Approaches to Quantum Gravity

The group field theory approach to Quantum Gravity 317
that meet at the vertex i. In particular, the transition amplitude (probability amplitude
for a certain scattering process) between certain boundary data represented by
two spin networks, of arbitrary combinatorial complexity, can be expressed as the
expectation value of the field operators having the same combinatorial structure of
the two spin networks [2]:
∫
〈 1 | 2 〉=
Dφ O 1 O 2 e −S(φ) =
∑
Ɣ/∂Ɣ=γ 1 ∪γ 2
λ N
sym[Ɣ] Z(Ɣ),
where the sum involves only 2-complexes (spin foams) with boundary given by the
two spin networks chosen.
The above perturbative expansion involves therefore two very different types of
sums: one is the sum over geometric data (group elements or representations of
G) which is the GFT analog of the integral over momenta or positions of usual
QFT; the other is the sum over Feynman diagrams. This includes a sum over all
triangulations for a given topology and a sum over all topologies (since all possible
gluings of D-simplices and face identifications are present by construction
in the GFT Feynman expansion). Both sums are potentially divergent. First of all
the naive definition of the Feynman amplitudes implies a certain degree of redundancy,
resulting from the symmetries of the defining GFT. A proper gauge fixing
of these symmetries, especially those whose group is non-compact, is needed to
avoid divergences [16]. Even after gauge fixing, the sum over geometric data has
a potential divergence for every “bubble” of the GFT Feynman diagram, i.e. for
every closed collection of 2-cells. This is the GFT analog of loop divergences of
the usual QFT. Of course, whether the GFT amplitudes are divergent or not depends
on the specific model. 3 In general a regularization and perturbative renormalization
procedure would be needed, but no systematic study of GFT renormalization has
been carried out to date, despite its obvious importance. The sum over Feynman
diagrams, on the other hand, is most certainly divergent. This is not surprising.
The sum over Feynman diagrams gives a sum over all triangulations for all topologies,
each weighted by a (discrete) Quantum Gravity sum-over-histories. That such
a sum can be defined constructively thanks to the simplicial and QFT setting is
already quite an achievement, and to ask for it to be finite would be really too
much! Also, from the strictly QFT perspective, it is to be expected that the expansion
in Feynman diagrams of a QFT would produce at most an asymptotic series
and not a convergent one. This is the case for all the interesting QFTs we know of.
What makes the usual QFT perturbative expansion useful in spite of its divergence
3 For example, while the most natural definition of the group field theory for the Barrett–Crane spin foam
model [7], presents indeed bubble divergences, a simple modification of it [18; 19], possesses finite Feynman
amplitudes, i.e. it is perturbatively finite without the need for any regularization.