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.
The book discusses fundamental aspects of Quantum Field Theory and of Gauge theories, with attention to mathematical consistency. Basic issues of the standard model of elementary particles (Higgs mechanism and chiral symmetry breaking in quantum Chromodynamics) are treated without relying on the perturbative expansion and on instanton calculus.
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.
This book contains a collection of essays written in honor of Wolfhart Zimmermann''s 80th birthday, most of them based on talks presented at a symposium in his honor.The book shows the unifying force of a subject (Quantum Field Theory) and a person (Zimmermann). It ranges from fundamental questions in quantum physics over applications to particle physics and noncommutative geometry to the latest developments in many body theory and dynamical systems. These key ideas are elucidated by worldwide-recognized experts including Faddeev, Becchi, Buchholz, Lowenstein and Salmhofer.Readers seeking examples on how a subject has evolved, diversified and deepened over the course of several decades and how a single person can influence this process can find here a perfect illustration. Altogether, readers are treated to a high-brow intellectual adventure.
This book gives a rigorous treatment of entanglement measures in the general context of quantum field theory. It covers a broad range of models and the use of fields allows us to properly take the localization of systems into account. The required mathematical techniques are introduced in a self-contained way.
Jacques Bros has greatly advanced our present understanding of rigorous quantum field theory through numerous contributions; this book arose from an international symposium held in honour of Bros on the occasion of his 70th birthday. Key topics in this volume include: Analytic structures of Quantum Field Theory (QFT), renormalization group methods, gauge QFT, stability properties and extension of the axiomatic framework, QFT on models of curved spacetimes, QFT on noncommutative Minkowski spacetime.
Arising out of the need for Quantum Mechanics (QM) to be part of the common education of mathematics students, this book formulates the mathematical structure of QM in terms of the C*-algebra of observables, which is argued on the basis of the operational definition of measurements and the duality between states and observables.
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.
An Introduction to Quantum Field Theory is a textbook intended for the graduate physics course covering relativistic quantum mechanics, quantum electrodynamics, and Feynman diagrams. The authors make these subjects accessible through carefully worked examples illustrating the technical aspects of the subject, and intuitive explanations of what is going on behind the mathematics. After presenting the basics of quantum electrodynamics, the authors discuss the theory of renormalization and its relation to statistical mechanics, and introduce the renormalization group. This discussion sets the stage for a discussion of the physical principles that underlie the fundamental interactions of elementary particle physics and their description by gauge field theories.
This book celebrates the life and work of the late Giovanni Morchio (1944–2021). It features scientific and anecdotal contributions written by his former colleagues, co-authors, and students, as well as senior scientists who were active witnesses to the dramatic advances in physics and in mathematics that took place during his 50-year-long career. The volume begins with a biographical introduction, detailing Giovanni Morchio’s life and his role as a physicist, mathematician, teacher, and scientist. The core of the book covers a vast spectrum of ideas, reflecting Dr Morchio’s scientific interests. Each chapter develops a specific topic of modern research, ranging from quantum mechanics and quantum field theory to additional themes such as the connection between general relativity and Newtonian gravitation. Every contribution provides a historical retrospective, a survey of advances, an outlook of future perspectives and challenges, and an updated bibliography. The last part collects the authors’ recollections of their professional and personal interactions with Dr Morchio, in recognition of his deep achievements, his exceptional pedagogical qualities, and his praiseworthy social and pro bono commitment. Authored by physicists of international calibre covering a broad range of subjects, the book will be a valuable reference for researchers and students of theoretical and mathematical physics.
