Dynamics for a 2-vertex Quantum Gravity Model

TL;DR

<3-5 sentence high-level summary>

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.

Figures (5)

• Figure 1: The 2-vertex graph: the two vertices $\alpha$ and $\beta$ are linked by $N$ edges.
• Figure 2: Holonomy operator $\chi^{ij}$ acting on the couple of edges $(ij)$, which sends the spins $(j_i,j_j)$ to the four possibilities $(j_i+\eta_i,j_j+\eta_j)$ with $\eta_i,\eta_j=\pm \frac{1}{2}$. To study the precise action of this holonomy operator, we focus on its action $\chi^{ij}_\alpha$ and $\chi^{ij}_\beta$ on the individual vertices $\alpha$ and $\beta$. Then we glue back the two resulting intertwiners while respecting the matching conditions.
• Figure 3: Plots in the critical regime for the coefficients $\alpha_J$ for values $N=4$, $\sigma=-1$ and two different frequencies $k=3$ on the top and $k=15$ below. The plots of the left are computed exactly using the recursion relation, while the plots of the right are given by the asymptotic formula. We notice that we have the right oscillatory behavior and the correct overall scaling in $1/\sqrt{J}$, but we still miss approximative predictions for the full amplitude and the initial off-shift of the oscillations.
• Figure 4: The 3-vertex graph: the three vertices $\alpha$, $\beta$, and $\Omega$ are linked by $N$ edges and we have $N$ additional (auxiliary) trivalent vertices.
• Figure 5: The action of the radiation operator on the left and the rotation-like operator on the right: we shift the spins on the legs $i,j$ around the vertices $\alpha$ and $\Omega$ without affecting the spins living on the legs around the vertex $\beta$.

Theorems & Definitions (5)

• proof
• proof
• proof
• Conjecture 1
• Conjecture 2