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

Abstract

2 the measure to obtain the divergence structure that fits the divergences one expects from diffeomorphism symmetry. This of course does not fully guarantee diffeomorphism symmetry or triangulation independence, as divergences might also arise due to other reasons.In Regge calculus, several measures have been proposed: in [6] Hamber and Williams propose a discretization of the formal continuum path integral, with a local discretization of the (DeWitt) measure [6,7], conflicting with the proposal by Menotti and Peirano [8], who mod out a subgroup of the continuum diffeomorphisms resulting in a highly non-local measure. A different discretization, also leading to a non-local measure due to discretizing first the DeWitt super metric [20] and then forming the determinant, was proposed in [9].In this work we will pick up the suggestion in [21], to choose a measure that, at least for the linearized (Regge) theory, leads to as much discretization invariance as possible. These considerations require to actually integrate out degrees of freedom and thus take the dynamics into account.The requirement of discretisation independence seems to be at odds with interacting theories, which possess local / propagating degrees of freedom. This apparent contradiction can be resolved by allowing a non-local action or nonlocal amplitudes for the quantum theory -which in fact are unavoidable if one coarse grains the theory. Non-local amplitudes are however difficult to deal with. We therefore ask in this paper the question, whether in the quantum theory we can retain as much symmetry as in the classical theory, with a choice of local measure. The classical 4D Regge action is known to be invariant under 5 − 1 moves and 4 − 2 moves, but not under 3 − 3 moves [21]. Here the non-invariance under the 3 − 3 moves -in fact the only move involving bulk curvature for the solution -allows the local Regge action to nevertheless lead to a theory with propagating degrees of freedom. We therefore ask whether it is possible to have a local measure for linearized Regge calculus that leads to invariance under 5 − 1 and / or 4 − 2 moves. Such a measure would therefore reproduce the symmetry properties of the Regge action. We will however show that such a local path integral measure does not exist.Our requirement of (maximal) discretisation independence is motivated by the 'perfect action / discretisation' approach [22] that targets to construct a discretisation, which 'perfectly' encodes the continuum dynamics and has a discrete remnant of the continuum diffeomorphism symmetry. Examples of such 'perfect discretisations' are 3D Regge calculus with and without a cosmological constant [23] and also 4D Regge calculus, if the boundary data impose a flat solution in the bulk. In these examples, the basic building blocks mimic the continuum dynamics, e.g. one takes constantly curved tetrahedra for 3D gravity with a cosmological constant. Such perfect discretisations can be constructed as the fixed point of a coarse graining scheme, see for instance [23][24][25...

3D holography: from discretum to continuum

“…As it is (at least formally) invariant under changes of triangulation [42][43][44][45], it also has a triangulation invariant measure built in. In the semiclassical limit it moreover coincides with the one found in [38,39]. The Ponzano-Regge model is the first example of a spin foam model.…”

“…2 Note that we can rewrite the Regge action as a sum over tetrahedra (plus terms that will not contribute to the Hessian of the action) Here Schläfli identity (3.4) guarantees that ∂θ σ e /∂l e = ∂θ σ e /∂l e . Bulk triangulation independence of the full action (or rather Hamilton's principal function) does also extend to the linearized case [38,39,67].…”

“…To make it also a topological, and for 3D Regge calculus, triangulation invariant theory in the quantum realm, one needs to identify the correct path integral measure. Such a measure is not known for the original quantum Regge calculus, but has been identified for the linearized theory in [38,39].…”

“…has been shown in [38,39] that invariance under changes of the bulk triangulation does fix the measure of 3D Regge calculus uniquely. This measure term does furthermore coincide with the asymptotics [79,80] of the Ponzano-Regge model [40,41], which can be understood as a quantized (first order) version of Regge calculus.…”

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

“…However such a discrete dynamics for four-dimensional gravity, which is implemented (via the 4-simpex gluing) in a local way, breaks four-dimensional diffeomorphism invariance [100][101][102][103]. To restore this symmetry one would have to consider a continuum limit [19,20,[104][105][106], which can be constructed via an auxiliary coarse graining flow [58].…”