A perturbative approach to Dirac observables and their space-time algebra

TL;DR

The paper develops a perturbative method to construct Dirac observables in general relativity by leveraging complete observables and an Abelianization of constraints around a fixed background, here the Minkowski space. Observables are expanded order-by-order in fluctuations, yielding a concrete link between full GR observables and linearized-field observables, with explicit first nontrivial corrections and a framework to study their space-time Poisson algebra. Dynamics are implemented via a time evolution generated by the ADM energy at infinity, and the second-order analysis includes gravity-matter interactions (notably a scalar field) and graviton backreaction, while the locality properties depend on the clock variables. The approach provides a structured path to gauge-invariant descriptions of gravitationally interacting fields and offers insights into the locality and causality of measurements in a background-plus-gravity setting, with potential implications for quantum gravity observables and their algebraic structure.

Abstract

We introduce a general approximation scheme in order to calculate gauge invariant observables in the canonical formulation of general relativity. Using this scheme we will show how the observables and the dynamics of field theories on a fixed background or equivalently the observables of the linearized theory can be understood as an approximation to the observables in full general relativity. Gauge invariant corrections can be calculated up to an arbitrary high order and we will explicitly calculate the first non--trivial correction. Furthermore we will make a first investigation into the Poisson algebra between observables corresponding to fields at different space--time points and consider the locality properties of the observables.