Approaches to Quantum Gravity

294 L. Freidel
under usual gauge transformation. The gauge symmetries of the continuum action
(16.1) are the Lorentz gauge symmetry
ω → g −1 dg + g −1 ωg, e → g −1 eg, (16.8)
locally parameterized by a group element g, and the translational symmetry locally
parameterized by a Lie algebra element φ
ω → ω, e → e + d ω φ (16.9)
and which holds due to the Bianchi identity d ω F = 0. The combination of these
symmetries is equivalent on-shell to diffeomorphism symmetry.
The discrete action (16.3) is invariant under discrete Lorentz gauge transformation
acting at each tetrahedra. This is the analog of the usual gauge symmetry of
lattice gauge theory. Since we consider here Euclidean gravity, the Lorentz group
is a compact group and this gauge symmetry is taken into account by using the
normalized Haar measure in (16.6).
Remarkably the discrete action (16.3) is also invariant under a discrete version
of the translational symmetry. Namely, it is possible to define a covariant derivative
∇ e acting on Lie algebra elements v associated to vertices of the triangulation,
such that the variation
δX e =∇ e (16.10)
leaves the action (16.3) invariant. The discrete covariant derivative reduces to
the usual derivative ∇ e ∼ se − te when the gauge field is Abelian and the
symmetry is due to the discrete Bianchi identity.
Since this symmetry is non-compact we need to gauge fix it in order to define
the partition function and expectation values of observables. A natural gauge fixing
consists of choosing a collection of edges T which form a tree (no loops) and which
is maximal (connected and which goes through all vertices). We then arbitarily fix
the value of X e for all edges e ∈ T. In the continuum this gauge fixing amounts to
choosing a vector field v (the tree) and fixing the value of eμ i vμ ,thatistochosean
“axial” gauge.
Taking this gauge fixing and the Faddev–Popov determinant into account in the
derivation (16.6, 16.7) we obtain the gauge fixed Ponzano–Regge model
Z ,T, j 0 = ∑ ∏ ∏
d je
{ j e } e e∈T
δ je , j
∏
e
0
(d j 0 e
) 2
As a consistency test it can be shown that Z ,T, j 0
choice of maximal tree T and gauge fixing parameter j 0 .
t
{ }
je1 j e2 j e3
. (16.11)
j e3 j e5 j e6
= Z GF
is independent of the