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 and quantum electrodynamics, as well as some problems in quantum information theory and pure mathematics. 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. 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. 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. One question concerns the simultaneous occurrence of solidity and superfluidity. Another is to validate Bogolubov's second term in the expansion of the ground state energy. A third is to see if the excitation spectrum of the Lieb-Liniger one-dimensional model can be established for realistic three-dimensional gases used in experiments. 2. Various topics and conjectures concerning magnetization and long-range order in the Hubbard model of correlated electrons will be investigated. In parallel, an attempt will be made to prove long-range order in the ground states of two-dimensional Heisenberg models. Both models are presently important in condensed matter physics. 3. A fundamental statistical-mechanical property of matter is the existence of the `thermodynamical limit of the free energy. A proof of this has been only partially accomplished so far when the electromagnetic radiation #eld is taken into account. It is intended to complete the demonstration. 4. To study the renormalization problem in quantum electrodynamics (QED), especially with regard to the many-body and bound state aspects. This continues a successful non-perturbative study begun under the previous grant in which we developed a model of relativistic QED that makes sense for bound states. A complete theory has not yet been invented, and it is important to try to do so because QED is one of our fundamental physical theories. 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 di#cult, even with computers, to be sure about ionization of atoms, but nature seems to be telling us a universal fact about fermions that it would be desirable to understand from first principles. 6. Several topics in quantum information theory and closely related #elds. In particular, a proof (or disproof) will be sought for Holevo's additivity of entropy conjecture. Success here will imply that entangled states cannot improve the capacity of quantum channels.