The Quantum Mechanical Many-body Problem and Statistical Mechanics

Princeton University, Princeton NJ

Investigators

Abstract

This research is devoted to various aspects of many-body theory in quantum mechanics, condensed matter physics, quantum information theory, and quantum electrodynamics, as well as some related mathematical problems. The underlying unity is that solutions in the various problem areas shed light on each other. The intellectual merit of this proposal is contained in the following partial list of specific research goals. 1. Low density Bose gases are now in the forefront of research experimentally and theoretically and it is intended to continue the previous successful work on the ground state energy, as well as the question of Bose-Einstein condensation and its relation to superfluidity. Specific questions are the validation of Bogolubovs second term in the expansion of the ground state energy and the calculation of the yrast line of a rotating gas. 2. Conjectures concerning magnetization and long-range order in the Hubbard model of correlated electrons will be investigated. Other goals in condensed matter physics are to understand the binding of polarons and to build models that can be rigorously demonstrated to have striped states. 3. A fundamental statistical-mechanical property of matter is the existence of the thermodynamic limit of the free energy. A proof of this has been only partially accomplished so far when the electromagnetic radiation field is taken into account. It is intended to complete the demonstration. 4. The Lieb-Thirring inequalities are important in a variety of physical and mathematical problems. The PI intends sharpening the constants and to extend the inequalities to a new domain with a background density of particles instead of a vacuum. 5. An attempt will be made to verify the long-standing conjecture that the maximum negative ionization of a large atom is only about one electron. Nature seems to be telling us about a universal fact about fermions that it would be desirable to understand from first principles. 6. Density functional theory for atoms and molecules is important practically and conceptually. Some open questions about the Muller theory will be investigated, as well as possible improvements to the Lieb-Oxford bound for the exchange-correlation energy. 7. Building on previous expertise in the area, topics in the increasingly important field of quantum information theory will be pursued. Broader impact: Students and postdocs will be involved in these projects, which will combine mentoring with research and help produce the next generation of mathematically knowledgeable physicists. It will also further the interdisciplinary bonds between the communities of mathematicians and physicists and establish the relevance of modern concepts of mathematical analysis to problems of condensed matter physics.