Diffeomorphism symmetry in quantum gravity models

TL;DR

This paper surveys how space-time diffeomorphism symmetry is realized, broken, or approximated in discretized quantum gravity frameworks such as Regge calculus and spin foams. It analyzes exact and approximate symmetries of the discretized action using Bianchi identities and Hessian tests, and connects covariant discretizations to canonical formulations through consistent discretization and simple mini-triangulation models. It discusses discrete hypersurface deformation algebras and their representations in 3d gravity and restricted 4d sectors, and considers implications for loop quantum gravity and spin foam amplitudes, including measure issues and projector properties. The overarching message is that recovering the correct continuum diffeomorphism symmetry requires careful treatment of triangulation dependence, constraint structure (including potential second-class constraints), and the continuum limit, with approaches such as master constraints or uniform discretization offering viable routes to a consistent quantum theory.

Abstract

We review and discuss the role of diffeomorphism symmetry in quantum gravity models. Such models often involve a discretization of the space-time manifold as a regularization method. Generically this leads to a breaking of the symmetries to approximate ones, however there are incidences in which the symmetries are exactly preserved. Both kind of symmetries have to be taken into account in covariant and canonical theories in order to ensure the correct continuum limit. We will sketch how to identify exact and approximate symmetries in the action and how to define a corresponding canonical theory in which such symmetries are reflected as exact and approximate constraints.

Figures (1)

• Figure 1: A three--valent vertex is evolved in discrete steps by erecting a pole and connecting the other vertices to the end of the pole. The different types of line indicate the different time steps.