Symplectic and semiclassical aspects of the Schläfli identity

H. M. Haggard, A. Hedeman, E.  Kur, and R. G Littlejohn

Journal of Physics A: Mathematical and Theoretical 48 (105203), 2015.

The Schläfli 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 Schläfli 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.

DOI: 10.1088/1751-8113/48/10/105203

Full text: HaHeKuLiSymplecticSemiclassicalAspectsSchlafliIdentity.pdf