Graphical Evolution of Spin Network States

TL;DR

This paper develops a method to describe the evolution of spin network states in loop quantum gravity using a scalar field as a time parameter, providing finite, diffeomorphism-invariant results through graphical techniques.

Contribution

It introduces a regulated Hamiltonian for spin network evolution and computes its action explicitly on edges and vertices using recoupling theory.

Findings

Action on edges corresponds to diffeomorphisms.

Action on vertices creates new edges and reroutes loops.

Results are finite and diffeomorphism invariant.

Abstract

The evolution of spin network states in loop quantum gravity can be described by introducing a time variable, defined by the surfaces of constant value of an auxiliary scalar field. We regulate the Hamiltonian, generating such an evolution, and evaluate its action both on edges and on vertices of the spin network states. The analytical computations are carried out completely to yield a finite, diffeomorphism invariant result. We use techniques from the recoupling theory of colored graphs with trivalent vertices to evaluate the graphical part of the Hamiltonian action. We show that the action on edges is equivalent to a diffeomorphism transformation, while the action on vertices adds new edges and re-routes the loops through the vertices.