Relating Covariant and Canonical Approaches to Triangulated Models of Quantum Gravity

TL;DR

This paper establishes a connection between covariant and canonical quantum gravity models, specifically dynamical triangulations and spin-foam models, by deriving covariant models from a canonical framework and analyzing the role of the Immirzi parameter.

Contribution

It introduces two regularizations of the Hamiltonian constraint projector to relate covariant and canonical approaches, clarifying the dynamics of quantum gravity models.

Findings

Recovered Hamiltonian constraint in dynamical triangulations

Linked spin-network moves to time evolution in spin-foam models

Highlighted the importance of the Immirzi parameter for continuum limits

Abstract

In this paper explore the relation between covariant and canonical approaches to quantum gravity and $BF$ theory. We will focus on the dynamical triangulation and spin-foam models, which have in common that they can be defined in terms of sums over space-time triangulations. Our aim is to show how we can recover these covariant models from a canonical framework by providing two regularisations of the projector onto the kernel of the Hamiltonian constraint. This link is important for the understanding of the dynamics of quantum gravity. In particular, we will see how in the simplest dynamical triangulations model we can recover the Hamiltonian constraint via our definition of the projector. Our discussion of spin-foam models will show how the elementary spin-network moves in loop quantum gravity, which were originally assumed to describe the Hamiltonian constraint action, are in fact related to the time-evolution generated by the constraint. We also show that the Immirzi parameter is important for the understanding of a continuum limit of the theory.