Geometric quantisation is a fundamental link between classical and quantum mechanics, and especially between the key roles of symmetry in these types of physics. It has been studied intensively since the 1980s, and has led to important new insights into the relation between classical and quantum mechanics, and the mathematics underlying these fields. However, the way geometric quantisation has been studied so far fundamentally only applies to classical phase spaces and symmetry groups that are compact. Compact sets are bounded and have other convenient properties that make geometric quantisation considerably easier to deal with, but preclude most applications in mathematics and physics.
A generalisation to the noncompact case would lead to a staggering potential for such applications, but also requires a completely new approach to the problem, as the techniques used traditionally become meaningless. There have been some first results using ad-hoc approaches that allow certain amounts of noncompactness, but none have the general and far-reaching applications that should be possible if a general theory of noncompact geometric quantisation is developed. This is exactly what I intend to do. The first steps taken have laid bare the challenges of noncompact geometric quantisation, and the time is now ripe to achieve ground-breaking results, using a completely new and general approach.
The key feature of this approach is the integration of the mathematical fields symplectic geometry (which describes classical mechanics), representation theory (which describes symmetry in quantum mechanics), and noncommutative geometry (which is a powerful tool for studying complex classical and quantum mechanical phase spaces). First results combining these fields indicate that this integration is extremely promising, and has attracted the interest of top researchers. The goal of this project is to fully exploit this link, thus obtaining exciting new applications in all three fields.
Fields of science
Call for proposal
See other projects for this call