Spinfoam tunneling of quantum geometries in angle variables

TL;DR

The paper investigates spinfoam tunneling in the holonomy (angle) representation within the Ponzano–Regge model of 3D Euclidean gravity to elucidate how classically forbidden quantum geometries contribute to amplitudes. It shows that, unlike in length-based pictures, the dominant tunneling contributions are non-classical geometries whose effects are captured by analytic continuation of the Regge action and by a scale-symmetric sum over holonomy data, revealing the mechanism behind extrinsic curvature flips relevant to black-to-white hole transitions. The authors demonstrate that using holonomy eigenstates clarifies the tunneling process and highlight limitations of coherent extrinsic boundary states, proposing a scale-fixing procedure via area Gram cofactors and suggesting a hybrid boundary state as a potential improvement. While performed in 3D Euclidean gravity, the results provide a controlled setting to generalize to 4D Lorentzian spinfoams, potentially impacting phenomenology of black hole transitions and broader quantum-gravity tunneling phenomena.

Abstract

Tunneling processes offer a promising path for finding signatures of quantum gravity. While tunneling of geometry has long been recognized in the literature, few detailed analyses in covariant Loop Quantum Gravity have been carried out. We investigate spinfoam transitions in the holonomy representation, which naturally encodes the extrinsic curvature of boundary states. To reduce technical complications to a minimum, we study these amplitudes within the simple framework of the Ponzano-Regge spinfoam model for three-dimensional Euclidean quantum gravity. We identify the geometries dominating the spinfoam path integral in the classically forbidden regime when formulated in terms of dihedral angles as boundary data. We characterize these non-classical geometries and show that their contributions to the spinfoam amplitude are exponentially suppressed in the semiclassical limit via analytic continuation of the discrete gravity action. We argue that they satisfy all the desired properties of tunneling processes. We also shed light on quantum black-to-white-hole transitions, in particular clarifying the origin of the exponential suppression of various quantum amplitudes, while at the same time laying the basis for a future complete calculation of the amplitude in covariant Loop Quantum Gravity.

Figures (2)

• Figure 1: Canonical evolution of a surface discretized by two triangles, shown at an initial stage (left pair) and a final stage (right pair), that is, you can think of "time" as flowing from left to right in each panel. Left panel – the boundary spin network is highlighted in blue, with spin labels $j_{ab}$. Right panel – the boundary spin network is highlighted in green, with holonomy labels $h_{ab}$.
• Figure 2: Classically allowed (blue) and forbidden transitions (red) between symmetric boundary data. We identify $\phi_{12} = \phi_{34}$ and $\phi_{13} = \phi_{24}$, fix $\phi_{14} = \phi_R= \arccos\left(-\tfrac{1}{3}\right)$, and determine $\phi_{23}$ using the closure condition. We select the solution compatible with the presence of the regular tetrahedron, highlighted by a gray triangle in the plot. The blue region corresponds to configurations where the transition is allowed (Euclidean signature), while the red background indicates classically forbidden configurations (Lorentzian signature). In the top-right corner, we shade the region where not all edge lengths are time-like and $l_{23}$ becomes spacelike. We extend the boundary between the two signature regions to include non-positive area vectors, shown as a dashed purple line.