The Geometry of Point Particles

Atiyah, M.F.; Sutcliffe, P.M.

Abstract

There is a very natural map from the configuration space of n distinct points in Euclidean 3-space into the flag manifold U(n)/U(1)n, which is compatible with the action of the symmetric group. The map is well defined for all configurations of points provided a certain conjecture holds, for which we provide numerical evidence. We propose some additional conjectures, which imply the first, and test these numerically. Motivated by the above map, we define a geometrical multi-particle energy function and compute the energy-minimizing configurations for up to 32 particles. These configurations comprise the vertices of polyhedral structures that are dual to those found in a number of complicated physical theories, such as Skyrmions and fullerenes. Comparisons with 2- and 3-particle energy functions are made. The planar restriction and the generalization to hyperbolic 3-space are also investigated.

Citation

Atiyah, M., & Sutcliffe, P. (2002). The Geometry of Point Particles. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 458(2021), 1089-1115. https://doi.org/10.1098/rspa.2001.0913

• Journal Article Type: Article
• Publication Date: May 8, 2002
• Deposit Date: May 1, 2007
• Journal: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
• Print ISSN: 1364-5021
• Electronic ISSN: 1471-2946
• Publisher: The Royal Society
• Peer Reviewed: Peer Reviewed
• Volume: 458
• Issue: 2021
• Pages: 1089-1115
• DOI: https://doi.org/10.1098/rspa.2001.0913
• Keywords: Point, Particles, Geometry, Energy, Minimization, Polyhedra.