Noncomputability Arising In Dynamical Triangulation Model Of Four-Dimensional Quantum Gravity

TL;DR

This paper demonstrates the nonexistence of an algorithm to generate all triangulations of four-dimensional manifolds with a given number of simplices, highlighting fundamental computational limitations in quantum gravity models.

Contribution

It proves that no algorithm can list all triangulations of 4D manifolds up to a certain size, revealing inherent noncomputability in quantum gravity computations.

Findings

No algorithm exists to generate all triangulations with ≤ N simplices.

Recursion-theoretic limitations affect approximate calculations in quantum gravity.

Highlights fundamental computational barriers in dynamical triangulation models.

Abstract

Computations in Dynamical Triangulation Models of Four-Dimensional Quantum Gravity involve weighted averaging over sets of all distinct triangulations of compact four-dimensional manifolds. In order to be able to perform such computations one needs an algorithm which for any given $N$ and a given compact four-dimensional manifold $M$ constructs all possible triangulations of $M$ with $\leq N$ simplices. Our first result is that such algorithm does not exist. Then we discuss recursion-theoretic limitations of any algorithm designed to perform approximate calculations of sums over all possible triangulations of a compact four-dimensional manifold.