The Geometry of Point Particles

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 which are dual to those found in a number of complicated physical theories, such as Skyrmions and fullerenes. Comparisons with 2-particle and 3-particle energy functions are made. The planar restriction and the generalization to hyperbolic 3-space are also investigated.

32 pages, including 8 figures

References in corpus (2)

• Skyrmions, Fullerenes and Rational Maps
• Solitonic fullerene structures in light atomic nuclei

Cited by in corpus (14)

• Central Configurations in Three Dimensions
• Icosahedral Skyrmions
• A proof of Atiyah's conjecture on configurations, of four points in Euclidean three-space
• Probability Density of the Multipole Vectors for a Gaussian Cosmic Microwave Background
• Anyons in Geometric Models of Matter
• Polyhedral Scattering of Fundamental Monopoles
• Configurations of Points and the Symplectic Berry-Robbins Problem
• On the Conjectures Regarding the 4-Point Atiyah Determinant
• A new proof of Atiyah's conjecture on configurations of four points
• Shapes of Quantum States
• Root Systems and the Atiyah-Sutcliffe Problem
• On the geometry of quantum indistinguishability
• Finite graphs and configurations of points
• On the Atiyah problem on hyperbolic configurations of four points