Relating Covariant and Canonical Approaches to Triangulated Models of Quantum Gravity

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.

28 pages, 10 figures

References in corpus (5)

• Dynamically Triangulating Lorentzian Quantum Gravity
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Cited by in corpus (9)

• Spin Foam Models for Quantum Gravity
• Three dimensional loop quantum gravity: physical scalar product and spin foam models
• Canonical simplicial gravity
• Implementing causality in the spin foam quantum geometry
• Positivity of Spin Foam Amplitudes
• On the causal Barrett--Crane model: measure, coupling constant, Wick rotation, symmetries and observables
• A causal perspective on random geometry
• Relating Spin Foams and Canonical Quantum Gravity: A Discrete Step Evolution Formulation of Spin Foams
• On the Classical Limit of Spin Network Gravity: Two Conjectures