Symmetry Breaking

Franco Strocchi (auth.)9783540735922, 3540735925

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

Table of contents :
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