Polymer Geometry at Planck Scale and Quantum Einstein Equations

Abstract

It is well known that quantum general relativity is perturbatively non-renormalizable. Particletheorists often take this to be a sufficient reason to abandon general relativity and seek an alternative which has a better ultraviolet behavior in perturbation theory. However, one is by no means forced to this route. For, there do exist a number of field theories which are perturbatively non-renormalizable but are exactly soluble. An outstanding example is the Gross-Neveau model in 3 dimensions, (GN)₃, which was recently shown to be exactly soluble rigorously [1]. Furthermore, the model does not exhibit any mathematical pathologies. For example, it was at first conjectured that the Wightman functions of a non-renormalizable theory would have a worse mathematical behavior. The solution to (GN)₃ showed that this is not the case; as in familiar renormalizable theories, they are tempered distributions. Thus, one can argue that, from a structural viewpoint, perturbative renormalizability is a luxury even in Minkowskian quantum field theories. Of course, it serves as a powerful guiding principle for selecting physically interesting theories since it ensures that the predictions of the theory at a certain length scale are independent of the potential complications at much smaller scales. But it is not a consistency check on the mathematical viability of a theory. Furthermore, in quantum gravity, one is interested precisely in the physics of the Planck scale; the short-distance complications are now the issues of primary interest. Therefore, it seems inappropriate to elevate perturbative renormalizability to a viability criterion.