Symplectic and Semiclassical Aspects of the Schl\"afli Identity

TL;DR

This paper explores the symplectic and semiclassical properties of the Schl"afli identity in flat 3D space, providing proofs, geometric insights, and discussing potential generalizations to curved spaces and quantum groups.

Contribution

It offers a symplectic geometric proof of the Schl"afli identity and discusses its semiclassical interpretation and possible extensions to curved geometries and quantum algebraic structures.

Findings

Symplectic proof of the Schl"afli identity in flat 3D space.

Discussion of symplectic and Lagrangian manifolds related to the identity.

Potential generalizations involving Poisson-Lie groups and q-deformed spin networks.

Abstract

The Schl\"afli identity, which is important in Regge calculus and loop quantum gravity, is examined from a symplectic and semiclassical standpoint in the special case of flat, 3-dimensional space. In this case a proof is given, based on symplectic geometry. A series of symplectic and Lagrangian manifolds related to the Schl\"afli identity, including several versions of a Lagrangian manifold of tetrahedra, are discussed. Semiclassical interpretations of the various steps are provided. Possible generalizations to 3-dimensional spaces of constant (nonzero) curvature, involving Poisson-Lie groups and q-deformed spin networks, are discussed.