iii) The classical counterpart of the Goldstone theorem is proved in Chap. 9,
which improves and partly corrects the heuristic perturbative arguments
of the literature.
The presentation emphasizes the general ideas (implemented in explicit examples) without indulging on the technical details, but also without derogating from
the mathematical soundness of the statements.
The second Part on “SYMMETRY BREAKING IN QUANTUM SYSTEMS”
tries to offer a presentation of the subject, which should be more mathematically
sounded and convincing than the popular accounts, but not too technical. The first
chapters are devoted to the general structures which arise in the quantum
description of infinitely extended systems with emphasis on the physical basis of
locality, asymptotic abelianess and cluster property and their mutual relations,
leading to a characterization of the pure phases.
Criteria of spontaneous symmetry breaking are discussed in Chap. 20 along
the lines of Wightman lectures at Coral Gables and their effectiveness and differences are explicitly worked ot and checked in the Ising model. The Bogoliubov
strategy is shown to provide a simple rigorous control of spontaneous symmetry
breaking in the free Bose gas as a possible alternative to Cannon and
Bratelli-Robinson treatment.
The Goldstone theorem is critically discussed in Chap. 25, especially for
non-relativistic systems or more generally for systems with long range delocalization. Such analysis, which does not seem to appear in textbooks, provides a
non-perturbative explanation of symmetry breaking with energy gap in
non-relativistic Coulomb systems and in the Higgs phenomenon and in our
opinion puts in a more convincing and rigorous perspective the analogies proposed
by Anderson. The Swieca conjecture about the role of the potential fall off is
checked by a perturbative expansion in time. Such an expansion also supports the
condition of integrability of the charge density commutators, which seems to be
overlooked in the standard treatments and plays a crucial role for the energy
spectrum of the Goldstone bosons. As a result of such an explicit analysis, the
critical decay of the potential for allowing “massive” Goldstone bosons turns out to
be that of the Coulomb potential, rather than the one power faster decay predicted
by Swieca condition.
The non-zero temperature version of the Goldstone theorem, discussed in
Chap. 26, corrects some wrong conclusions of the literature. An extension of the
Goldstone theorem to non-symmetric Hamiltonians is discussed in Chap. 28 with
the derivation of non-trivial (non-perturbative) information on the energy gap of the
modified Goldstone spectrum.
The symmetry breaking in gauge theories, in particular the Higgs phenomenon
which is at the basis of the standard model of elementary particles, is analyzed in
Chap. 29. The problems of the perturbative explanation of the evasion of the
Goldstone theorem are pointed out and a non-perturbative account is presented. In
the local renormalizable gauges, the absence of physical Goldstone bosons follows
from the Gauss law constraint or subsidiary condition on the physical states. In the
viii
Preface to Previous Edition
which improves and partly corrects the heuristic perturbative arguments
of the literature.
The presentation emphasizes the general ideas (implemented in explicit examples) without indulging on the technical details, but also without derogating from
the mathematical soundness of the statements.
The second Part on “SYMMETRY BREAKING IN QUANTUM SYSTEMS”
tries to offer a presentation of the subject, which should be more mathematically
sounded and convincing than the popular accounts, but not too technical. The first
chapters are devoted to the general structures which arise in the quantum
description of infinitely extended systems with emphasis on the physical basis of
locality, asymptotic abelianess and cluster property and their mutual relations,
leading to a characterization of the pure phases.
Criteria of spontaneous symmetry breaking are discussed in Chap. 20 along
the lines of Wightman lectures at Coral Gables and their effectiveness and differences are explicitly worked ot and checked in the Ising model. The Bogoliubov
strategy is shown to provide a simple rigorous control of spontaneous symmetry
breaking in the free Bose gas as a possible alternative to Cannon and
Bratelli-Robinson treatment.
The Goldstone theorem is critically discussed in Chap. 25, especially for
non-relativistic systems or more generally for systems with long range delocalization. Such analysis, which does not seem to appear in textbooks, provides a
non-perturbative explanation of symmetry breaking with energy gap in
non-relativistic Coulomb systems and in the Higgs phenomenon and in our
opinion puts in a more convincing and rigorous perspective the analogies proposed
by Anderson. The Swieca conjecture about the role of the potential fall off is
checked by a perturbative expansion in time. Such an expansion also supports the
condition of integrability of the charge density commutators, which seems to be
overlooked in the standard treatments and plays a crucial role for the energy
spectrum of the Goldstone bosons. As a result of such an explicit analysis, the
critical decay of the potential for allowing “massive” Goldstone bosons turns out to
be that of the Coulomb potential, rather than the one power faster decay predicted
by Swieca condition.
The non-zero temperature version of the Goldstone theorem, discussed in
Chap. 26, corrects some wrong conclusions of the literature. An extension of the
Goldstone theorem to non-symmetric Hamiltonians is discussed in Chap. 28 with
the derivation of non-trivial (non-perturbative) information on the energy gap of the
modified Goldstone spectrum.
The symmetry breaking in gauge theories, in particular the Higgs phenomenon
which is at the basis of the standard model of elementary particles, is analyzed in
Chap. 29. The problems of the perturbative explanation of the evasion of the
Goldstone theorem are pointed out and a non-perturbative account is presented. In
the local renormalizable gauges, the absence of physical Goldstone bosons follows
from the Gauss law constraint or subsidiary condition on the physical states. In the
viii
Preface to Previous Edition