Symmetry Breaking (Lecture Notes in Physics)

Spontaneous symmetry breaking is a revolutionary new idea of thinking in terms of symmetry and the corresponding properties of physical systems. The first part of this book develops the mathematical understanding of spontaneous symmetry breaking on the basis of classical field theory. The author addresses the existence of sectors, stability, and presents an improved Noether theorem, as well as the classical counterpart of the Goldstone theorem. The second part deals with quantum systems. Here criteria for spontaneous symmetry breaking are discussed in detail as well as Goldstone's theorem, gauge theories are included, and the theorem on the Higgs mechanism is presented. The author keeps the mathematical details to the minimum required to make the book accessible to students with basic knowledge of Hilbert space structures. Much of the material appears here for the first time in book form. Introduction to Part I....Pages 3-6 1 Symmetries of a Classical System....Pages 7-8 2 Spontaneous Symmetry Breaking....Pages 9-11 3 Symmetries in Classical Field Theory....Pages 13-16 4 General Properties of Solutions of Classical Field Equations....Pages 17-20 5 Stable Structures, Hilbert Sectors, Phases....Pages 21-28 6 Sectors with Energy-Momentum Density....Pages 29-31 7 An Improved Noether Theorem. Spontaneous Symmetry Breaking....Pages 33-37 8 Examples....Pages 39-43 9 The Goldstone Theorem....Pages 45-49 10 Appendix....Pages 51-60 Introduction to Part II....Pages 63-66 1 Quantum Mechanics. Algebraic Structure and States....Pages 67-71 2 Fock Representation....Pages 73-79 3 Non-Fock Representations....Pages 81-87 4 Mathematical Description of Infinitely Extended Systems....Pages 89-93 5 Physically Relevant Representations....Pages 95-98 6 Cluster Property and Pure Phases....Pages 99-103 7 Examples....Pages 105-113 8 Symmetry Breaking in Quantum Systems....Pages 115-122 9 Examples....Pages 123-125 10 Constructive Symmetry Breaking....Pages 127-130 11 Symmetry Breaking in the Ising Mode....Pages 131-138 12 * Thermal States....Pages 139-150 13 Fermi and Bose Gas at Non-zero Temperature....Pages 151-157 14 Quantum Fields at Non-zero Temperature....Pages 159-160 15 Breaking of Continuous Symmetries. Goldstone’s Theorem....Pages 161-176 16 * The Goldstone Theorem at Non-zero Temperature....Pages 177-179 17 The Goldstone Theorem for Relativistic Local Fields....Pages 181-188 18 An Extension of Goldstone Theorem to Non-symmetric Hamiltonians....Pages 189-192 19 The Higgs Mechanism: A Theorem....Pages 193-196 The Intriguing Mechanism Of Spontaneous Symmetry Breaking Is A Powerful Innovative Idea At The Basis Of Most Of The Recent Developments In Theoretical Physics, From Statistical Mechanics To Many-body Theory To Elementary Particles Theory; For Infinitely Extended Systems A Symmetric Hamiltonian Can Account For Non Symmetric Behaviours, Giving Rise To Non Symmetric Realizations Of A Physical System. In The First Part Of This Book, Devoted To Classical Field Theory, Such A Mechanism Is Explained In Terms Of The Occurrence Of Disjoint Sectors And Their Stability Properties And Of An Improved Version Of The Noether Theorem. For Infinitely Extended Quantum Systems, Discussed In The Second Part, The Mechanism Is Related To The Occurrence Of Disjoint Pure Phases And Characterized By A Symmetry Breaking Order Parameter, For Which Non Perturbative Criteria Are Discussed, Following Wightman, And Contrasted With The Standard Goldstone Perturbative Strategy. The Goldstone Theorem Is Discussed With A Critical Look At The Hypotheses That Emphasizes The Crucial Role Of The Dynamical Delocalization Induced By The Interaction Range. The Higgs Mechanism In Local Gauges Is Explained In Terms Of The Gauss Law Constraint On The Physical States. The Mathematical Details Are Kept To The Minimum Required To Make The Book Accessible To Students With Basic Knowledge Of Hilbert Space Structures. Much Of The Material Has Not Appeared In Other Textbooks. Pt. I. Symmetry Breaking In Classical Systems -- 1. Symmetries Of A Classical System -- 2. Spontaneous Symmetry Breaking -- 3. Symmetries In Classical Field Theory -- 4. General Properties Of Solutions Of Classical Field Equations -- 5. Stable Structures, Hilbert Sectors, Phases -- 6. Sectors With Energy-momentum Density -- 7. An Improved Noether Theorem : Spontaneous Symmetry Breaking -- 8. Examples -- 9. The Goldstone Theorem -- 10. Appendix -- Pt. Ii. Symmetry Breaking In Quantum Systems -- 1. Quantum Mechanics : Algebraic Structure And States -- 2. Fock Representation -- 3. Non-fock Representations -- 4. Mathematical Description Of Infinitely Extended Systems -- 5. Physically Relevant Representations -- 6. Cluster Property And Pure Phases -- 7. Examples -- 8. Symmetry Breaking In Quantum Systems -- 9. Examples -- 10. Constructive Symmetry Breaking -- 11. Symmetry Breaking In The Ising Model -- 12. Thermal States -- 13. Fermi And Bose Gas At Non-zero Temperature -- 14. Qauntum Fields At Non-zero Temperature -- 15. Breaking Of Continuous Symmetries : Goldstone's Theorem -- 16. The Goldstone Theorem At Non-zero Temperature -- 17. The Goldstone Theorem For Relativistic Local Fields -- 18. An Extension Of Goldstone Theorem To Non-symmetric Hamiltonians -- 19. The Higgs Mechanism : A Theorem. Franco Strocchi. Includes Bibliographical References (p. [197]-203).