The constraints of post-quantum classical gravity

TL;DR

This paper explores a class of classical-quantum gravity theories that avoid no-go theorems, analyzing their constraints, algebra, and implications for back-action of quantum fields on spacetime.

Contribution

It introduces a methodology to derive generalized Hamiltonian and momentum constraints in classical-quantum gravity theories and analyzes their algebraic structure.

Findings

The constraint algebra generally does not close without additional constraints.

The theories recover general relativity in the classical limit.

Additional constraints do not necessarily reduce local degrees of freedom.

Abstract

We study a class of theories in which space-time is treated classically, while interacting with quantum fields. These circumvent various no-go theorems and the pathologies of semi-classical gravity, by being linear in the density matrix and phase-space density. The theory can either be considered fundamental or as an effective theory where the classical limit is taken of space-time. The theories have the dynamics of general relativity as their classical limit and provide a way to study the back-action of quantum fields on the space-time metric. The theory is invariant under spatial diffeomorphisms, and here, we provide a methodology to derive the constraint equations of such a theory by imposing invariance of the dynamics under time-reparametrization invariance. This leads to generalisations of the Hamiltonian and momentum constraints. We compute the constraint algebra for a wide class of realisations of the theory (the "discrete class") in the case of a quantum scalar field interacting with gravity. We find that the algebra doesn't close without additional constraints, although these do not necessarily reduce the number of local degrees of freedom.