Approaches to Quantum Gravity

Three-dimensional spin foam Quantum Gravity 291
model. We show how this definition can be related to the discretization of the
gravity action, we then show that a residual gauge symmetry still present in the
system should be gauge fixed and that this eliminates unwanted symmetry; we then
describe the coupling of matter to 3d gravity and show how it can be effectively
described in terms of a non-commutative braided quantum field theory.
16.2 Classical gravity and matter
In the first order formalism, 3d gravity is described in term of a frame field eμ i dxμ
and a spin connection ωμ i dxμ .TheyarebothvaluedintheLiealgebraso(3) for the
Euclidean theory, while they would be in so(2, 1) in the Lorentzian theory. Both
indices i and μ run from 0 to 2. The action is defined as:
S[e,ω]= 1
16πG
∫
e i ∧ F i [w], (16.1)
where F ≡ dω + ω ∧ ω is the curvature tensor of the 1-form ω. The equation of
motion for pure gravity then simply imposes that the connection is flat,
F[ω] =0.
The second equation of motion imposes that the torsion vanishes, T = d ω e = 0.
Spinless particles are introduced as a source of curvature (the spin would be
introduced as a source of torsion):
F i [ω] =4πGp i δ(x).
Outside the particle, the space-time remains flat and the particle simply creates a
conical singularity with deficit angle related to the particle’s mass [6; 7]:
θ = κm, κ ≡ 4πG. (16.2)
This deficit angle gives the back reaction of the particle on the space-time
geometry.
Since the deficit angle is obviously bounded by 2π, we have a maximal mass
(for a point-particle) which defines the Planck mass:
m ≤ m max = 2π κ = 1
2G = m P.
The fact that the Planck mass m P in three space-time dimensions depends only
on the Newton constant G and does not depend on the Planck constant is an
essential feature of 3d Quantum Gravity and explains why 3d gravity possesses
such surprising features as an ADM energy bounded from below and above.