Path integral measure and triangulation independence in discrete gravity

A path integral measure for gravity should also preserve the fundamental symmetry of general relativity, which is diffeomorphism symmetry. In previous work, we argued that a successful implementation of this symmetry into discrete quantum gravity models would imply discretization independence. We therefore consider the requirement of triangulation independence for the measure in (linearized) Regge calculus, which is a discrete model for quantum gravity, appearing in the semi–classical limit of spin foam models. To this end we develop a technique to evaluate the linearized Regge action associated to Pachner moves in 3D and 4D and show that it has a simple, factorized structure. We succeed in finding a local measure for 3D (linearized) Regge calculus that leads to triangulation independence. This measure factor coincides with the asymptotics of the Ponzano Regge Model, a 3D spin foam model for gravity. We furthermore discuss to which extent one can find a triangulation independent measure for 4D Regge calculus and how such a measure would be related to a quantum model for 4D flat space. To this end, we also determine the dependence of classical Regge calculus on the choice of triangulation in 3D and 4D.

Figure 1

$3-2$ move. The two tetrahedra can be split into three by connecting the two vertices separated by the shared triangle. The dashed line in the three tetrahedra configuration is the dynamical edge.

Figure 2

$4-1$ move. The tetrahedron is split into four by placing one additional vertex inside the tetrahedron and connecting it to the remaining vertices in the boundary giving four internal edges (dashed).

Figure 3

$4-2$ move. By connecting the vertices (0) and (1) the two 4-simplices are split into four with one bulk edge, here drawn dashed.

Figure 4

$5-1$ move. The 4-simplex is split into five 4-simplices by placing one vertex inside the 4-simplex and connecting it to the boundary vertices, hence obtaining five bulk edges (dashed lines).

Figure 5

$3-3$ move. Three 4-simplices sharing the triangle (012) and not containing (345) are rebuilt into three 4-simplices sharing the triangle (345) and not including triangle (012). The shared triangles are drawn dashed in this figure. Note that all edges are boundary edges and are contained in both configurations, so the configurations are determined by the shared triangle.

Figure 6

As one increases the edge lengths $l_{03}$, one also has to increase $l_{01}$ in order to keep the triangulation flat, i.e. ${\omega }_{01}=0$.

Figure 7

As one increases the edge lengths $l_{23}$, one has to decrease $l_{01}$ in order to keep the triangulation flat, i.e. ${\omega }_{01}=0$.