Symmetry Breaking (Lecture Notes in Physics, 732)

This new edition of Prof. Strocchi's well received primer on rigorous aspects of symmetry breaking presents a more detailed and thorough discussion of the mechanism of symmetry breaking in classical field theory in relation with the Noether theorem. Moreover, the link between symmetry breaking without massless Goldstone bosons in Coulomb systems and in gauge theories is made more explicit in terms of the delocalized Coulomb dynamics. Furthermore, the chapter on the Higgs mechanism has been significantly expanded with a non-perturbative treatment of the Higgs phenomenon, at the basis of the standard model of particle physics, in the local and in the Coulomb gauges. Last but not least, a subject index has been added and a number of misprints have been corrected. From the reviews of the first edition: The notion of spontaneous symmetry breaking has proven extremely valuable, the problem is that most derivations are perturbative and heuristic. Yet mathematically precise versions do exist, but are not widely known. It is precisely the aim of his book to correct this unbalance. - It is remarkable to see how much material can actually be presented in a rigorous way (incidentally, many of the results presented are due to Strocchi himself), yet this is largely ignored, the original heuristic derivations being, as a rule, more popular. - At each step he strongly emphasizes the physical meaning and motivation of the various notions introduced [...] a book that fills a conspicuous gap in the literature, and does it rather well. It could also be a good basis for a graduate course in mathematical physics. It can be recommended to physicists as well and, of course, for physics/mathematics libraries. __J.-P. Antoine, Physicalia 28/2, 2006__ Strocchi's main emphasis is on the fact that the loss of symmetric behaviour requires both the non-symmetric ground states and the infinite extension of the system. It is written in a pleasant style at a level suitable for graduate students in theoretical physics. While mathematically proper, it is not forbidding for a physics readership; the author is always aware this subject is a branch of physics. It should make profitable reading for many theoretical physicists. __L.H. Ryder, J. Phys. A, 38 (2005) 9719-9730__In these notes the author explores the phenomenon of spontaneous symmetry breaking as it arises in classical and quantum systems. Despite many accounts in popular textbooks and a widespread belief, the phenomenon is rather subtle, requires an infinite set of degrees of freedom and an advanced mathematical setting of the system under investigation. [...] The mathematically oriented graduate student will certainly benefit from this thorough, rigorous and detailed investigation. __G. Roepstorff, Zentralblatt MATH, Vol. 1075, 2006__ This new edition of Prof. Strocchi's well received primer on rigorous aspects of symmetry breaking presents a more detailed and thorough discussion of the mechanism of symmetry breaking in classical field theory in relation with the Noether theorem. Moreover, the link between symmetry breaking without massless Goldstone bosons in Coulomb systems and in gauge theories is made more explicit in terms of the delocalized Coulomb dynamics. Furthermore, the chapter on the Higgs mechanism has been significantly expanded with a non-perturbative treatment of the Higgs phenomenon, at the basis of the standard model of particle physics, in the local and in the Coulomb gauges. Last but not least, a subject index has been added and a number of misprints have been corrected. From the reviews of the first edition: The notion of spontaneous symmetry breaking has proven extremely valuable, the problem is that most derivations are perturbative and heuristic. Yet mathematically precise versions do exist, but are not widely known. It is precisely the aim of his book to correct this unbalance. - It is remarkable to see how much material can actually be presented in a rigorous way (incidentally, many of the results presented are due to Strocchi himself), yet this is largely ignored, the original heuristic derivations being, as a rule, more popular. - At each step he strongly emphasizes the physical meaning and motivation of the various notions introduced [...] a book that fills a conspicuous gap in the literature, and does it rather well. It could also be a good basis for a graduate course in mathematical physics. It can be recommended to physicists as well and, of course, for physics/mathematics libraries. J.-P. Antoine, Physicalia 28/2, 2006 Strocchi's main emphasis is on the fact that the loss of symmetric behaviour requires both the non-symmetric ground states and the infinite extension of the system. It is written in a pleasant style at a level suitable for graduate students in theoretical physics. While mathematically proper, it is not forbidding for a physics readership; the author is always aware this subject is a branch of physics. It should make profitable reading for many theoretical physicists. L.H. Ryder, J. Phys. A, 38 (2005) 9719-9730 In these notes the author explores the phenomenon of spontaneous symmetry breaking as it arises in classical and quantum systems. Despite many accounts in popular textbooks and a widespread belief, the phenomenon is rather subtle, requires an infinite set of degrees of freedom and an advanced mathematical setting of the system under investigation. [...] The mathematically oriented graduate student will certainly benefit from this thorough, rigorous and detailed investigation. G. Roepstorff, Zentralblatt MATH, Vol. 1075, 2006 Front Matter....Pages I-X Front Matter....Pages 1-6 Symmetries of a Classical System....Pages 7-8 Spontaneous Symmetry Breaking....Pages 9-11 Symmetries in Classical Field Theory....Pages 13-16 General Properties of Solutions of Classical Field Equations....Pages 17-20 Stable Structures, Hilbert Sectors, Phases....Pages 21-28 Stability under Space Translations. Positive Energy....Pages 29-32 Noether Theorem and Symmetry Breaking....Pages 33-37 Examples....Pages 39-43 The Goldstone Theorem....Pages 45-49 Appendix....Pages 51-60 Front Matter....Pages 61-66 Quantum Mechanics. Algebraic Structure and States....Pages 67-71 Fock Representation....Pages 73-79 Non-Fock Representations....Pages 81-87 Mathematical Description of Infinitely Extended Quantum Systems....Pages 89-93 Physically Relevant Representations....Pages 95-98 Cluster Property and Pure Phases....Pages 99-103 Examples....Pages 105-113 Symmetry Breaking in Quantum Systems....Pages 115-122 Examples....Pages 123-125 Constructive Symmetry Breaking....Pages 127-130 Front Matter....Pages 61-66 Symmetry Breaking in the Ising Model....Pages 131-138 Thermal States....Pages 139-150 Fermi and Bose Gas at Non-zero Temperature....Pages 151-157 Quantum Fields at Non-zero Temperature....Pages 159-160 Breaking of Continuous Symmetries. Goldstone’s Theorem....Pages 161-176 The Goldstone Theorem at Non-zero Temperature....Pages 177-179 The Goldstone Theorem for Relativistic Local Fields....Pages 181-188 An Extension of Goldstone Theorem to Non-symmetric Hamiltonians....Pages 189-192 Symmetry Breaking in Gauge Theories....Pages 193-206 Erratum....Pages 217-217 Back Matter....Pages 207-216 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). The new edition of this well received primer on rigorous aspects of symmetry breaking presents a more detailed and thorough discussion of the mechanism of symmetry breaking in classical field theory in relation with the Noether theorem. Moreover, the link between symmetry breaking without massless Goldstone bosons in Coulomb systems and in gauge theories is made more explicit. A subject index has been added and a number of misprints have been corrected.