This project investigates mathematical problems arising from the foundations of quantum mechanics and from the foundations of statistical mechanics. The research on the foundations of quantum mechanics is concerned primarily with Bohmian mechanics -- a deterministic theory of particle motion naturally emerging as a completion of the Schrodinger dynamics for the wave function -- and related models. Among the problems proposed for exploration are (1) the extent to which the Bohmian dynamics can be utilized to facilitate the solution to the time-dependent Schrodinger equation, and (2) the development of mathematical structures providing a natural completion of quantum field theory. The research on the foundations of statistical mechanics is concerned primarily with the second law of thermodynamics and the status of the Boltzmann entropy as a nonequilibrium entropy, even for systems far from equilibrium and far from the low-density regime governed by the Boltzmann equation. These two lines of research will be merged through careful analysis of the validity of Boltzmann's ideas in quantum mechanics.
Our world is a quantum world: the laws of nature describing the deepest levels of reality are quantum mechanical laws. While the validity of the predictions of quantum mechanics should not be seriously questioned, its standard formulation is conceptually inadequate. This project concerns the analysis of mathematical structures relevant to a nonstandard formulation of quantum mechanics called Bohmian mechanics, a formulation that is at once simpler than the standard one and conceptually much more coherent. Quantum mechanics has been the basis of remarkable technological innovations in recent years. Examples include quantum computers, which in principle would allow computations of a complexity previously regarded as impossible, and quantum cryptography, which affords a level of security that would otherwise be inconceivable. The research with which this project is concerned will lead to a deeper understanding of quantum theory and will facilitate the technological breakthroughs that quantum theory continues to make possible. The research is anticipated to generate novel mathematical structures, and it may lead to the resolution of longstanding puzzles and paradoxes that plague the foundations of quantum theory.
