldstone theorem is worked out in detail beyond the Swieca conjecture. By using
a perturbative expansion in time, the critical decay of the potential for the absence
of massless Goldstone bosons and the occurrence of an energy gap turns out to be
that of the Coulomb potential rather than the one power faster decay predicted by
Swieca condition. Such an analysis clarifies the link between spontaneous symmetry
breaking in non-relativistic Coulomb systems and in (positive) gauge theories (Higgs
phenomenon and U (1) problem in QC D); in particular it explains the occurrence
of massive Goldstone bosons associated to symmetry breaking as a consequence of
a Coulomb-like delocalization induced by the dynamics in both cases.
The non-zero temperature version of the Goldstone theorem is discussed in Chap.
26, with a careful handling of the distributional problems of the zero momentum
limit, which actually gives rise to derivatives of the Dirac delta function. The extension of the Goldstone theorem to the more general case in which the Hamiltonian and
the generators of the symmetry group generate a Lie algebra (non-symmetric Hamiltonians) provides non-perturbative information on the energy gap of the modified
Goldstone spectrum (Chap. 28).
A version of the Goldstone theorem for gauge symmetries in local gauge theories,
which accounts for the absence of physical Goldstone bosons (Higgs mechanism), is
presented in Chap. 29, by exploiting Gauss’ law and an extension of the Goldstone
theorem for relativistic local fields, which does not use positivity.
In conclusion the aim of these lectures is to provide an introduction to the quantum
mechanics of infinitely extended systems and to the fascinating and important subject
of spontaneous symmetry breaking. No pretension of completeness is made about
the subject, which has a vast physical and mathematical literature. Notwithstanding,
the basic mechanism of spontaneous symmetry breaking, apart from the popular
accounts which do not convey the relevant mathematical structures, does not seem
to be part of the common education of theoretical physics students.
The background knowledge required is reduced to the basic elements of the theory
of Hilbert space operators and to the foundations of ordinary quantum mechanics.
The presentation does not indulge in the mathematical details, while respecting the
71
General criteria and non-perturbative constructive approaches to spontaneous
symmetry breaking are briefly discussed in Chaps. 18–20 and applied to simple
examples in Chaps. 21 and 23. In particular the Ising model displays the discrepancy
between the non-perturbative (Ruelle and Bogoliubov) approaches and the perturbative (Goldstone) criterion.
The modification of the general structure for systems at non-zero temperature and
the basic role of the KMS condition is reviewed in Chap. 22 and applied to simple
examples of many-body systems and of quantum fields.
The spontaneous breaking of continuous symmetries and the implication on the
energy spectrum are discussed in detail in Chap. 25. The Goldstone theorem is carefully discussed with a critical analysis of its hypotheses. In particular, the integrability
of the charge density commutators and the localization properties of the dynamics
are argued to be the relevant ingredients for a clear and mathematical control of the
Goldstone theorem for non-relativistic systems. The relation between the range of the
potential and the critical delocalization of the dynamics leading to an evasion of the
Goldstone theorem is worked out in detail beyond the Swieca conjecture. By using
a perturbative expansion in time, the critical decay of the potential for the absence
of massless Goldstone bosons and the occurrence of an energy gap turns out to be
that of the Coulomb potential rather than the one power faster decay predicted by
Swieca condition. Such an analysis clarifies the link between spontaneous symmetry
breaking in non-relativistic Coulomb systems and in (positive) gauge theories (Higgs
phenomenon and U (1) problem in QC D); in particular it explains the occurrence
of massive Goldstone bosons associated to symmetry breaking as a consequence of
a Coulomb-like delocalization induced by the dynamics in both cases.
The non-zero temperature version of the Goldstone theorem is discussed in Chap.
26, with a careful handling of the distributional problems of the zero momentum
limit, which actually gives rise to derivatives of the Dirac delta function. The extension of the Goldstone theorem to the more general case in which the Hamiltonian and
the generators of the symmetry group generate a Lie algebra (non-symmetric Hamiltonians) provides non-perturbative information on the energy gap of the modified
Goldstone spectrum (Chap. 28).
A version of the Goldstone theorem for gauge symmetries in local gauge theories,
which accounts for the absence of physical Goldstone bosons (Higgs mechanism), is
presented in Chap. 29, by exploiting Gauss’ law and an extension of the Goldstone
theorem for relativistic local fields, which does not use positivity.
In conclusion the aim of these lectures is to provide an introduction to the quantum
mechanics of infinitely extended systems and to the fascinating and important subject
of spontaneous symmetry breaking. No pretension of completeness is made about
the subject, which has a vast physical and mathematical literature. Notwithstanding,
the basic mechanism of spontaneous symmetry breaking, apart from the popular
accounts which do not convey the relevant mathematical structures, does not seem
to be part of the common education of theoretical physics students.
The background knowledge required is reduced to the basic elements of the theory
of Hilbert space operators and to the foundations of ordinary quantum mechanics.
The presentation does not indulge in the mathematical details, while respecting the