Complexity in quantum gravity

Abstract

There are two difficulties with quantum gravity. One is that the theory has to be quantum mechanical, and the other is that it has to be invariant under general coordinate transformations. Presumably the resolution of these difficulties will lie in certain delicate revisions that will have to be considered for both disciplines. Often, the fact that perturbative quantum gravity is in excellent shape, is not sufficiently appreciated. The theory is admittedly nonrenormalizable, which means that at every order in the perturbative expansion, new uncalculable “constants of nature” emerge. The number of uncalculable constants at every order is quite small however, certainly in comparison with the amount of information that the calculations could provide. There are two reasons why this theory is usually completely dismissed. One is its tremendous complexity, as already at low orders the number of algebraical manipulations needed in the calculations is gigantic. Secondly, of course, the emergence of uncomputable numbers (nonrenormalizability) renders the theory useless at the Planck scale. It is clear that a nonperturbative formulation of quantum gravity will have to be entirely different, but it is important to observe that most of the proposed alternatives to perturbative quantum gravity are actually much less predictive than the simple perturbation theory. (Super)string theory is making rather vociferous claims for the status of “only consistent theory of quantum gravity,” but this theory, too, is formulated perturbatively; the expansion, here, is one in topological complexity of string diagrams, and this expansion is as hopelessly divergent as the ordinary perturbative theory. However, the fact that, at each given order, there are no unknown counter terms strongly suggests a more powerful nonperturbative underlying system waiting to be uncovered. Yet there is a danger in such expectations, which can be illustrated by observing that there are many ordinary quantum field theories that have unique perturbative expensions but show completely new physics at the nonperturbative level. Therefore, what is needed is a fundamentally nonperturbative formulation of a theory.