Noncomputability arising in dynamical triangulation model of four-dimensional Quantum Gravity

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 ≤ 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.

Further evidence that the transition of 4D dynamical triangulation is 1st order

“…We have simulated the system at volumes of 32, 000 and 64, 000 simplices at several values of κ 2 close to the phase transition. There is no known method to use ergodic moves that keep the volume constant and probably no such method can exist [8]. It is known that such a method cannot exist for manifolds that are unrecognizable and also that some 4-manifolds indeed are unrecognizable, although for the 4-sphere used in our simulations this is not known.…”

Computational ergodicity of S4

“…The algorithmic unrecognizability of M 0 means that there exists no algorithm which allows us to decide whether another manifold M, again finitely presented by a triangulation T (M) is combinatorially equivalent to M 0 . When this is combined with the existence of the finite set of local moves which are able to connect any two triangulations of M 0 in a finite number of steps, but where this number is a function of the chosen triangulations, one can prove the following theorem [8]:…”

“…In ref. [8] it was conjectured that it will not be the case and some plausibility arguments in favor of the conjecture were given. If the conjecture is correct, a Monte Carlo method based on the finite set of local moves will never get around effectively in the class of triangulations of M 0 .…”

The Geometry of Dynamical Triangulations

“…It is also interesting to note that according to relation (42) the incidence numbers {q(σ n−2 )} of a simplicial manifold cannot be provided by a monotonically increasing sequence of integers, for otherwise Σ σ n−2 q(σ n−2 ) would be O(N 2 n ) in plain contrast with (42).…”

“…In particular, from (42) its is immediate to prove that if we let q max . = sup T (N,B) sup σ n−2 {q(σ n−2 )} denote the maximum value of q(σ n−2 ), as σ n−2 varies in the set of triangulations with N simplices σ n and given value of b(n, n − 2), then, for N n−2 >> 1,…”