In looking for an alternate theory for the unification of quantum phenomena and gravitation, it is conventient to enumerate a set of basic principles that are held onto, to distinguish them from other phenomena that can be more lightly dismissed.

This is an easy argument to make: it fits very well with the first-order quantization atomic physics, and it also is central to the second quantization of field theory (in the form of Feynmann Path Integrals). In addition, it's notion of "paths" coexists perfectly with notions of geometric manifolds, thus, its in harmony with general realtivity. Finally, recent results from statistical mechanics/chaos theory shows how to go from a collection of paths (which may be differentiable nowhere, as the Feynmann integral implies) to a set of smooth, differentiable densities that break time symmetry (results of Driebe et al).

One of the "problems" of attempts to unify gravity with quantum mechanics is the need to define "tunnelling" between different topologies; that is, how to define a quantum mechanical smearing between different topologies. But one can use this to make an argument that such an approach is fundamentally wrong. An alternate, and more beleivable argument, is that baryons and leptons owe thier stability due to topological constraints. Thus, tunneling between topologies would seem to destroy the stability of baryons and leptons. This argument can be put another way: Classical General Relativity does not "explain" or "admit" spin-1/2 particles. This is because the space-like symmetry subgroup of special relativity (Lorentz transformations) is the rotation group O(3), whereas we know from experiment that spin-1/2 particles are governed by the symmetries of SU(2). But this "problem" is in fact suggestive of an alternate explanation: it is precisely the mis-match between the space-like subgroup symmetry O(3) of the Lorentz group and the SU(2) symmetries of spin-1/2 particles that lends them a topological stability in real life. Thus, in searching for a unified theory, one cannot (easily) make topological stability arguments if one entertains theories that allow "tunneling" between different geometries.

The idea that microscopic space-time needs to be four dimensional, five dimensional (Kaluza-Klein) or higher. Its not at all obvious that these apparent symmetries might not be macroscopic manifestations a very a very different underlying dimensionality, such as an extremely topologically interconnected two-dimensional manifold, or of a progression of random matrices.

For example, imagine a three-dimensional cube. Its faces are two dimensional surfaces. The surface of a cube is topologically equivalent to the two-sphere, and a metric on its surface is two-dimensional. Now imagine connecting opposite surfaces with two-dimensional "wormholes" or topological handles. This object now resembles a three-dimensional periodic (cubic) lattice: that is, differential equations in it would be subjected to periodic boundary conditions of a cubic lattice. Subject to small perturbations, one might expect "phonons", and one might find that the resulting dispersion relations take on a three-dimensional flavour, as determined by the toplogy, rather than a two-dimensional character as determined by the local surface metric. By similar arguments, one might expect that such a highly-connected two-dimensional surface might have an "effective topology" of Minkowski space, or of some higher dimensional manifold. It would not be a bad ansatz to take the local 2D metric to be that of the complex plane: the complex plane alone has quite a rich set of analytic tools and features.

By the same arguments, we can conclude that another discardable notion is that the unified theory must be super-symmetric in some way, or that it must involve strings. Clearly, one can make many interesting algebraic statements with such theories, but that should not be mistaken for a basic tenet.

Copyright (c) 2002 Linas Vepstas <linas@linas.org>