Quantum Field Theory (QFT) has proved to be the most useful strategy for the description of elementary particle interactions and as such is regarded as a fundamental part of modern theoretical physics. In most presentations, the emphasis is on the effectiveness of the theory in producing experimentally testable predictions, which at present essentially means Perturbative QFT. However, after more than fifty years of QFT, we still are in the embarrassing situation of not knowing a single non-trivial (even non-realistic) model of QFT in 3+1 dimensions, allowing a non-perturbative control. As a reaction to these consistency problems one may take the position that they are related to our ignorance of the physics of small distances and that QFT is only an effective theory, so that radically new ideas are needed for a consistent quantum theory of relativistic interactions (in 3+1 dimensions).The book starts by discussing the conflict between locality or hyperbolicity and positivity of the energy for relativistic wave equations, which marks the origin of quantum field theory, and the mathematical problems of the perturbative expansion (canonical quantization, interaction picture, non-Fock representation, asymptotic convergence of the series etc.). The general physical principles of positivity of the energy, Poincare' covariance and locality provide a substitute for canonical quantization, qualify the non-perturbative foundation and lead to very relevant results, like the Spin-statistics theorem, TCP symmetry, a substitute for canonical quantization, non-canonical behaviour, the euclidean formulation at the basis of the functional integral approach, the non-perturbative definition of the S-matrix (LSZ, Haag-Ruelle-Buchholz theory).A characteristic feature of gauge field theories is Gauss' law constraint. It is responsible for the conflict between locality of the charged fields and positivity, it yields the superselection of the (unbroken) gauge charges, provides a non-perturbative explanation of the Higgs mechanism in the local gauges, implies the infraparticle structure of the charged particles in QED and the breaking of the Lorentz group in the charged sectors.A non-perturbative proof of the Higgs mechanism is discussed in the Coulomb gauge: the vector bosons corresponding to the broken generators are massive and their two point function dominates the Goldstone spectrum, thus excluding the occurrence of massless Goldstone bosons.The solution of the U(1) problem in QCD, the theta vacuum structure and the inevitable breaking of the chiral symmetry in each theta sector are derived solely from the topology of the gauge group, without relying on the semiclassical instanton approximation.

An Introduction to Non-Perturbative Foundations of Quantum Field Theory (International Series of Monographs on Physics)

Science & Math

Strocchi, Franco

Physics

Oxford University Press

Quantum Field Theory (QFT) has proved to be the most useful strategy for the description of elementary particle interactions and as such is regarded as a fundamental part of modern theoretical physics. In most presentations, the emphasis is on the effectiveness of the theory in producing experimentally testable predictions, which at present essentially means Perturbative QFT. However, after more than fifty years of QFT, we still are in the embarrassing situation of not knowing a single non-trivial (even non-realistic) model of QFT in 3+1 dimensions, allowing a non-perturbative control. As a reaction to these consistency problems one may take the position that they are related to our ignorance of the physics of small distances and that QFT is only an effective theory, so that radically new ideas are needed for a consistent quantum theory of relativistic interactions (in 3+1 dimensions). The book starts by discussing the conflict between locality or hyperbolicity and positivity of the energy for relativistic wave equations, which marks the origin of quantum field theory, and the mathematical problems of the perturbative expansion (canonical quantization, interaction picture, non-Fock representation, asymptotic convergence of the series etc.). The general physical principles of positivity of the energy, Poincare' covariance and locality provide a substitute for canonical quantization, qualify the non-perturbative foundation and lead to very relevant results, like the Spin-statistics theorem, TCP symmetry, a substitute for canonical quantization, non-canonical behaviour, the euclidean formulation at the basis of the functional integral approach, the non-perturbative definition of the S-matrix (LSZ, Haag-Ruelle-Buchholz theory). A characteristic feature of gauge field theories is Gauss' law constraint. It is responsible for the conflict between locality of the charged fields and positivity, it yields the superselection of the (unbroken) gauge charges, provides a non-perturbative explanation of the Higgs mechanism in the local gauges, implies the infraparticle structure of the charged particles in QED and the breaking of the Lorentz group in the charged sectors. A non-perturbative proof of the Higgs mechanism is discussed in the Coulomb gauge: the vector bosons corresponding to the broken generators are massive and their two point function dominates the Goldstone spectrum, thus excluding the occurrence of massless Goldstone bosons. The solution of the U(1) problem in QCD, the theta vacuum structure and the inevitable breaking of the chiral symmetry in each theta sector are derived solely from the topology of the gauge group, without relying on the semiclassical instanton approximation.

