Bouncing evolution in a model of loop quantum gravity

To understand the dynamics of loop quantum gravity, the deparametrized model of gravity coupled to a scalar field is studied in a simple case, where the graph underlying the spin network basis is one loop based at a single vertex. The Hamiltonian operator ${\widehat{H}}_v$ is chosen to be graph-preserving, and the matrix elements of ${\widehat{H}}_v$ are explicitly worked out in a suitable basis. The nontrivial Euclidean part ${\widehat{H}}_v^E$ of ${\widehat{H}}_v$ is studied in details. It turns out that by choosing a specific symmetrization of ${\widehat{H}}_v^E$, the dynamics driven by the Hamiltonian give a picture of bouncing evolution. Our result in the model of full loop quantum gravity gives a significant echo of the well-known quantum bounce in the symmetry-reduced model of loop quantum cosmology, which indicates a closed relation between singularity resolution and quantum geometry.

Figure 1

Behaviors of ${\phi }_{\omega }(j)$ as a function of $\omega$ for various spins $j$. The values of $j$ from the upper left and right to the lower left and right are $j=75/2$, $j=75$, $j=225/2$ and $j=150$ respectively. As shown in the plots, when $\omega$ is far from the maximal and minimal roots of the polynomial ${\phi }_{\omega }(j)$, ${\phi }_{\omega }(j)$ oscillates.

Figure 2

The numerical results of ${\phi }_{\omega }(j)$ in Eq. (42) with initial data ${\phi }_{\omega }(1/2)=1$. We let $\omega =10$ for the upper panel and $\omega =30$ for the other. According to the results, ${\phi }_{\omega }(j)$ converges to two different functions $f_{\omega }^{\pm }(j)$ as $j\to \infty$.

Figure 3

A comparison between ${\phi }_{\omega }(j)$ and $f_{\omega }^{\pm }(j)$ for $\omega =20$. The upper panel shows the numerical results of $|{\phi }_{\omega }(j)|/\sqrt{{\chi }^{\prime }(j)}$ (blue circles) and $|f_{\omega }^{\pm }(j)|/\sqrt{{\chi }^{\prime }(j)}$ (red dots) respectively. The lower panel shows the numerical results of $j|{\phi }_{\omega }(j)-f_{\omega }^{\pm }(j)|$. The envelope of $j|{\phi }_{\omega }(j)-f_{\omega }^{\pm }(j)|$ decreases to 0 as $j$ goes to infinity, and hence ${\phi }_{\omega }(j)$ is well estimated by $f_{\omega }^{\pm }(j)$ up to terms of order $1/j^{\mathrm{\Delta }}$ with $\mathrm{\Delta }>1$ for large $j$.

Figure 4

Plot of normalized ${\phi }_{\omega }(j)$ as a function of $\omega$. As shown by the black line, the initial data ${\phi }_{\omega }(1/2)$ is a Gaussian-like function of $\mathcal{N}{e}^{-\mathfrak{J}(\omega )}$.

Figure 5

Plot of eigenfunction ${\phi }_{\omega }(j)$. For each $\omega$, there is always a barrier at some fixed spin $j$ where the bounce happens.

Figure 6

Plot of evolutions of the expectation value $\sum _{j}j|{\psi }_t(j){|}^2$ with the state ${\psi }_t(j)$ given in Eq. (60) (up panel) and wave function $|{\phi }_{{\omega }_0}(j)|$ (down panel). The parameters are chosen as: $\sigma =1$, ${\omega }_0=100$. The bounce happens close to the “barrier.”

Figure 7

Numerical results of $I_j≔\sqrt{{\chi }^{\prime }(j)}{\psi }_t(j)$, visualized from different viewing angles, where we choose $\sigma =1$, ${\omega }_0=100$. The $\tau$ axis corresponds to the phase $(\varphi -\frac{t}{2\sqrt{{\omega }_0}})$ in Eq. (60).