Help me write my thesis...

So, you guys should read my introduction (so far) and tell me what you think. At least up to this point I think it should be accessible to everyone. This is probably the best summary of what I'm doing I've written so far, although in truth it says almost nothing about what I've done :) This is more the context stuff. I'll have another section after it dealing with the details of my project.


It is widely known that there is something `wrong' with fundamental physics. We find ourselves with two excellent theories, each predicting the results of experiments with stunning accuracy, but neither of which seems to correctly describe nature in the ultra-high energy regimes where they are expected to intermingle. General relativity gives us gravity through its unique perspective on the structure of spacetime, but the nature of spacetime singularities remains mysterious. Quantum field theory, through the Standard Model, gives us the nuclear forces and electromagnetism, but it is described by a framework in which the matter fields evolve on a fixed, flat, background spacetime. Extensions to curved space can at best be considered semi-classical approximations, for the spacetime structure is not treated on the same footing as the matter fields.
Field theories inevitably require an infinite selection of coupling constants to describe the coupling of fields in infinitely small regions of space. At energy scales well below the Planck scale only finitely many of these coupling constants are required to correctly describe physics, but as we probe higher and higher energies it becomes less valid to make such an approximation. If we try to evaluate the energy of the fields in infinitely small regions of space the energy density also becomes infinite, which according to general relativity should result in the gravitational collapse of the entire universe. This does not seem to happen, so we have a problem.
One road to the resolution of this problem lies in the discretisation of spacetime. If spacetime is divided into discrete elements at or above the Planck scale then a natural cut-off to the summations exists, meaning we no longer have to sum over infinitely small regions. There have of course been many attempts to do this but as yet none of them have succeeded satisfactorily. It would be nice if Planck scale gravity could be described by a field theory of the kind seen in the Standard Model, but such theories run into even worse renormalisation problems than the other fields. Aside from this spin-2 field theories of gravitons are essentially formulated with the gravitons being the quanta of the linearised gravitation field, but because they are based on a linearisation of gravity they must certainly fail at energies when the initial linearising approximations fail, i.e. they fail when the gravitions have sufficient energy to significantly affect the background space of which they are considered to be perturbations. Quantum theories of gravity have thus been aimed at stepping back further and trying to find simpler and more fundamental ingredients which may make up spacetime and possibly everything else.
General relativity is able to describe the structure of spacetime with very few presuppositions about its nature and there is no need for anything to exist `beyond' the resultant manifold: it is entirely `background independent'. Quantum theories of gravity often attempt to duplicate this property: the archetypal string theory is formulated on an essentially Minkowski background space, though this can be entirely altered by fundamental `graviton' strings; dynamical triangulations treat spacetime as composed of fundamental Minkowski `blocks' which may fit together to produce extended spacetimes of essentially any geometry; and loop quantum gravity attempts to rebuild a notion of spacetime from as few ingredients as possible, imposing only the notion of an affine connection. It is required that in the low energy limit these theories generate a nearly-flat, nearly-classical spacetime.
These theories have certainly been developed much more thoroughly than those we will advocate herein, but at the same time they have yet to demonstrate a satisfactory level of self-consistency let alone make any testable predictions. It is thus entirely possible that more radical action is required. In the words of Misner, Thorne and Wheeler Misner:1972 "No-one would dream of studying the laws of elasticity to uncover the principles of quantum mechanics. Neither would anyone investigate the work hardening of a metal to learn about atomic physics...one had to know about atoms to conceive of dislocations, and had to know about dislocations to understand work-hardening. Is it not likewise hopeless to go from the `elasticity of geometry' to an understanding of particle physics, and from particle physics to the uncovering of pre-geometry?"
In this spirit we investigate the fundamental structure of approaches to quantum gravity based on graph theory. These are undoubtedly more radical than general approaches to quantum gravity; the attempt is made to rebuild a concept of spacetime from nothing more than a set of relations. There is no fundamental concept of space, time or even an affine connection, let alone such things as metrics or geometry. The universe can be pictured as nothing more than a mathematical abstraction at this level. The space of possibilities that such a theory generates is staggeringly large so it is almost certain that somewhere in this space there will be a set of relations that mimick classical spacetime. The difficulty lies in explaining why one such configuration may dominate over another.
Theories of this kind have had numerous incarnations. It is believed that Wheeler himself was one of the first to consider this kind of pregeometry Misner:1972, describing the concept as a "sewing machine" stitching together the fabric of spacetime. If we have an appropriate fundamental quantum ingredient we can stitch together whatever geometry we like. Wheeler found that such models did not appeal to his preferred notions of simplicity, suggesting instead the philosophy that, since physics reduces to mathematics and mathematics reduces to logic, the structure of the universe may be fundamentally associated with logical propositions; however it turns out that quantum graphs have connections to quantum information theory -the fundamental building blocks are exactly qubits- so it may be that logical propositions are more related to quantum graphs than Wheeler imagined. We do not investigate this particular question any further, focusing instead on the "stitching" process and the nature of the fundamental ingredients.