Spin and the Symplectic Flag Manifold

TL;DR

This paper develops a group-theoretic framework for particle interactions using symplectic geometry, showing that the symplectic group Sp(n) governs the isometries of spinor spaces and that particle interactions correspond to quaternionic flag varieties.

Contribution

It introduces a novel application of symplectic groups and quaternionic flag varieties to model particle interactions in spin systems.

Findings

Sp(n) is the largest isometry group for Pauli spinors.

Interactions are represented by quaternionic flag varieties Sp(n)/Sp(1)^n.

A group-theoretic approach to particle interactions is established.

Abstract

A theory for the transitive action of a group on the configuration space of a system of particles is shown to lead to the conclusion that interactions can be represented by the action of cosets of the group. By application of this principle to Pauli spinors, the symplectic group Sp(n) is shown to be the largest group of isometries of the space. Interactions between particles are represented by the complete quaternionic flag variety Sp(n)/Sp(1)^n.