Coherent 3j-symbol representation for the loop quantum gravity intertwiner space

TL;DR

This paper presents a novel coherent state-based technique for representing intertwiners in loop quantum gravity, enabling geometric and semiclassical analysis of the Hamiltonian's matrix elements.

Contribution

It introduces a new coherent state projection method for intertwiners, reformulating Hamiltonian operators as differential operators on complex polynomials.

Findings

Explicit expressions for intertwiner projections on coherent states

Hamiltonian operators expressed as differential operators

Facilitates semiclassical limit analysis of quantum gravity dynamics

Abstract

We introduce a new technique for dealing with the matrix elements of the Hamiltonian operator in loop quantum gravity, based on the use of intertwiners projected on coherent states of angular momentum. We give explicit expressions for the projections of intertwiners on the spin coherent states in terms of complex numbers describing the unit vectors which label the coherent states. Operators such as the Hamiltonian can then be reformulated as differential operators acting on polynomials of these complex numbers. This makes it possible to describe the action of the Hamiltonian geometrically, in terms of the unit vectors originating from the angular momentum coherent states, and opens up a way towards investigating the semiclassical limit of the dynamics via asymptotic approximation methods.