Pinched geometries in $\mathbf{2}$D Lorentzian quantum Regge calculus

TL;DR

This work investigates pinched geometries in a 2D Lorentzian quantum Regge calculus using tensor renormalization group techniques. By representing the partition function as a tensor network and discretizing with generalized Gauss-Laguerre quadrature, the authors test the finiteness of higher moments of time-like edge lengths across different measures and triangulations. Across both regular and 8-4 triangulations, and for multiple integration measures, they find that pinched geometries remain finite and are strongly suppressed as the number of triangles grows, supporting a potential emergence of smooth geometries and suggesting a degree of universality. The results demonstrate the efficacy of TRG methods for Lorentzian quantum gravity models and point toward future explorations with matter fields and triangulation-invariant constructions.

Abstract

We investigate pinched geometries in a two-dimensional Lorentzian model of quantum Regge calculus (QRC) using the tensor renormalization group (TRG) method. A pinched geometry refers to a configuration with an infinitely long temporal extent, even when the total spacetime area is fixed. We examine several choices of integration measures and triangulations to study whether such geometries can dominate in the limit of infinitely many triangles. Our results indicate that pinched geometries are strongly suppressed, and this suppression is observed across different integral measures and triangulations. These results suggest the possible emergence of smooth geometries as well as a sort of universality for infinitely many triangles.

Figures (16)

• Figure 1: A regular triangulation with a time foliation and a single light-cone at each vertex: Thick blue lines, thin red lines and dashed black lines indicate space-like edges, time-like edges and light rays, respectively.
• Figure 2: A triangle defined in the $2$D Minkowski space.
• Figure 3: An illustration of the spike.
• Figure 4: An illustration of the pinched geometry.
• Figure 5: A dual graph of a Lorentzian triangle: Each triangle with the edge lengths, $\sigma$, $\tau_1$ and $\tau_2$, in the triangulation corresponds to a trivalent vertex whose edges respectively carry continuous indices, $z=\sigma^2$, $x=\tau^2_1$ and $y= \tau^2_2$, in the dual picture, and the rank-$3$ tensor $S_{xyz}$ is assigned to each trivalent vertex.