Dynamics for a 2-vertex Quantum Gravity Model

TL;DR

This paper employs the U(N) framework to analyze the dynamics of a simple two-vertex spin network in loop quantum gravity, revealing algebraic structures, symmetries, and connections to quantum cosmology.

Contribution

It introduces a U(N) invariant Hamiltonian for a two-vertex quantum gravity model and explores its dynamics and symmetries, linking it to loop quantum cosmology.

Findings

Algebraic structure of the spin network Hilbert space described.

U(N) invariance restricts the model to isotropic/homogeneous sectors.

Proposed Hamiltonian induces dynamics consistent with quantum cosmology.

Abstract

We use the recently introduced U(N) framework for loop quantum gravity to study the dynamics of spin network states on the simplest class of graphs: two vertices linked with an arbitrary number N of edges. Such graphs represent two regions, in and out, separated by a boundary surface. We study the algebraic structure of the Hilbert space of spin networks from the U(N) perspective. In particular, we describe the algebra of operators acting on that space and discuss their relation to the standard holonomy operator of loop quantum gravity. Furthermore, we show that it is possible to make the restriction to the isotropic/homogeneous sector of the model by imposing the invariance under a global U(N) symmetry. We then propose a U(N) invariant Hamiltonian operator and study the induced dynamics. Finally, we explore the analogies between this model and loop quantum cosmology and sketch some possible generalizations of it.