Discrete approaches to quantum gravity in four dimensions

TL;DR

The paper surveys three major discrete approaches to four-dimensional quantum gravity—gauge-theoretic lattice discretizations, quantum Regge calculus, and dynamical triangulations—with emphasis on strictly four-dimensional, discrete, and quantum implementations. Across these frameworks, the common themes are the need for a suitable nonperturbative measure, the appearance of distinct geometric phases (crumpled vs. elongated), and the absence so far of convincing evidence for a nontrivial continuum limit or a second-order phase transition that would yield a realistic interacting theory. While each approach develops sophisticated tools (e.g., Ashtekar variables, Regge actions, and state-sum ensembles) and yields insights into phase structure, correlation functions, and coupling to matter, none has produced a robust, predictive continuum quantum gravity. The findings highlight the central role of the lattice measure and discreteness in shaping the phase diagram and suggest that modifications to measure, inclusion of additional matter, or new nonperturbative frameworks may be required to realize a viable quantum gravity theory.

Abstract

The construction of a consistent theory of quantum gravity is a problem in theoretical physics that has so far defied all attempts at resolution. One ansatz to try to obtain a non-trivial quantum theory proceeds via a discretization of space-time and the Einstein action. I review here three major areas of research: gauge-theoretic approaches, both in a path-integral and a Hamiltonian formulation, quantum Regge calculus, and the method of dynamical triangulations, confining attention to work that is strictly four-dimensional, strictly discrete, and strictly quantum in nature.