Discrete Approaches to Quantum Gravity in Four Dimensions

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.

Semiclassical regime of Regge calculus and spin foams

“…The quantum theory is normally defined, for a given triangulation, by the path integral over all edge lengths in the triangulation 3 [37,38,39]:…”

“…These are the questions we set out to investigate in this paper, using Regge calculus [36,37,38,39] as a testing ground. The Regge equivalent of the spin foam boundary observable correlation (1), for a given arbitrary triangulation, is:…”

Unexpected spin-off from quantum gravity

“…This is inspired by quantum gravity, where the "path integral" (a nonperturbative quantum superposition of all spacetime geometries, which is central to the quan-tum dynamics) can be defined via a statistical, weighted sum over triangulated random geometries. The original approach of (Euclidean) Dynamical Triangulations (EDT -see [4,5] for reviews) turns out to be unsuitable for our purposes, because the contributing triangulated lattices are highly curved, with an effectively fractal structure, for any dimension d ≥ 2. Their geometry is so radically different from the usual flat lattices that it alters the universal behaviour of matter systems defined on them, as has been well documented for two-dimensional spin systems (see [6] and references therein).…”

Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners

“…The assumption constitutes the basis of several discrete methods [1], such as dynamical triangulations and Regge calculus, but it also implicitly underlies the older Euclidean path integral approach [2,3] and the somewhat more indirect arguments which suggest that there may exist a nontrivial fixed point of the renormalisation group [4][5][6]. Finally, it is the key assumption which underlies loop and spin foam quantum gravity.…”

“…1 These approaches are background independent in the sense that they do not presuppose the existence of a given background metric. In comparison to the older geometrodynamics approach (which is also formally background independent) they make use of many new conceptual and technical ingredients.…”