The geometry of classical particles.

*(English)*Zbl 1050.55502
Yau, S.T. (ed.), Surveys in differential geometry. Papers dedicated to Atiyah, Bott, Hirzebruch and Singer . Somerville, MA: International Press (ISBN 1-57146-069-1/hbk). Surv. Differ. Geom., Suppl. J. Differ. Geom. 7, 1-15 (2000).

From the introduction: In a recent paper [Proc. R. Soc. Lond., Ser. A 453, 1771–1790 (1997; Zbl 0892.46084)], M. V. Berry and J. M. Robbins have described a classical approach to the spin-statistics theorem of quantum physics. In the course of their investigation they were led to a purely geometrical question in 3-dimensional Euclidean space. This paper grew out of an attempt to answer their question and to understand its significance. The Berry-Robbins problem concerns two very well-known spaces:

(i) the configuration space \(C_n(\mathbb{R}^3)\), parametrizing \(n\) distinct ordered points in \(\mathbb{R}^3\);

(ii) the flag manifold \(U(n)/T^n\), parametrizing ‘flags’ i.e., \(n\) ordered mutually orthogonal one-dimensional vector subspaces of \(\mathbb{C}^n\) (here \(U(n)\) is the unitary group, \(T^n\) the diagonal subgroup fixing a given flag). The symmetric group \(\Sigma_n\) acts freely on both these spaces by permuting the points or the subspaces. The problem is the following: (1.1) Does there exist (for all \(n)\) a continuous map \(f_n:\mathbb{C}_n(\mathbb{R}^3)\to U(n)/T^n\) which is compatible with the action of \(\Sigma^n\)? In this paper I shall give a positive answer to this problem using only elementary geometry. However, this solution has some unsatisfactory features and a much more elegant solution may exist. This depends on a conjecture which is remarkably simple to state but appears to be difficult to settle.

For the entire collection see [Zbl 1044.53002].

(i) the configuration space \(C_n(\mathbb{R}^3)\), parametrizing \(n\) distinct ordered points in \(\mathbb{R}^3\);

(ii) the flag manifold \(U(n)/T^n\), parametrizing ‘flags’ i.e., \(n\) ordered mutually orthogonal one-dimensional vector subspaces of \(\mathbb{C}^n\) (here \(U(n)\) is the unitary group, \(T^n\) the diagonal subgroup fixing a given flag). The symmetric group \(\Sigma_n\) acts freely on both these spaces by permuting the points or the subspaces. The problem is the following: (1.1) Does there exist (for all \(n)\) a continuous map \(f_n:\mathbb{C}_n(\mathbb{R}^3)\to U(n)/T^n\) which is compatible with the action of \(\Sigma^n\)? In this paper I shall give a positive answer to this problem using only elementary geometry. However, this solution has some unsatisfactory features and a much more elegant solution may exist. This depends on a conjecture which is remarkably simple to state but appears to be difficult to settle.

For the entire collection see [Zbl 1044.53002].