Over the past several years the PI has developed a theory of quantum metric spaces, within the setting of algebras of operators on Hilbert space. My theory includes an analog of the classical Gromov-Hausdorff distance between metric spaces. I gave several applications of these ideas, notably to the convergence of matrix algebras to coadjoint orbits of compact Lie groups. I propose to continue to strengthen this theory, and to apply it in several directions suggested by the many situations in the physics and mathematics of quantization where one has a sequence of quantum spaces which appear to be converging to another space, either quantum or classical. As a major new direction I will try to develop an analogous theory for the quantum versions of the superstructure of vector bundles, connections, Yang-Mills actions, etc. I will also try to extend my theory beyond the quantum analog of locally compact spaces, so as to attempt to deal with the approximations of quantum field-theory models, especially those of integrable systems, say by quantum lattice models.
Our nation's technological and economic success has at its foundation the mathematical models of the world around us which scientists develop in order to understand how to use the flood of data which flows from the laboratories of the experimental scientists. But human beings and computers can only deal with finite collections of numbers at a time. Thus in applying these mathematical models it is almost always necessary to approximate the infinite variability of our world by finite collections of numbers. It is then crucial to understand how valid any given approximation is. With respect to individual calculations this matter has received extensive study. But less study has been made of how complex models as a whole can be approximated well by simpler models as a whole. Relatively little is known about such "global" approximations in the case of the models of quantum physics, which is the part of physics which governs chemical and biochemical reactions, the functioning of semi-conductors, and many other key technologies. In the classical realm there is an important form of global approximation called Gromov-Hausdorff distance. I have developed a quantum analog of it, and successfully applied it to a few examples. I propose to strengthen this theory, and to apply it to a broader class of examples, so as to better understand how to effectively approximate various models of quantum phenomena of current interest.
The main objective of this grant was to continue my project of developing a theory of quantum metric spaces, that is, of quantum spaces equipped with the analog of the notion of a distance, or "metric", on an ordinary space. I do this within the setting of algebras of operators on Hilbert space, which is the setting used by physicists to model quantum systems. A central part in my theory consists of developing a quantum analog of the classical Gromov-Hausdorff distance between metric spaces. I had earlier given several applications of these ideas, notably to the convergence of matrix algebras to coadjoint orbits of compact Lie groups, which is one of the main examples of interest to physicists. In my research supported by this grant I made significant progress in extending my quantum Gromov-Hausdorff distance so as to treat also the analogs of vector bundles over quantum metric spaces. Vectors bundles provide the foundation for the gauge theories widely used by physicists. I also made significant progress in understanding how to express quantum distances in terms of "Dirac operators", with the technical advantages that Dirac operators bring. Dirac operators are of much importance in quantum physics. During the grant period two of the doctoral students I was mentoring (who were supported in part by this grant) wrote fine doctoral dissertations that made substantial advances related to the main objective of this grant. Two other doctoral students that I have been mentoring (also supported in part by this grant) have already made nice original discoveries related to the main objective of this grant, and they may receive their doctoral degrees next Spring. The technological and economic success of our nation has at its foundation the mathematical models of the world around us which scientists develop in order to understand how to use the flood of data which flows from the laboratories of the experimental scientists. But human beings and computers can only deal with finite collections of numbers at a time. Thus in applying these mathematical models it is almost always necessary to approximate the infinite variability of our world by finite collections of numbers. It is then crucial to understand how valid any given approximation is. With respect to individual calculations this matter has received extensive study. But much less study has been made of how complex models as a whole can be approximated well by simpler models as a whole. Relatively little is known in quantitative terms concerning such "global" approximations in the case of the models of quantum physics. Since quantum physics is the part of physics which governs chemical and biochemical reactions, the functioning of semi-conductors, and many other key technologies, it is of much importance to understand how well complex quantum models can be approximated by simpler models. In the classical realm the global notion of approximation called Gromov-Hausdorff distance has been developed and found useful. I have been developing a quantum analog of Gromov-Hausdorff distance between quantum metric spaces in order to use it to better understand how to effectively approximate various models of quantum phenomena of current importance.
