Complex critical points and curved geometries in four-dimensional Lorentzian spinfoam quantum gravity

This paper focuses on the semiclassical behavior of the spinfoam quantum gravity in four dimensions. There has been long-standing confusion, known as the flatness problem, about whether the curved geometry exists in the semiclassical regime of the spinfoam amplitude. The confusion is resolved by the present work. By numerical computations, we explicitly find curved Regge geometries that contribute dominantly to the large-$j$ Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam amplitudes on triangulations. These curved geometries are with small deficit angles and relate to the complex critical points of the amplitude. The dominant contribution from the curved geometry to the spinfoam amplitude is proportional to $e^{i\mathcal{I}}$, where $\mathcal{I}$ is the Regge action of the geometry plus corrections of higher order in curvature. As a result, in the semiclassical regime, the spinfoam amplitude reduces to an integral over Regge geometries weighted by $e^{i\mathcal{I}}$, where $\mathcal{I}$ is the Regge action plus corrections of higher order in curvature. As a by-product, our result also provides a mechanism to relax the cosine problem in the spinfoam model. Our results provide important evidence supporting the semiclassical consistency of the spinfoam quantum gravity.

Figure 1

(a) The ${\mathrm{\Delta }}_3$ triangulation (the center panel) made by gluing three 4-simplices (in blue, red, and purple). The internal triangle (135) is highlighted in red. (b) The triangulation ${\sigma }_{1-5}$ made by the 1-5 Pachner move dividing a 4-simplex into five 4-simplices. ${\sigma }_{1-5}$ has ten internal triangles and five internal segments $I=1,\dots ,5$ (red). (c) The real and complex critical points $\stackrel{\circ }{x}$ and $Z(r)$. $\mathcal{S}(r,z)$ is analytic extended from the real axis to the complex neighborhood illustrated by the red disk.

Figure 2

(a) plots $e^{\lambda \mathrm{Re}(\mathcal{S})}$ versus the deficit angle ${\delta }_h$ at $\lambda ={10}^{11}$ and $\gamma =0.1$ in $A({\mathrm{\Delta }}_3)$, and (b) plots $e^{\lambda \mathrm{Re}(\mathcal{S})}$ versus the deficit angle $\delta =\sqrt{\frac{1}{10}{\sum }_{h=1}^{10}{\delta }_h^2}$ at $\lambda ={10}^{11}$ and $\gamma =1$ in ${\mathcal{Z}}_{{\sigma }_{1-5}}$. These two plots show the numerical data of curved geometries (red points) and the best fits (5.8) and (6.4) (blue curve). (c) and (d) are the contour plots of $e^{\lambda \mathrm{Re}(\mathcal{S})}$ as functions of $(\lambda ,{\delta }_h)$ at $\gamma =0.1$ and of $(\gamma ,{\delta }_h)$ at $\lambda =5\times {10}^{10}$ in $A({\mathrm{\Delta }}_3)$. (e) and (f) are the contour plots of $e^{\lambda \mathrm{Re}(\mathcal{S})}$ as functions of $(\lambda ,\delta )$ at $\gamma =1$ and of $(\gamma ,\delta )$ at $\lambda =5\times {10}^{10}$ in ${\mathcal{Z}}_{{\sigma }_{1-5}}$. They demonstrate the (nonblue) regime of curved geometries where the spinfoam amplitude is not suppressed.

Figure 3

The log-log plot of $|{\delta }_h^s|$ for different $s=\{s_v{\}}_v$ when varying $l_{26}={\stackrel{\circ }{l}}_{26}+\delta l_{26}$.

Figure 4

(a). The dual cable diagram of the ${\mathrm{\Delta }}_3$ spinfoam amplitude: The boxes correspond to tetrahedra carrying $g_a\in \mathrm{SL}(2,\mathbb{C})$. The strands stand for triangles carrying spins $j_f$. The strand with the same color belonging to different dual vertex corresponds to the triangle shared by the different 4-simplices. The circles as the end points of the strands carry boundary states $|j_b,{\xi }_{eb}⟩$. The arrows represent orientations. This figure is adapted from [19]. (b). The dual cable diagram of the 1-5 Pachner move amplitude. The internal faces are colored loops carrying internal spins $j_h$. The boundary faces are black strands carrying boundary spins $j_b$. The arrows represent orientations. This figure is adapted from [32].