Partial and Complete Observables for Canonical General Relativity

TL;DR

The paper develops a formalism of partial and complete observables for canonical general relativity, enabling the construction of Dirac observables by reducing the problem to a minimal set of constraints. By using space-time scalar partial observables and introducing weakly Abelian diffeomorphism-invariant Hamiltonian constraints, the authors show how GR observables can be computed in a staged fashion and, in favorable choices, reduced to a single constraint. They establish a concrete link between space-time (covariant) observables and canonical (Dirac) observables, and demonstrate this framework with gravity coupled to scalar clocks. The approach provides a path toward manageable quantization by aligning covariant and canonical pictures and clarifying the role of clock variables in defining observables. These results offer a principled method to derive gauge-invariant quantities and pave the way for applying complete observables in quantum gravity contexts.

Abstract

In this work we will consider the concepts of partial and complete observables for canonical general relativity. These concepts provide a method to calculate Dirac observables. The central result of this work is that one can compute Dirac observables for general relativity by dealing with just one constraint. For this we have to introduce spatial diffeomorphism invariant Hamiltonian constraints. It will turn out that these can be made to be Abelian. Furthermore the methods outlined here provide a connection between observables in the space--time picture, i.e. quantities invariant under space--time diffeomorphisms, and Dirac observables in the canonical picture.