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 oversimpli? ed (non-rigorous) treatments, often con? ned to speci? c models. One of the main points is to emphasize that the radical loss of symmetric behaviour requiresboth the existence of non-symmetric ground states and the in? nite extension of the system. The ? rst Part on SYMMETRY BREAKING IN CLASSICAL SYSTEMS is devoted to the mathematical understanding of spontaneous symmetry breaking on the basis of classical ? eld theory. The main points, which do not seem to appear in textbooks, are the following. i) ExistenceofdisjointHilbertspacesectors, stable under time e- lution in the set of solutions of the classical (non-linear) ? eld equations. Theyarethestrictanalogsofthedi? erentphasesofstatisticalmechanical systems and/or of the inequivalent representations of local ? eld algebras in quantum ? eld theory (QFT). As in QFT, such structures rely on the concepts of locality (or localization) and stability, (see Chap. 5), with emphasis on the physicalmotivations of the mathematicalconcepts; such structures have the physical meaning of disjoint physical worlds, disjoint phases etc. which can be associated to a given non-linear ? eld equation. The result of Theorem 5. 2 may be regarded as a generalization of the criterium of stability to in? nite dimensional systems and it links such n stability to elliptic problems inR with non-trivial boundary conditions at in? nity (Appendix E).
Inhaltsverzeichnis
Symmetry Breaking in Classical Systems. - Symmetries of a Classical System. - Spontaneous Symmetry Breaking. - Symmetries in Classical Field Theory. - General Properties of Solutions of Classical Field Equations. - Stable Structures, Hilbert Sectors, Phases. - Stability under Space Translations. Positive Energy. - Noether Theorem and Symmetry Breaking. - Examples. - The Goldstone Theorem. - Symmetry Breaking in Quantum Systems. - Quantum Mechanics. Algebraic Structure and States. - Fock Representation. - Non-Fock Representations. - Mathematical Description of Infinitely Extended Quantum Systems. - Physically Relevant Representations. - Cluster Property and Pure Phases. - Examples. - Symmetry Breaking in Quantum Systems. - Examples. - Constructive Symmetry Breaking. - Symmetry Breaking in the Ising Model. - Thermal States. - Fermi and Bose Gas at Non-zero Temperature. - Quantum Fields at Non-zero Temperature. - Breaking of Continuous Symmetries. Goldstone s Theorem. - The Goldstone Theorem at Non-zero Temperature. - The Goldstone Theorem for Relativistic Local Fields. - An Extension of Goldstone Theorem to Non-symmetric Hamiltonians. - Symmetry Breaking in Gauge Theories. - Erratum. - Erratum.
