Diffeomorphisms from finite triangulations and absence of 'local' degrees of freedom

TL;DR

This paper discusses how the implementation of diffeomorphism symmetry in quantum gravity leads to a topologically invariant partition function, implying no local degrees of freedom and that the Planck scale cutoff is a dynamical effect.

Contribution

It shows that the partition function in quantum gravity can be computed from a triangulation that encodes topology and differentiable structure without introducing physical cut-offs.

Findings

Partition function is a topological invariant of smooth manifolds.

Triangulations can be refined arbitrarily without changing physical predictions.

Physical degrees of freedom are associated only with boundary refinements.

Abstract

If the diffeomorphism symmetry of general relativity is fully implemented into a path integral quantum theory, the path integral leads to a partition function which is an invariant of smooth manifolds. We comment on the physical implications of results on the classification of smooth and piecewise-linear 4-manifolds which show that the partition function can already be computed from a triangulation of space-time. Such a triangulation characterizes the topology and the differentiable structure, but is completely unrelated to any physical cut-off. It can be arbitrarily refined without affecting the physical predictions and without increasing the number of degrees of freedom proportionally to the volume. Only refinements at the boundary have a physical significance as long as the experimenters who observe through this boundary, can increase the resolution of their measurements. All these are consequences of the symmetries. The Planck scale cut-off expected in quantum gravity is rather a dynamical effect.