M.F. Atiyah
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. |
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