A Primer of Analytical Mechanics (UNITEXT for Physics)

Springer

The book gives an introduction to Weyl non-regular quantization suitable for the description of physically interesting quantum systems, where the traditional Dirac-Heisenberg quantization is not applicable. The latter implicitly assumes that the canonical variables describe observables, entailing necessarily the regularity of their exponentials (Weyl operators). However, in physically interesting cases -- typically in the presence of a gauge symmetry -- non-observable canonical variables are introduced for the description of the states, namely of the relevant representations of the observable algebra.
In general, a gauge invariant ground state defines a non-regular representation of the gauge dependent Weyl operators, providing a mathematically consistent treatment of familiar quantum systems -- such as the electron in a periodic potential (Bloch electron), the Quantum Hall electron, or the quantum particle on a circle -- where the gauge transformations are, respectively, the lattice translations, the magnetic translations and the rotations of 2π. 
Relevant examples are also provided by quantum gauge field theory models, in particular by the temporal gauge of Quantum Electrodynamics, avoiding the conflict between the Gauss law constraint and the Dirac-Heisenberg canonical quantization. The same applies to Quantum Chromodynamics, where the non-regular quantization of the temporal gauge provides a simple solution of the U(1) problem and a simple link between the vacuum structure and the topology of the gauge group.
Last but not least, Weyl non-regular quantization is briefly discussed from the perspective of the so-called polymer representations proposed for Loop Quantum Gravity in connection with diffeomorphism invariant vacuum states.

Gauge Invariance and Weyl-polymer Quantization (Lecture Notes in Physics, 904)

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.

Symmetry Breaking (Lecture Notes in Physics, 732)

Mathematics

About the Author\nFranco Strocchi, after graduating with maximum cum laude at the University of Pisa and at Scuola Normale Superiore in 1961, was lecturer at both Institutions, research associate and visiting professor at Princeton University, visiting Schroedinger professor at Wien University, visiting scientist at CERN theory division, full professor of theoretical physics at the International School of Advanced Studies (SISSA, Trieste) and professor at Scuola Normale Superiore. He gave invited lectures at various meetings and International Schools and he is the author of several books on the subjects of his research works (Elements of Quantum Mechanics of Infinite Systems, World Scientific 1985, Selected Topics on the General Properties of Quantum Field Theory, World Scientific 1993, Symmetry Breaking, Springer 2005, 2008, An Introduction to the Mathematical Structure of Quantum Mechanics, World Scientific 2008, 2010, An Introduction to Non-Perturbative Foundations of Quantum Field Theory, Oxford University Press, International Series of Monographs in Physics, 2013, 2016, Gauge Invariance and Weyl-Polymer Quantization, Springer 2016, A primer of Analytical Mechanics, Springer 2018) and of more than one hundred scientific papers.\nThe third edition of the by now classic reference on rigorous analysis of symmetry breaking in both classical and quantum field theories adds new topics of relevance, in particular the effect of dynamical Coulomb delocalization, by which boundary conditions give rise to volume effects and to energy/mass gap in the Goldstone spectrum (plasmon spectrum, Anderson superconductivity, Higgs phenomenon). The book closes with a discussion of the physical meaning of global and local gauge symmetries and their breaking, with attention to the effect of gauge group topology in QCD. 
From the reviews of the first edition: 
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. J.-P. Antoine, Physicalia 28/2, 2006 
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 
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. S. Hajjawi, Mathematical Reviews, 2008\nFrom the Back Cover\nThe third edition of the by now classic reference on rigorous analysis of symmetry breaking in both classical and quantum field theories adds new topics of relevance, in particular the effect of dynamical Coulomb delocalization, by which boundary conditions give rise to volume effects and to energy/mass gap in the Goldstone spectrum (plasmon spectrum, Anderson superconductivity, Higgs phenomenon). The book closes with a discussion of the physical meaning of global and local gauge symmetries and their break

