Graphical evolution of spin network states

The evolution of spin network states in loop quantum gravity can be defined with respect to a time variable, given 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 reroutes the loops through the vertices. A remaining unresolved problem is to take the square root of the infinite-dimensional matrix of the Hamiltonian constraint and to obtain the eigenspectrum of the “clock field” Hamiltonian.