Discretization independence implies non-locality in 4D discrete quantum gravity

TL;DR

The paper demonstrates that in 4D discrete quantum gravity, a local path integral measure invariant under certain triangulation moves cannot exist due to non-factorizable Hessian determinants, implying non-locality in the theory.

Contribution

It proves the non-existence of a local invariant measure for 4D linearized Regge theory and introduces a geometric interpretation of the Hessian to understand its properties.

Findings

Hessian determinant does not factorize over simplices.

Invariant local measure under Pachner moves does not exist.

Non-local measure factors can absorb non-local Hessian parts.

Abstract

The 4D Regge action is invariant under 5--1 and 4--2 Pachner moves, which define a subset of (local) changes of the triangulation. Given this fact one might hope to find a local path integral measure that makes the quantum theory invariant under these moves and hence makes the theory partially triangulation invariant. We show that such a local invariant path integral measure does not exist for the 4D linearized Regge theory. To this end we uncover an interesting geometric interpretation for the Hessian of the 4D Regge action. This geometric interpretation will allow us to prove that the determinant of the Hessian of the 4D Regge action does not factorize over 4--simplices or subsimplices. It furthermore allows to determine configurations where this Hessian vanishes, which only appears to be the case in degenerate backgrounds or if one allows for different orientations of the simplices. We suggest a non--local measure factor that absorbs the non--local part of the determinant of the Hessian under 5--1 moves as well as a local measure factor that is preserved for very special configurations.