Relational Observables in Gravity: a Review

TL;DR

The paper surveys relational observables in gravity, focusing on the complete observable formalism and the role of dynamical reference frames to address the problem of time. It outlines a general construction for gauge-invariant observables and discusses perturbative approaches, as well as de-parameterizable toy-models using dust or scalar fields to obtain explicit observables. It then discusses the quantum theory, contrasting Dirac versus reduced phase space quantization, and the status of spectra of geometric operators in loop quantum gravity. The work highlights how relational and gauge-invariant formalisms may yield physically meaningful predictions in background-independent quantum gravity and identifies key open problems and directions for development.

Abstract

We present an overview on relational observables in gravity mainly from a loop quantum gravity perspective. The gauge group of general relativity is the diffeomorphism group of the underlying manifold. Consequently, general relativity is a totally constrained theory with vanishing canonical Hamiltonian. This fact, often referred to as the problem of time, provides the main conceptual difficulty towards the construction of gauge-invariant local observables. Nevertheless, within the framework of complete observables, that encode relations between dynamical fields, progress has been made during the last 20 years. Although analytic control over observables for full gravity is still lacking, perturbative calculations have been performed and within de-parameterizable toy models it was possible for the first time to construct a full set of gauge invariant observables for a background independent field theory. We review these developments and comment on their implications for quantum gravity.