The Stability of Matter in Quantum Mechanics , Elliott H. Lieb and Robert Seiringer

Cambridge U. Press, New York, 2010. $50.00 (293 pp.). ISBN 978-0-521-19118-0

Why is the matter around us stable? By “stability” I am not simply referring to the absolute limit on the amount of energy of an atom; every student who has taken a quantum mechanics course has solved the fundamental example of atomic hydrogen. Rather, I mean stability that makes the amount of energy proportional to the number of atomic particles and leads to the fact that two liters of fuel contain twice as much energy as one liter.

The stability of matter should primarily be an outcome of nonrelativistic quantum mechanics, since nuclear forces, radiative terms, and other non-Coulomb interactions contribute only tiny corrections to the binding energies of atoms and molecules. Quantum mechanics—given an appropriate formalism of the uncertainty principle—prevents an electron from falling into the nucleus. In addition, the distinction between fermions and bosons becomes important for systems with large numbers of particles. We now know that the binding energy would increase too rapidly with the number of negatively charged bosons and therefore violate the required energy bound, rendering bosons unsuitable for ordinary matter.

The rigorous proof showing that nonrelativistic quantum mechanics predicts stability of matter is a highlight of the application of modern mathematics to fundamental problems in physics. With their outstanding book, *The Stability of Matter in Quantum Mechanics *, mathematical physicists Elliott Lieb and Robert Seiringer provide a complete, self-contained summary of five decades of research, primarily by Lieb and his collaborators, into the stability of matter in various physical situations. Both authors are leaders in that domain.

Going beyond the stability problem in nonrelativistic quantum mechanics, the authors also model the corresponding quantum mechanical systems with relativistic kinematics. Although only toy models, they are frequently used for calculations of atomic and molecular energies. In relativistic quantum mechanics, a new feature occurs: The product of the charge of the nucleus and the fine structure constant must be bounded to ensure the finiteness of the energy. The stability of large systems also implies a bound on the fine structure constant, which characterizes the strength of the electromagnetic interaction. The authors also take into account gravitational interactions, in which can be seen an even more spectacular result: Stars collapse under gravity, and their critical mass—above which they become unstable—depends on the gravitational constant.

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 marvelous experience as they learn from *The Stability of Matter in Quantum Mechanics. *