Tecniche Simpliciali in Gravita' Quantistica: Spin Networks e Spinfoams

The modern research of a quantum theory of the gravitational interaction sees many possible lines of promising developments. A possible strategy is to follow the path determined by a conservative approach, starting from the assumptions that the two most reliable physical theories, i.e. General Relativity and Quantum Mechanics, only apparently disagree. This implies to focus on the possible common characteristics. From a mathematical point of view, General Relativity is constructed by Differential Geometry, while Quantum Mechanics is written in the language of Group Representation Theory. It could be a good idea trying to redefine General Relativity with the mathematical language of Groups. An insightful methodology concerns to treating the geometrical aspects of spacetime just as intrinsically combinatorial. All the components of the spacetime, the arena of physical events, are deprived of any metric property and equipped with labels, a spin representation for the surfaces and an invariant tensor for volumes. The geometrical spacetime is replaced by an abstract network of relations and connections between these labels. In this sense, the spacetime is discretized, and in a wider sense quantized. The building blocks of this new spacetime are simplexes, the generalization in n dimensions of the common triangle. One can develop a set of techniques to make calculations on this new framework, interpreting the gravitational interaction and the curvature of spacetime as a well-defined combinatorial disposition of labels. This is the simplicial approach to Quantum Gravity, which replaces the differential geometry working on the spacetime interpreted as a manifold, with the group representation theory, the language of Quantum Mechanics. The classical manifold and geometry emerge from this combinatorial world in the classical limit (at least one hope they can be shown to emerge!). The strategy adopted to explain the theory and to solve exercises consists to follow two parallel paths: algebraic and the graphic.