Approaches to Quantum Gravity

The group field theory approach to Quantum Gravity 321
perturbative expansion is Borel summable. The modification amounts to adding
another vertex term to the original one, given by:
+ λδ 6∏
∫
dg i [ φ(g 1 , g 2 , g 3 )φ(g 3 , g 4 , g 5 )φ(g 4 , g 2 , g 6 )φ(g 6 , g 5 , g 1 )] | δ |< 1.
4!
i=1
(17.5)
The new term corresponds simply to a slightly different recoupling of the
group/representation variables at each vertex of interaction, geometrically to the
only other possible way of gluing four triangles to form a closed surface. This
result is interesting for more than one reason: (1) it shows that it is possible to control
the sum over triangulations of all topologies appearing in the GFT perturbative
expansion; (2) even if it has no clear physical interpretation yet from the Quantum
Gravity point of view, it is indeed a very mild modification, and most importantly
one likely to be forced upon us by renormalization group arguments, that usually
require us to include in the action of our field theory all possible terms that are
compatible with the symmetries. The restriction of the 3D Boulatov model for a
real field to the homogeneous space SO(3)/SO(2) ≃ S 2 [25], and with the global
SO(3) invariance having been dropped, gives a generalization of the tensor model
(17.3) with action:
S[φ] = ∑ 1
2 φ j 1 j 2 j 3
α 1 α 2 α 3
φ j 1 j 2 j 3
α 1 α 2 α 3
+ λ ∑
φ j 1 j 2 j 3
α
4!
1 α 2 α 3
φ j 3 j 4 j 5
α 3 α 4 α 5
φ j 5 j 2 j 6
α 5 α 2 α 6
φ j 6 j 4 j 1
α 6 α 4 α 1
j i ,α i j i ,α i
where the indices α i run over a basis of vectors in the representation space j i , and
its partition function is: Z = ∑ ∑ ∏
Ɣ
j f f (2 j f + 1), with f being the faces
(−λ) n Ɣ
sym(Ɣ)
of the 2-complex/Feynman graph, which is divergent and has to be regularized.
There are three ways of doing it, all a tensor model as a result: (1) simply dropping
the sum over the representations j f by fixing them to equal a given J; (2) placing
a cut-off on the sum by restricting j i < N, obtaining Z = ∑ (−λ) nv(Ɣ)
Ɣ sym(Ɣ) [(N +
1) 2 ] n f (Ɣ) ; (3) equivalently, but more elegantly, by defining the model not on S 2
but on the non-commutative 2-sphere SN 2 , which also carries a representation of
SU(2) but implies a bounded decomposition in spherical harmonics (labeled by
j < N), thus giving the same result for the partition function. We recognize in the
above result the partition function for the tensor model (17.3) and for a dynamical
triangulations model [14].
Let us now discuss the 4D case. Here GFT model building has followed the
development of spin foam models for 4D Quantum Gravity (see chapter 15 by
Perez). The guiding idea has been the fact that classical gravity can be written as a
constrained version of a BF theory for the Lorentz group. The Barrett–Crane spin
foam models in fact [8] amount roughly to a restriction of spin foam models for
BF theories to involve only simple representations of the Lorentz group (SO(4)