Symmetry Breaking (Theoretical and Mathematical Physics)

This is a collection of the following articles: F. Strocchi, Classification of solutions of non-linear hyperbolic equations and non-linear elliptic problems, E.H. Zarantonello, Conical spectral theory, E. De Giorgi, Generalized limits in calculus of variations, G. Dal Maso and L. Modica, A general theory of variational functionals.

Topics in functional analysis 1980-81 (Publications of the Scuola Normale Superiore)

Dal Maso, Gianni

De Giorgi, Ennio

Zarantonello, Eduardo H.

Modica, Luciano

Edizioni della Normale

The book provides a non-perturbative approach to the symmetry breaking in the standard model, in this way&nbsp;avoiding the critical issues which affect the standard presentations. The debated empirical meaning of global&nbsp;and local gauge symmetries is clarified. The absence of Goldstone bosons in the Higgs mechanism is non-perturbatively&nbsp;explained by the validity of Gauss laws obeyed by the currents which generate the related <div>global gauge symmetry. The solution of the U(1) problem and the vacuum structure in&nbsp;quantum chromodynamics&nbsp;(QCD) are obtained&nbsp;without recourse to the problematic semiclassical instanton approximation, by rather exploiting the topology&nbsp;of the gauge group.</div>

Symmetry Breaking in the Standard Model: A Non-Perturbative Outlook (Publications of the Scuola Normale Superiore, 19)

SELECTED TOPICS ON THE GENERAL PROPERTIES OF QUANTUM FIELD THEORY: LECTURE NOTES (World Scientific Lecture Notes in Physics)

This book provides a readable account of the foundations of QFT, in particular of the Euclidean formulation with emphasis on the interplay between physical requirements and mathematical structures. The general structures underlying the conventional local (renormalizable) formulation of gauge QFT are discussed also on the basis of simple models. The mechanism of confinement, non-trivial topology and ?-vacua, chiral symmetry breaking and solution of the U(1) problem are clarified through a careful analysis of the Schwinger model, which settles unclear or debated points.

World Scientific Publishing Company

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Books by Strocchi, Franco

See the best price to sell, buy, or rent books by Strocchi, Franco. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

An Introduction to Non-Perturbative Foundations of Quantum Field Theory (International Series of Monographs on Physics)

An Introduction to Non-Perturbative Foundations of Quantum Field Theory (International Series of Monographs on Physics)

A Primer of Analytical Mechanics (UNITEXT for Physics)

Gauge Invariance and Weyl-polymer Quantization (Lecture Notes in Physics, 904)

Symmetry Breaking (Lecture Notes in Physics, 732)

Symmetry Breaking (Theoretical and Mathematical Physics)

Topics in functional analysis 1980-81 (Publications of the Scuola Normale Superiore)

Symmetry Breaking in the Standard Model: A Non-Perturbative Outlook (Publications of the Scuola Normale Superiore, 19)

SELECTED TOPICS ON THE GENERAL PROPERTIES OF QUANTUM FIELD THEORY: LECTURE NOTES (World Scientific Lecture Notes in Physics)

SELECTED TOPICS ON THE GENERAL PROPERTIES OF QUANTUM FIELD THEORY: LECTURE NOTES (World Scientific Lecture Notes in Physics)