The Quantum Mechanical Many-Body Problem and Statistical Mechanics

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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. These various topics have grown out of a body of research over several decades and the underlying unity is that solutions in the various problem areas shed light on each other. The intellectual merit is contained in the following partial list of specific research goals. 1. Building on the recent success in proving the 34 year old conjecture about the Wehrl entropy for spin coherent states, the extension from SU(2) to other Lie groups of interest to mathematics and physics will be investigated. 2. Various topics and conjectures concerning magnetization and long-range order in the Hubbard model of correlated electrons will be investigated. Other problems 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. One goal is to find sharper constants. 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. It is very difficult, even with computers, to be sure about ionization of atoms, but 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 that will include the concept of local kinetic energy. 7. Building on previous expertise in the area, the increasingly important field of quantum information theory will be pursued. In particular, the recent success in understanding measures of entanglement as well as the the uncertainty principle for bipartite systems will used to try to solve the bipartite density-matrix reconstruction problem. A broader impact of this activity is that 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 promote the relevance of modern concepts of mathematical analysis to problems of condensed matter physics.