Zenith Grant Awardee

Dr. John Baez

University of California at Riverside

Project Title

Categorifying Fundamental Physics

Project Summary

In ordinary mathematics, and physics as well, equations are fundamental. However, every equation is a half-truth: after all, if the two sides of the equation look different, why are we saying they're the same? 'Categorification' is a fancy name for coming clean on this issue: instead of merely saying that two things are the same, we specify a way of regarding them as the same.

This has surprising consequences. For example, we usually think of quantities like energy as continuously variable, or 'analog'. Quantum mechanics shows there is a certain discreteness built into the world, but it still uses analog ideas. Using categorification, we can phrase large portions of quantum mechanics in a purely discrete way. We want to know how far we can push this.

In addition to thinking about physics in new ways, we shall explore new methods of carrying out research. We intend to share not just our results, but the process by which we find them. We will do this by incorporating a wide range of multi-media into our research, including videos of lectures and seminars made publicly available online.

Technical Abstract

The goal of this program is to develop a radically new understanding of the mathematics underlying physical theories. Our main tool will be categorification. Traditional mathematics takes the concept of equality as fundamental. Categorification replaces equations by isomorphisms: instead of merely saying that two things are the same, we specify a way of regarding them as the same. This is a surprisingly powerful change of viewpoint.

Our program has three components. First, we are developing a version of quantum mechanics in which Hilbert spaces are replaced by purely combinatorial structures. Second, we are categorifying classical mechanics and geometric quantization, which leads naturally to a generalization of these ideas from point particles to extended objects. Third, we are studying the role of 'exceptional' algebraic structures, such as the octonions and exceptional Lie groups, in of particle physics.

In addition to thinking about physics in new ways, we shall explore new methods of carrying out research. We intend to share not just our results, but the process by which we find them. We will do this by incorporating a wide range of multi-media into our research, including videos of lectures and seminars made publicly available online.