Discrete gravitational diagram technique in the soft synchronous gauge

TL;DR

The paper develops a discrete gravity framework using Regge calculus on a hypercubic lattice with a soft synchronous gauge to construct a tractable perturbative expansion. By maximizing a bell-shaped functional measure over edge lengths, it dynamically fixes an edge-length scale b_s ∼ l_Pl√((η−9)/(2π)) with η large, and fixes the timelike scale via a principal-value gauge prescription. The authors derive a refined finite-difference action and transform the measure to enable a Lebesgue-type perturbation, showing that gauge-fixing contributions vanish in the ε → 0 limit and that the discrete theory reproduces continuum GR structures for non-Planckian momenta, while providing a UV cutoff ~ π/b_s. New vertices arising from the metric parametrization enrich the one-loop sector, including corrections to the Newtonian potential, and the framework remains well-behaved when expansions are conducted near the measure’s maximum. Overall, the work offers a concrete, gauge-regularized diagrammatic toolkit for discrete gravity with controlled continuum correspondence and finite quantum corrections at Planck-scale discreteness.

Abstract

This paper develops our work on the consequences of the Regge calculus, where some edge length scale arises as an optimal starting point of the perturbative expansion with taking into account a bell-shaped form of the measure obtained using functional integration over connection. A "hypercubic" structure is considered (some variables are frozen), it is described by the metric $g_{λμ}$ at the sites. The edge length scale as some maximum point of the measure is $\sim η^{1 / 2}$, where $η$ defines the free factor like $ ( - \det \| g_{λμ} \| )^{ η/ 2}$ in the measure and should be a large parameter to ensure true action upon integration over connection. A priori, the perturbative expansion may contain increasing powers of $η$, but this does not happen for the starting point inside some neighborhood of the maximum point of the measure, and it does happen outside this neighborhood. This appears to be a dynamic mechanism for establishing the edge length scale. We use a discrete version of the soft synchronous gauge in the principal value type prescription we discuss in a recent paper arXiv:2601.02181. This allows one to fix the timelike length scale at a low level for which the measure is known in closed form. This gauge is considered together with a refined finite-difference form of the action to match the analytical properties of the propagator to the continuum case.

Figures (1)

• Figure 1: A typical dependence of the measure on the edge length scale $b$ if its maximum is at $b = b_{\rm s}$. An interval of the possible initial $b$ for which there is no a large parameter in the perturbative expansion. The fuzziness of its boundaries means the order-of-magnitude accuracy of estimating their location $O ( 1 )$ relative to $b_{\rm s}$.