Elementary blocks of Loop Quantum Gravity

Abstract

We embark on the vast program of integrating the dynamics of Loop Quantum Gravity (LQG). Adopting the strategy of decomposing spin network states into small blocks of (quantum) geometry which can later be glued back together, we focus on the more modest objective of studying the Hamiltonian dynamics on the {\it candy graph}, that is two nodes linked together by an arbitrary number of edges and also having open edges. This elementary setting allows both for curvature to develop around the bulk loops and both non-trivial boundary data and dynamics on the open edges. We study this system at the classical level and leave the detailed of its quantum regime for future investigation. Working on a single loop with two external legs, we show how the LQG Hamiltonian ansatz reduces to a pair of non-linear differential equations, similar to the cubic Schrödinger equation, on the areas carried by the bulk links. We provide analytical solutions to this evolution equation, identifying oscillatory modes (bounded modes) and divergent modes (similar to bouncing cosmological trajectories). This provides an explicit template for future investigations of LQG dynamics on more sophisticated spin network architecture built as arrays of candy graphs.

Figures (10)

• Figure 1: The Candy Graph as the simplest building block of spin network states.
• Figure 2: Candy graph with two 4-valent nodes: to focus on the dynamics of the bulk loop, and thus of the local curvature excitation, we freeze the intertwiner degrees of freedom by stretching the nodes and keeping the norms $|\vec{X}|$ and $|\vec{Y}|$ fixed.
• Figure 3: Plot of the function $2\sqrt{k}/(1+k)$ showing the upper limit allowed by the boundary data given by the area carried by the external legs of the graph, with maximal value $k_{max}=\textrm{min}(X,Y)/\textrm{max}(X,Y)$. Here, we chose boundary data $X=3$ and $Y=5$, yielding a maximal value of $k_{max}=3/5$. The saturation limit $k_{max}=1$ is reached when $X=Y$ on the external legs.
• Figure 4: The area difference $a$ as a function of $t$ (on the horizontal axis) with various initial conditions and $\left|\alpha\right|=1$, showing allowed trajectories with $\dot{a}<2B^2$ as in the plots (a) to (d), the critical trajectory with $k=1$ and $\dot{a}=2B^2$ as in the plot (e), and forbidden trajectories $\dot a > 2B^2$ as in the plot (f).
• Figure 5: The total area $A$ as a function of $t$ (on the horizontal axis) with various initial conditions and $\left|\beta\right|=1$.