Quantum gravity for piecewise flat spacetimes

TL;DR

This paper develops a quantum gravity theory based on piecewise linear 4-manifolds, using edge lengths as fundamental variables, and demonstrates how the observed cosmological constant can emerge from the effective action.

Contribution

It introduces a finite Regge path integral approach with a specific measure, connecting classical Regge calculus to quantum field theory and addressing the cosmological constant problem.

Findings

The semi-classical limit reproduces classical Regge gravity.

The effective cosmological constant matches observed values.

Large triangulations lead to a quantum field theory approximation.

Abstract

We describe a theory of quantum gravity which is based on the assumption that the spacetime structure at small distances is given by a piecewise linear (PL) 4-manifold corresponding to a triangulation of a smooth 4-manifold. The fundamental degrees of freedom are the edge lengths of the triangulation. One can work with finitely many edge lengths, so that the corresponding Regge path integral can be made finite by using an appropriate path-integral measure. The semi-classical limit is computed by using the effective action formalism, and the existence of a semi-classical effective action restricts the choice of the path-integral measure. The classical limit is given by the Regge action, so that one has a quantum gravity theory for a piecewise-flat general relativity. By using the effective action formalism we show that the observed value of the cosmological constant can be recovered from the effective cosmological constant. When the number of 4-simplices in the spacetime triangulation is large, then the PL effective action is well approximated by a quantum field theory effective action with a physical cutoff determined by the smallest edge length.