Hidden quantum gravity in 3D Feynman diagrams

In this work we show that 3D Feynman amplitudes of standard quantum field theory in flat and homogeneous space can be naturally expressed as expectation values of a specific topological spin foam model. The main interest of the paper is to set up a framework which gives a background-independent perspective on usual field theories and can also be applied in higher dimensions. We also show that this Feynman graph spin foam model, which encodes the geometry of flat spacetime, can be purely expressed in terms of algebraic data associated with the Poincare group. This spin foam model turns out to be the spin foam quantization of a BF theory based on the Poincare group, and as such is related to a quantization of 3D gravity in the limit G(N)-> 0. We investigate the 4D case in a companion paper (Baratin A and Freidel L 2007 Class. Quantum Grav. 24 2027) where the strategy proposed here leads to similar results.