Approaches to Quantum Gravity

306 L. Freidel
choice of statistics. In order to reproduce the Quantum Gravity amplitudes we need
to choose a non-trivial statistics where the Fourier modes of the fields are assumed
to obey the exchange relation:
˜φ(g 1 ) ˜φ(g 2 ) = ˜φ(g 2 ) ˜φ(g −1
2 g 1g 2 ). (16.51)
This exchange relation is in fact naturally determined by our choice of star product
and the duality between space and time (plane waves). Indeed, let us look at the
product of two identical fields:
∫
φ⋆φ(X) =
dg 1 dg 2 e 1
2κ tr(Xg 1g 2 )˜φ(g 1 )˜φ(g 2 ). (16.52)
We can ‘move’ ˜φ(g 2 ) to the left by making the following change of variables g 1 →
g 2 and g 2 → g −1
2 g 1g 2 , the star product reads
∫
φ⋆φ(X) = dg 1 dg 2 e 2κ 1 tr(Xg 1g 2 )˜φ(g 2 )˜φ(g −1
2 g 1g 2 ). (16.53)
The identification of the Fourier modes of φ⋆φ(X) leads to the exchange relation
(16.51).
This commutation relation is exactly the one arising from the braiding of two
particles coupled to Quantum Gravity. This braiding was first proposed in [24]and
computed in the spin foam model in [2]. It is encoded into a braiding matrix
R · ˜φ(g 1 ) ˜φ(g 2 ) = ˜φ(g 2 ) ˜φ(g −1
2 g 1g 2 ). (16.54)
This is the R matrix of the κ-deformation of the Poincaré group [24]. We see that
the non-trivial statistics imposed by the study of our non-commutative field theory
is related to the braiding of particles in three spacetime dimensions. This non-trivial
braiding accounts for the non-trivial gravitational scattering between two matter
particles. Such field theories with non-trivial braided statistics are usually simply
called braided non-commutative field theories and were first introduced in [25].
16.8 Generalizations and conclusion
Our results naturally extend to the Lorentzian theory. Although a direct derivation
of the spin foam model from the continuum theory is still lacking, a Lorentzian
version of the Ponzano–Regge model has been written down [26; 27] and the topological
state sum is formulated in terms of the {6 j} symbols of SU(1, 1). One can
already apply existing gauge fixing techniques [2; 28] to regularize the amplitudes
based on a non-compact gauge group. Moreover, particles are once again inserted
as topological defects creating conical singularities and a similar (almost identical)
effective non-commutative field theory can be derived from the spin foam
amplitudes.