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.

Highlights

  • We study a class of theories in which space-time is treated classically, while interacting with quantum fields

  • 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

  • We here restrict ourselves to the case where the CQ equation has the purely classical evolution generated by the ADM Hamiltonian, the pure quantum evolution generated by the KleinGordon (KG) Hamiltonian and the interaction term a CQ dynamics, whose first moment is such that it approximates the Hamiltonian formulation of gravity in the classical limit

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Summary

CQ theory

We review classical-quantum dynamics and introduce the formalism which provides a basis for the rest of the paper. For a more detailed discussion on CQ dynamics we refer the reader to [24, 28]. We shall first introduce CQ dynamics in its full generality, before focusing on CQ dynamics which reproduces Hamiltonian evolution in the classical limit [28]. We will sketch the derivation of the general form of the master-equation, but the reader can skip directly to it, at equation (2.9) and its simpler form equation (2.10)

Classical-quantum dynamics
Hybrid dynamics with a Hamiltonian limit
CQ theory which reproduces Einstein gravity
Liouville formulation of classical gravity
CQ theories of gravity
Deriving constraints from gauge conditions
Dirac argument for
Deriving the constraint surface of GR from the Dirac argument
Dirac argument for CQ master equations
Deriving constraints in post-quantum theories of gravity
A general method of arriving at constraints
A CQ theory of gravity coupled to a scalar field
Conservation of the momentum constraint Ca for the scalar field
Discussion
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