NONCOMPUTABILITY ARISING IN DYNAMICAL TRIANGULATION MODEL OF 4-DIMENSIONAL QUANTUM-GRAVITY

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 less-than-or-equal-to 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.