Approaches to Quantum Gravity

The group field theory approach to Quantum Gravity 315The Feynman amplitudes can be constructed easily after identification of the propagatorand the vertex amplitude. Each edge of the Feynman diagram is made of Dstrands, one for each argument of the field, and each one is then re-routed at theinteraction vertex, with the combinatorial structure of an D-simplex, following thepairing of field arguments in the vertex operator. This is shown diagrammaticallyas follows.123123KD − 1DD − 1D1D + 12V3Each strand goes through several vertices, coming back to where it started, forclosed Feynman diagrams, and therefore identifies a 2-cell. Each Feynman diagramƔ is then a collection of 2-cells (faces), edges and vertices, i.e. a 2-complex, that,because of the chosen combinatorics for the arguments of the field in the action,is topologically dual to a D-dimensional simplicial complex [7; 17]. Clearly, theresulting complexes/triangulations can have arbitrary topology, each correspondingto a particular scattering process of the fundamental building blocks of space, i.e.(D − 1)-simplices. The D-dimensional triangulation dual to the 2-complex, arisingas a GFT Feynman diagram, would not necessarily be a simplicial manifold,asthedata in the GFT Feynman diagrams do not constrain the neighborhoods of simplicesof dimensions from (D − 3) downwards to be spheres. In the general case,the resulting simplicial complex, obtained by gluing D-simplices along their (D −1)-faces, would correspond to a pseudo-manifold, i.e. to a manifold with conicalsingularities [7; 17; 37]. A precise set of conditions under which the GFT Feynman