Generating Functions for Coherent Intertwiners

TL;DR

This paper develops generating functions for scalar products of SU(2) coherent intertwiners, explores their summability and geometric interpretation, and derives differential equations for SU(2) flatness on spin networks.

Contribution

It introduces exactly summable generating functions for coherent intertwiners and generalizes Wheeler-DeWitt equations for arbitrary graphs in SU(2) quantum gravity.

Findings

Identified a distinguished combinatorial weight with geometric meaning

Derived partial differential equations for SU(2) flatness

Provided explicit solutions for these equations

Abstract

We study generating functions for the scalar products of SU(2) coherent intertwiners, which can be interpreted as coherent spin network evaluations on a 2-vertex graph. We show that these generating functions are exactly summable for different choices of combinatorial weights. Moreover, we identify one choice of weight distinguished thanks to its geometric interpretation. As an example of dynamics, we consider the simple case of SU(2) flatness and describe the corresponding Hamiltonian constraint whose quantization on coherent intertwiners leads to partial differential equations that we solve. Furthermore, we generalize explicitly these Wheeler-DeWitt equations for SU(2) flatness on coherent spin networks for arbitrary graphs.