Symmetry Breaking

1. Front Matter

   Pages I-X

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2. Symmetry Breaking in Classical Systems

   1. Front Matter

      Pages 1-6

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   2. Symmetries of a Classical System

      • Franco Strocchi

      Pages 7-8

   3. Spontaneous Symmetry Breaking

      • Franco Strocchi

      Pages 9-11

   4. Symmetries in Classical Field Theory

      • Franco Strocchi

      Pages 13-16

   5. General Properties of Solutions of Classical Field Equations

      • Franco Strocchi

      Pages 17-20

   6. Stable Structures, Hilbert Sectors, Phases

      • Franco Strocchi

      Pages 21-28

   7. Stability under Space Translations. Positive Energy

      • Franco Strocchi

      Pages 29-32

   8. Noether Theorem and Symmetry Breaking

      • Franco Strocchi

      Pages 33-37

   9. Examples

      • Franco Strocchi

      Pages 39-43

   10. The Goldstone Theorem

       • Franco Strocchi

       Pages 45-49

   11. Appendix

       • Franco Strocchi

       Pages 51-60

3. Symmetry Breaking in Quantum Systems

   1. Front Matter

      Pages 61-66

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   2. Quantum Mechanics. Algebraic Structure and States

      • Franco Strocchi

      Pages 67-71

   3. Fock Representation

      • Franco Strocchi

      Pages 73-79

   4. Non-Fock Representations

      • Franco Strocchi

      Pages 81-87

   5. Mathematical Description of Infinitely Extended Quantum Systems

      • Franco Strocchi

      Pages 89-93

   6. Physically Relevant Representations

      • Franco Strocchi

      Pages 95-98

   7. Cluster Property and Pure Phases

      • Franco Strocchi

      Pages 99-103

   8. Examples

      • Franco Strocchi

      Pages 105-113

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).

• Algebraic structure
• Gauge theory
• Goldstone theorem
• Higgs mechanism
• Particle Physics
• Theoretical physics
• mathematical physics
• quantum mechanics
• spontaneous symmetry breaking

From the reviews of the second edition:

"This second edition of Strocchi’s Symmetry breaking presents a complete, generalized and highly rigorous discussion of the subject, based on a formal analysis of conditions necessary for the mechanism of spontaneous symmetry breaking to occur in classical systems, as well as in quantum systems. … this book is specifically recommended for mathematical physicists interested in a deeper and rigorous understanding of the subject, and it should be mandatory for researchers studying the mechanism of spontaneous symmetry breaking." (Serena Hajjawi, Mathematical Reviews, 2008)

"The monograph under review is devoted to the mathematical understanding of spontaneous symmetry breaking (SSB) … . The monograph gives a strong mathematical base to the SSB phenomena in applied mathematics and physics. Therefore it will be very useful for mathematicians and physicists, students of relevant specialties." (Boris V. Loginov, Zentralblatt MATH, Vol. 1145, 2008)

• Classe di Scienze, Scuola Normale Superiore, Pisa, Italy

  Franco Strocchi

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