Causality and matter propagation in 3D spin foam quantum gravity

In this paper we tackle the issue of causality in quantum gravity, in the context of 3d spin foam models. We identify the correct procedure for implementing the causality/orientation dependence restriction that reduces the path integral for BF theory to that of quantum gravity in first order form. We construct explicitly the resulting causal spin foam model. We then add matter degrees of freedom to it and construct a causal spin foam model for 3d quantum gravity coupled to matter fields. Finally, we show that the corresponding spin foam amplitudes admit a natural approximation as the Feynman amplitudes of a noncommutative quantum field theory, with the appropriate Feynman propagators weighting the lines of propagation, and that this effective field theory reduces to the usual quantum field theory in flat space in the no-gravity limit.

Figure 1

This is the Riemann surface associated with $\frac{1}{(1-w^2{)}^{3/2}}$. The arrowed lines are the contours of integration in the three cases mentioned in the text.

Figure 2

Another picture of the contours of integration. As one approaches the cut (thick line joining $-1$ and 1) from above one gets positive roots, while approaching from below gives negative ones.