How to construct diffeomorphism symmetry on the lattice

TL;DR

Diffeomorphism invariance is generally broken by lattice discretizations in gravity-like theories; the paper develops a perturbative framework to realize an exact diffeomorphism symmetry at a chosen order $\epsilon$ through a set of consistency conditions and a coarse graining procedure. It applies this to discretized parametrized fields, deriving zeroth, first, and second order improvements that define effective actions on coarse lattices and constrain the form of gauge generators. The discrete parametrized harmonic oscillator serves as a concrete testbed, showing that only certain discretizations are perturbatively consistent at first order while second-order consistency imposes stronger constraints; nonetheless a perfect action can be approached via coarse graining, illustrating a path to triangulation independent, potentially nonlocal discretizations. The findings inform both classical canonical formulations and quantum gravity discretizations by highlighting how exact diffeomorphism invariance can emerge from discrete dynamics and guiding the search for fixed-point, symmetry preserving discretizations.

Abstract

Diffeomorphism symmetry, the fundamental invariance of general relativity, is generically broken under discretization. After discussing the meaning and implications of diffeomorphism symmetry in the discrete, in particular for the continuum limit, we introduce a perturbative framework to construct discretizations with an exact notion of diffeomorphism symmetry. We will see that for such a perturbative framework consistency conditions need to be satisfied which enforce the preservation of the gauge symmetry to the perturbative order under discussion. These consistency conditions will allow structural investigations of diffeomorphism invariant discretizations.