Feynman diagrammatic approach to spinfoams

Abstract

The spinfoams for people without the 3D/4D imagination could be an alternative title of our work. We derive spinfoams from operator spin-network diagrams. Our diagrams are the spin-network analog of the Feynman diagrams. Their framework is compatible with the framework of loop quantum gravity (LQG). For every operator spin-network diagram, we construct a corresponding operator spinfoam. Admitting all the spin-networks of LQG and all possible diagrams leads to a clearly defined large class of operator spinfoams. This way our framework provides a proposal for a class of 2-cell complexes that should be used in the spinfoam theories of LQG. Within this class, our diagrams are just equivalent to the spinfoams. The advantage, however, in the diagram framework is that it is self contained and all the amplitudes can be calculated directly from the diagrams without explicit visualization of the corresponding spinfoams. The spin-network diagram operators and amplitudes are consistently defined on their own. Each diagram encodes all the combinatorial information. We illustrate the applications of our diagrams: we introduce a diagram definition of Rovelli’s surface amplitudes as well as of the canonical transition amplitudes. Importantly, our operator spin-network diagrams are defined in a way general enough to accommodate all the versions of the Engle–Pereira–Rovelli–Livine or the Freidel–Krasnov model, as well as other possible models. The diagrams are also compatible with the structure of the LQG Hamiltonian operators, which is an additional advantage. Finally, a scheme for a complete definition of a spinfoam theory by declaring a set of interaction vertices emerges from the examples presented at the end of the paper.