Dynamics in Geometric Models of Matter

Ever since Einstein's interpretation of gravity as spacetime geometry in his theory of general relativity, physicists have searched for a unified and geometric description of all known forces. Shortly after Einstein invented general relativity, Theodor Kaluza and Oscar Klein proposed a unified geometric model of gravity and electromagnetism, with electromagnetic forces being interpreted in terms of curvature into a fifth dimension (in addition to time and the three spatial dimensions). The fifth dimension had to be assumed to be tightly curled up into small circles. Under certain assumptions on the five-dimensional geometry, the Kaluza-Klein model reproduces classical electromagnetism together with general relativity.

In a recent paper Atiyah, Manton and Schroers propose a geometric model of matter which is akin to the Kalzua-Klein model, but exchanges the roles of the electric and magnetic fields. Modifying and relaxing Kaluza and Klein's assumptions, they arrive models for elementary particles like the electron and the proton in terms of four-dimensional geometries. These models are static (time is not included) and capture qualitative properties of the particles in question.

The goal of this project is develop the geometric description of matter by clarifying how it accounts for the spin of elementary particles and how one can describe interactions in purely geometric terms. This is ambitious and adventurous research which draws on many different research areas in mathematics and physics. It could pave the way for a radically new mathematical language for elementary particle, nuclear and atomic physics.

Impact via Training of the Research Assistant: The postdoctoral research assistant working on this project will develop fluency in a wide range of mathematical techniques. By the end of the appointment, he or she will have experience in working in an interdisciplinary research team and networking with researchers from different countries and different disciplines. The resulting profile, combining outstanding analytical and numerical skills with interdisciplinary and international team work experience, is highly sought by employers in both industry and academia.

International Network: The series of sandpit events and the conference travel by the PI,CoI and PDRA will create an international network of leading researchers that will be well-placed to pursue both the ramifications and applications of the proposed research in mathematics and physics.

Medium-term Impact on Technology: Topological and geometrical techniques like those addressed in this proposal are increasingly used in applied science and technology, e.g. in the study of graphene or in topological quantum computing. Some of the planned sandpit events will be specifically devoted to exploring such applications, for example a discussion meeting on the use of index theorems in computing zero-modes of graphene configurations with leading researchers in that field, like Jiannis Pachos at Leeds.

Long-term Impact on the Foundations of Science: The proposed research targets fundamental questions in nuclear and particle physics. While the impact on these is only likely to be realised in the long term, even partial success in re-formulating our current understanding of matter has profound implication for all of science and technology. A new understanding of the electron would revolutionise particle and atomic physics and a new picture of the proton and neutron would affect all of nuclear physics, with profound implications for associated technologies and industries.

Public Engagement: Questions related to the foundations of physics like the ones asked in this proposal continue to fascinate the wider public. Results from the research will be communicated to the general public through talks and a dedicated web page in which the objectives and progress will be explained in non-specialist terms, tailored to the audience. The Co-Investigator, Sir Michael Atiyah, is an internationally famous mathematician and well-known for his ability to communicate mathematics and physics to non-experts, as demonstrated recently by his lecture on the Youthful Spirit of Mathematics, organised by the Cambridge Philosophical Society, or his interview on the Beauty of Mathematics for the Simons Foundation, available on YouTube. Both the research assistant and the Principal Investigator will undertake communication and media training as part of this project.