On the initial value problem for semiclassical gravity without and with quantum state collapses
Authors: Juárez-Aubry Benito A. , Kay Bernard S. , Miramontes Tonatiuh , Sudarsky Daniel 
Published: January 2023

Abstract
Semiclassical gravity is the theory in which the classical Einstein tensor of a spacetime is coupled to quantum matter fields propagating on the spacetime via the expectation value of their renormalized stress-energy tensor in a quantum state. We explore two issues, taking the Klein Gordon equation as our model quantum field theory. The first is the provision of a suitable initial value formulation for the theory. Towards this, we address the question, for given initial data consisting of the classical metric and its first three 'time' derivatives off the surface together with a choice of initial quantum state, of what is an appropriate 'surface Hadamard' condition such that, for initial data for which it is satisfied it is reasonable to conjecture that there will be a Cauchy development whose quantum state is Hadamard. This requires dealing with the fact that, given two points on an initial surface, the spacetime geodesic between them does not, in general, lie on that surface. So the (squared) geodesic distance that occurs in the Hadamard subtraction differs from that intrinsic to the initial surface. We handle this complication by expanding the former as a suitable 3-dimensional covariant Taylor expansion in the latter. Moreover the renormalized expectation value of the stress-energy tensor in the initial surface depends explicitly on the fourth, 'time', derivative of the metric, which is not part of the initial data, but which we argue is given, implicitly, by the semiclassical Einstein equations on the initial surface. (The rôle played by those equations also entails that the surface Hadamard condition subsumes the constraints.) We also introduce the notion of physical solutions, which, inspired by a 1993 proposal of Parker and Simon, we define to be solutions which are smooth in ħ at ħ = 0. We conjecture that for these solutions the second and third time derivatives of the metric will be determined
once the first and second time derivatives are specified. We point out that a simpler treatment of the initial value problem can be had if we adopt yet more of the spirit of Parker and Simon and content ourselves with solutions to order ħ which are Hadamard to order ħ. A further simplification occurs if we consider semiclassical gravity to order ħ
0. This resembles classical general relativity in that it is free from the complications of higher derivative terms and does not require any Hadamard condition. But it can still incorporate nontrivial quantum features such as superpositions of classical-like quantum states of the matter fields. Our second issue concerns the prospects for combining semiclassical gravity with theories of spontaneous quantum state collapse. We will focus our attention on proposals involving abrupt changes in the quantum field state which occur on certain (random, non-intersecting) Cauchy surfaces according to some — yet to be developed — generally covariant objective collapse model but that, in between such collapse surfaces, we have a physical semiclassical solution (or a solution of order O(ħ) or a solution of order O(ħ
0)). On each collapse surface, the semiclassical gravity equations will necessarily be violated and, as Page and Geilker pointed out in 1981, there will therefore necessarily be a discontinuity in the expectation value of the renormalized stress-energy tensor. Nevertheless, we argue, based on our conjecture about the well-posedness of the initial value problem for physical solutions, that, with a suitable rule for the jump in the metric and/or the extrinsic curvature, the time evolution will still be uniquely determined. We tentatively argue that a natural jump rule would be one in which the metric itself and the transverse traceless part of the extrinsic curvature will be continuous and the jump will be confined to the remaining parts of the extrinsic curvature. We aid and complement our discussion by studying our two issues also in the simpler cases of a semiclassical scalars model and semiclassical electrodynamics.