Quantum geometry from phase space reduction

TL;DR

This paper establishes an explicit connection between spin network states and classical tetrahedral geometries through phase space reduction, providing new insights into quantum geometry and spin foam models.

Contribution

It introduces an explicit isomorphism linking spin network basis to classical tetrahedra via phase space reduction, illustrating the commutation of quantization and reduction.

Findings

Explicit formula for SU(2) invariant states as integrals over classical tetrahedra

Representation of the FK spin foam model as an integral over classical tetrahedra

Determination of the asymptotic behavior of the vertex amplitude

Abstract

In this work we give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra. The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly, or equivalently, as an integral over the space of classical tetrahedra. This provides an explicit realization of theorems by Guillemin--Sternberg and Hall that describe the commutation of quantization and reduction. In the final part of the paper, we use our result to express the FK spin foam model as an integral over classical tetrahedra and the asymptotics of the vertex amplitude is determined.