Quantum Gravity and Regge Calculus

TL;DR

The paper surveys canonical formulations of general relativity as a gauge theory, emphasizing the connection approach and the loop/spin-net program, and contrasts continuum canonical quantization with discretized schemes inspired by Regge calculus. It identifies a persistent gap between the geometric content of GR and the quantum constructions, centering on the Barbero–Immirzi parameter $\beta$ and the related discretization issues, including the measure and reality conditions for complex connections. It reviews Thiemann's proposals for a well-defined Hamiltonian constraint and the interpretation of area and volume spectra, highlighting a $\beta$-dependent discreteness that cannot alone fix the physical scale. It then analyzes Regge-based discretization in detail, showing how a lattice with area vectors and defect angles leads to challenging dynamics and a potential route via Wick rotation or coherent-state maps to connect $SU(2)$ to $SL(2,\mathbb{C})$, though a fully consistent quantum Regge theory remains elusive.

Abstract

This is an informal review of the formulation of canonical general relativity and of its implications for quantum gravity; the various versions are compared, both in the continuum and in a discretized approximation suggested by Regge calculus. I also show that the weakness of the link with the geometric content of the theory gives rise to what I think is a serious flaw in the claimed derivation of a discrete structure for space at the quantum level.