A Topological Approach to Quantum Gravity

The aim of this work is to deal with the problem of the formulation of a theory of Quantum Gravity based on the synthesis of General Relativity and the Quantum Theory. A new formalism is introduced, which is meant to be of mathematical aid to the author's approach, and some axioms are given in order to find simultaneously proof of quantum and gravitational predictions. The relationship between these axioms and the separability of spacetime and the measurability of physical quantities at small scales has been studied keeping in mind that this approach is pre-metrical and therefore there is no notion of distance. Hausdorff separability at small distances and spacetime discreteness is lost while, on the contrary, modern approaches take these properties as basilar. The main result that emerges from this study is the following: the topology of the space is such that the space itself is not Hausdorff but may have only weaker separation properties that does not agree with the usual continuity and discreteness of the space as assumed in modern theories. The proposed physical interpretation is that step by step at small scales the metric becomes meaningless and a granular and quantized microscopic structure emerges. At large scales this structure would lose significance and a commonly accepted notion of metric (and further separability) could be recovered. If possible, the metric types would be of those type as the solutions to General relativity equations. A simple quantum scalar massless eld in a 1+1 space time model in which there are infinite non-separable regions has been studied: specific field boundary conditions are the mathematical translation of the model properties. Thus model can be confronted with a Casimir one and analogies and differences are sought. We introduce two parameters that describe the segment probability distribution and the length growth and study the dependence of the energy density and energy shift on these two parameters. We evaluate the difference between the usual Casimir effect as presented by Ford and the current model where topological holes impose further boundary conditions.