Research into the stability of matter has been one of the most successful chapters in mathematical physics, and is a prime example of how modern mathematics can be applied to problems in physics. A unique account of the subject, this book provides a complete, self-contained description of research on the stability of matter problem. It introduces the necessary quantum mechanics to mathematicians, and aspects of functional analysis to physicists. The topics covered include electrodynamics of classical and quantized fields, Lieb-Thirring and other inequalities in spectral theory, inequalities in electrostatics, stability of large Coulomb systems, gravitational stability of stars, basics of equilibrium statistical mechanics, and the existence of the thermodynamic limit. The book is an up-to-date account for researchers, and its pedagogical style makes it suitable for advanced undergraduate and graduate courses in mathematical physics.

• Co-authored by one of the main contributors to research on the stability of matter • Self-contained summary of research carried out over the last 40 years which removes the need to buy numerous books on the subject • Written to be understandable to both physicists and mathematicians, bridging the gap between the two subjects

### Contents

Preface; 1. Prologue; 2. Introduction to elementary quantum mechanics and stability of the first kind; 3. Many-particle systems and stability of the second kind; 4. Lieb–Thirring and related inequalities; 5. Electrostatic inequalities; 6. An estimation of the indirect part of the Coulomb energy; 7. Stability of non-relativistic matter; 8. Stability of relativistic matter; 9. Magnetic fields and the Pauli operator; 10. The Dirac operator and the Brown–Ravenhall model; 11. Quantized electromagnetic fields and stability of matter; 12. The ionization problem, and the dependence of the energy on N and M separately; 13. Gravitational stability of white dwarfs and neutron stars; 14. The thermodynamic limit for Coulomb systems; References; Index.

### Reviews

'This is an outstanding book which will be used both for research and for teaching. It will make an excellent text for a graduate course in either a physics or mathematics department. Physics students will learn to appreciate the beauty and relevance of mathematics and vice versa. The authors are leaders in the field. Their book not only describes important results but also makes them exciting.' Joel Lebowitz, Rutgers University

'The stability of matter - in the sense that the binding energy of any agglomerate of particles never exceeds their rest energy - is one of the important contributions of quantum mechanics to the functioning of our cosmos. But quantum mechanics alone is not enough. It is necessary to distinguish the difference between the two kinds of elementary particles, fermions and bosons, for the tremendous increase of the binding energy with the number of bosonic particles violates the required energy bound and makes them unsuitable for ordinary matter. Maybe that's why we live in a fermionic world. These subtleties and much more are hidden in the innocent looking Schroedinger equation. To distil that out you need the appropriate mathematical tools, as provided in this magnificent book where on each page you can feel the hands of masters of the subject.' Walter Thirring, University of Vienna

'Why does matter, from the size of atoms to stars avoid collapse? The Stability of Matter in Quantum Mechanics gives an impeccably written, self-contained introduction to the gems of this subject and the beautiful work of Elliott Lieb and coworkers over the past several decades. Every argument is ideally polished in this concise masterpiece. This book is an absolute must for any graduate students and active researchers, both mathematicians and physicists, interested in how a powerful yet elegant mathematics has answered one of the fundamental problems in mathematics and physics.' S.-T. Yau, Harvard University

'The discussions of those and other topics make the book a rich source for research into related fields. However, The Stability of Matter in Quantum Mechanics is also for students of mathematics and physics, not just for researchers. Since deep and beautiful mathematical techniques and results are needed, the required mathematical level is certainly high. But students should not be discouraged because the book's pedagogical style carefully guides them through the physical concepts and relevant mathematics before putting all the pieces together. Students and teachers alike will enjoy a marvellous experience as they learn from The Stability of Matter in Quantum Mechanics.' Physics Today

'… certain to become a standard reference for both researchers as well as students in the stability of matter for many years to come.' Mathematical Reviews

'… written at a mathematically sophisticated level and the authors are clearly masters of their subject. … subtle and fascinating definitive results obtained. … The authors invoke a classical energetic argument to inspire their approach … a comprehensive compendium of theorems and proofs in what is now a mature field that deserves to be more widely known. …heartily recommended to all physicists and mathematicians interested in the fundamental structure of matter … an up-to-date account of the mathematical theory of quantum stability.' Contemporary Physics