Preface to Previous Edition
The main motivation for such lecture notes is the importance of the concept and
mechanism of spontaneous symmetry breaking in modern theoretical physics and
the relevance of a textbook exposition at the graduate student level beyond the
oversimplified (non-rigorous) treatments, often confined to specific models. One
of the main points is to emphasize that the radical loss of symmetric behaviour
requires both the existence of non-symmetric ground states and the infinite
extension of the system.
The first Part on SYMMETRY BREAKING IN CLASSICAL SYSTEMS is
devoted to the mathematical understanding of spontaneous symmetry breaking on
the basis of classical field theory. The main points, which do not seem to appear in
textbooks, are the following.
i) Existence of disjoint Hilbert space sectors, stable under time evolution, in
the set of solutions of the classical (non-linear) field equations. They are the
strict analogs of the different phases of statistical mechanical systems and/or
of the inequivalent representations of local field algebras in quantum field
theory (QFT). As in QFT, such structures rely on the concepts of locality (or
localization) and stability, (see Chap. 5), with emphasis on the physical
motivations of the mathematical concepts; such structures have the physical
meaning of disjoint physical worlds, disjoint phases etc. which can be
associated to a given non-linear field equation. The result of Theorem 5.2
may be regarded as a generalization of the criterium of stability to infinite
dimensional systems and it links such stability to elliptic problems in R
n with
non-trivial boundary conditions at infinity (Appendix 10.5).
ii) Such structures allow to reconcile the classical Noether theorem with
spontaneous symmetry breaking, through a discussion of a mechanism
which accounts for (and explains) the breaking of the symmetry group (of the
equations of motion), in a given Hilbert space sector H, down to the subgroup which leaves H stable (Theorem 7.2).
vii
The main motivation for such lecture notes is the importance of the concept and
mechanism of spontaneous symmetry breaking in modern theoretical physics and
the relevance of a textbook exposition at the graduate student level beyond the
oversimplified (non-rigorous) treatments, often confined to specific models. One
of the main points is to emphasize that the radical loss of symmetric behaviour
requires both the existence of non-symmetric ground states and the infinite
extension of the system.
The first Part on SYMMETRY BREAKING IN CLASSICAL SYSTEMS is
devoted to the mathematical understanding of spontaneous symmetry breaking on
the basis of classical field theory. The main points, which do not seem to appear in
textbooks, are the following.
i) Existence of disjoint Hilbert space sectors, stable under time evolution, in
the set of solutions of the classical (non-linear) field equations. They are the
strict analogs of the different phases of statistical mechanical systems and/or
of the inequivalent representations of local field algebras in quantum field
theory (QFT). As in QFT, such structures rely on the concepts of locality (or
localization) and stability, (see Chap. 5), with emphasis on the physical
motivations of the mathematical concepts; such structures have the physical
meaning of disjoint physical worlds, disjoint phases etc. which can be
associated to a given non-linear field equation. The result of Theorem 5.2
may be regarded as a generalization of the criterium of stability to infinite
dimensional systems and it links such stability to elliptic problems in R
n with
non-trivial boundary conditions at infinity (Appendix 10.5).
ii) Such structures allow to reconcile the classical Noether theorem with
spontaneous symmetry breaking, through a discussion of a mechanism
which accounts for (and explains) the breaking of the symmetry group (of the
equations of motion), in a given Hilbert space sector H, down to the subgroup which leaves H stable (Theorem 7.2).
vii