Canonical Gravity, Diffeomorphisms and Objective Histories

TL;DR

This paper explores how diffeomorphism invariance in Hamiltonian formulations of General Relativity influences the structure of constraints and the interpretation of objective histories, with implications for Loop Quantum Gravity.

Contribution

It clarifies the role of diffeomorphism invariance in Hamiltonian gravity and emphasizes the importance of the transformation properties of basic fields for objective histories.

Findings

Diffeomorphisms are generated by constraints reflecting the algebra of the diffeomorphism group.

Poisson brackets of basic fields with generators encode space-time transformation properties.

Giving up certain invariance properties affects the objectivity of histories in classical and quantum gravity.

Abstract

This paper discusses the implementation of diffeomorphism invariance in purely Hamiltonian formulations of General Relativity. We observe that, if a constrained Hamiltonian formulation derives from a manifestly covariant Lagrangian, the diffeomorphism invariance of the Lagrangian results in the following properties of the constrained Hamiltonian theory: the diffeomorphisms are generated by constraints on the phase space so that a) The algebra of the generators reflects the algebra of the diffeomorphism group. b) The Poisson brackets of the basic fields with the generators reflects the space-time transformation properties of these basic fields. This suggests that in a purely Hamiltonian approach the requirement of diffeomorphism invariance should be interpreted to include b) and not just a) as one might naively suppose. Giving up b) amounts to giving up objective histories, even at the classical level. This observation has implications for Loop Quantum Gravity which are spelled out in a companion paper. We also describe an analogy between canonical gravity and Relativistic particle dynamics to illustrate our main point.