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 quantummechanics, condensed matter physics and quantum electrodynamics, as well as someproblems in quantum information theory and pure mathematics. These various topicshave grown out of a body of research over several decades and the underlying unity isthat solutions in the various problem areas shed light on each other. A broader impactof this activity is that students and postdocs will be involved in these projects, which willcombine mentoring with research and help produce the next generation ofmathematically knowledgeable physicists. It will also further the interdisciplinary bondsbetween the communities of mathematicians and physicists and promote the relevanceof 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 specificresearch goals.1. Low density Bose gases are now in the forefront of research {experimentally andtheoretically) and it is intended to continue the previous successful work on the groundstate energy, as well as the question of Bose-Einstein condensation and its relation tosuperfluidity. One question concerns the simultaneous occurrence of solidity andsuperfluidity. Another is to validateBogolubov's second term in the expansion of the ground state energy. A third is to see ifthe excitation spectrum of the Lieb-Liniger one-dimensional model can be establishedfor realistic three-dimensional gases used in experiments.2. Various topics and conjectures concerning magnetization and long-range order in theHubbard model of correlated electrons will be investigated. In parallel, an attempt will bemade to prove long-range order in the ground states of two-dimensional Heisenbergmodels. 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 partiallyaccomplished so far when the electromagnetic radiation #eld is taken into account. It isintended to complete the demonstration.4. To study the renormalization problem in quantum electrodynamics (QED), especiallywith regard to the many-body and bound state aspects. This continues a successfulnon-perturbative study begun under the previous grant in which we developed a modelof relativistic QED that makes sense for bound states. A complete theory has not yetbeen invented, and it is important to try to do so because QED is one of ourfundamental physical theories.5. An attempt will be made to verify the long-standing conjecture that the maximumnegative ionization of a large atom is only about one electron. It is very di#cult, evenwith computers, to be sure about ionization of atoms, but nature seems to be telling us auniversal fact about fermions that it would be desirable to understand from firstprinciples.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. Successhere will imply that entangled states cannot improve the capacity of quantum channels.