Effective Spin Foam Models for Four-Dimensional Quantum Gravity

A number of approaches to four-dimensional quantum gravity, such as loop quantum gravity and holography, situate areas as their fundamental variables. However, this choice of kinematics can easily lead to gravitational dynamics peaked on flat spacetimes. We show that this is due to how regions are glued in the gravitational path integral via a discrete spin foam model. We introduce a family of “effective” spin foam models that incorporate a quantum area spectrum, impose gluing constraints as strongly as possible, and leverage the discrete general relativity action to specify amplitudes. These effective spin foam models avoid flatness in a restricted regime of the parameter space.

Figure 1

The $G$ factors (dashed lines), which impose the matching conditions weakly, and the real part of the product of the amplitude factors ${\mathcal{A}}_t$ and ${\mathcal{A}}_{\sigma }$ as functions of the bulk spin $j_{\mathrm{blk}}$ for various $\gamma$ (solid lines) and for an intermediate (a) and a large (b) spin value. Here, the $G$ factors peak on a curvature value $\epsilon \approx 0.5$. Larger $\gamma$’s lead to a more oscillatory behavior. These examples are detailed in the Supplemental Material [45].