Approaches to Quantum Gravity

Three-dimensional spin foam Quantum Gravity 305
Ponzano–Regge model as shown in [2]. In [4] we have shown that this result can
be extended to an arbitrary Feynman diagram Ɣ embedded in R 3 .
More precisely this means that we can evaluate explicitly the amplitude of the
non-planar diagram coupled to Quantum Gravity in terms of a set of local Feynman
rules provided we add to the usual Feynman rules an additional one for each
crossing of the diagrams. The set of Feynman rules is summarized in Fig.16.1.
For each edge of Ɣ we insert a propagator ˜K (g), for each trivalent vertex we
insert a conservation rule δ(g 1 g 2 g 3 ) where g i labels the incoming group valued
momenta at the vertex, and for each crossing of the diagram we associate a weight
δ(g 1 g 2 g
1 ′ −1 g −1 2 ) where g 2 is labeling the edge which is over crossing and the g 1 s
are labelling the edge undercrossing (see Fig.16.1). The Feynman diagram amplitude
for a closed Feynman diagram is then obtained by integrating over all group
momenta.
This completes the description of the Feynman rules and it can be easily shown
that these rules do not depend on the choices of projection and representative
edges.
These Quantum Gravity Feynman rules are exactly the Feynman rules of the
non-commutative field theory introduced above provided that the field entering
the definition of the action (16.49) obeys a non-trivial statistics. Indeed when
we compute the Feynman amplitude from field theory one first has to expand the
expectation value of a product of free field in terms of two point functions using
the Wick theorem. In order to do this operation we first need to exchange the order
of Fourier modes φ(g) ˜ before using the Wick theorem. If the diagram is planar
no exchange of Fourier mode is needed but such exchanges are necessary in the
non-planar case. The specification of the rules of exchange of Fourier modes is a
g 1
º δ(g 1 g 2 g 3 )
g 2 g 3
g 1
º K m (g 1 )
g 1
g 1 g 2
º δ(g 1
g 2
g 1
¢ – 1 g 2
¢ – 1 ) δ(g 2
g 2
¢ – 1 )
g' 2 g' 1
Fig. 16.1. Feynman rules for particles propagation in the Ponzano–Regge model.