Geometrical measurements in three-dimensional quantum gravity

TL;DR

The paper develops geometrical observables for three-dimensional quantum gravity in the Turaev–Viro framework with positive cosmological constant, focusing on distance measurements between points as discrete spin observables. Distances are encoded by spins with probabilities $P_j = \frac{(\dim_q j)^2}{N}$, linking quantum dimensions to a topological, triangulation-invariant state sum and a geometric interpretation on the 3-sphere $S^3$ in the appropriate limit. A Fourier-transform relation is established between TV invariants and Yetter's relativistic spin-network invariants, enabling a momentum-like representation of embedded graphs and yielding identities among $6j$-symbols via concrete cases (e.g., the tetrahedron). The work connects discrete quantum geometry to classical $S^3$ geometry, clarifies the role of knotting/linking in observables, and provides a framework for higher-point geometrical measurements and their continuum limits, with potential implications for understanding quantum spacetime structure.

Abstract

A set of observables is described for the topological quantum field theory which describes quantum gravity in three space-time dimensions with positive signature and positive cosmological constant. The simplest examples measure the distances between points, giving spectra and probabilities which have a geometrical interpretation. The observables are related to the evaluation of relativistic spin networks by a Fourier transform.

Figures (5)

• Figure 2: State for a triangle
• Figure 3: Possible orbits for $q$
• Figure 4: Geometry for a triangle
• Figure 5: Geodesic triangle on a sphere
• Figure :

Theorems & Definitions (1)

• Theorem