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On the geometry of loop quantum gravity on a graph

Carlo Rovelli, Simone Speziale

TL;DR

The paper investigates how to assign geometry to loop quantum gravity states living on a fixed graph by examining interpolating discrete geometries. It shows that twisted geometries provide a natural, piecewise-flat interpolation of holonomy-flux data and, in the special case where the data are compatible with a 4D Regge geometry, the twisted-geometry variables precisely reproduce the holonomy-flux computations on that Regge geometry, with the extrinsic-curvature content captured by the relation $\xi = \gamma\,\theta - \alpha$. Moreover, Regge geometries are shown to be a special case within the broader twisted-geometry framework, while generic holonomy-flux data require a more general, potentially discontinuous interpolation and larger phase spaces, including a twistorial extension when area matching is relaxed. This clarifies the relationship between LQG on graphs and discrete gravity formalisms, demonstrating that holonomies and fluxes contain more information than Regge variables and that twisted geometries offer a robust geometric intuition for semiclassical analyses and coherent-state constructions across arbitrary graphs.

Abstract

We discuss the meaning of geometrical constructions associated to loop quantum gravity states on a graph. In particular, we discuss the "twisted geometries" and derive a simple relation between these and Regge geometries.

On the geometry of loop quantum gravity on a graph

TL;DR

The paper investigates how to assign geometry to loop quantum gravity states living on a fixed graph by examining interpolating discrete geometries. It shows that twisted geometries provide a natural, piecewise-flat interpolation of holonomy-flux data and, in the special case where the data are compatible with a 4D Regge geometry, the twisted-geometry variables precisely reproduce the holonomy-flux computations on that Regge geometry, with the extrinsic-curvature content captured by the relation . Moreover, Regge geometries are shown to be a special case within the broader twisted-geometry framework, while generic holonomy-flux data require a more general, potentially discontinuous interpolation and larger phase spaces, including a twistorial extension when area matching is relaxed. This clarifies the relationship between LQG on graphs and discrete gravity formalisms, demonstrating that holonomies and fluxes contain more information than Regge variables and that twisted geometries offer a robust geometric intuition for semiclassical analyses and coherent-state constructions across arbitrary graphs.

Abstract

We discuss the meaning of geometrical constructions associated to loop quantum gravity states on a graph. In particular, we discuss the "twisted geometries" and derive a simple relation between these and Regge geometries.

Paper Structure

This paper contains 5 sections, 20 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Assigning geometries to states
  3. Twisted geometries from Regge geometries
  4. Twisted geometries that are not Regge geometries
  5. Conclusions

Figures (1)

  • Figure 1: A data set and various interpolating functions: polynomial (cfr. mode expansion in cosmology), piecewise linear (cfr. Regge geometries) and piecewise flat (discontinuous, cfr. twisted geometries for generic holonomy-fluxes).