On the initial value problem for semiclassical gravity without and with quantum state collapses

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 expec-tation 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 ad-dress 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 satis-fied 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 role 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 h at h = 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 h which are Hadamard to order h. A further simplification occurs if we consider semiclassical gravity to order h degrees. 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 pro-posals 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(h) or a solution of order O(h degrees)). 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.

Automated summary

Semiclassical gravity is a theory that combines classical Einstein tensor with quantum matter fields in a quantum state. The article explores two issues, focusing on the Klein Gordon equation as the model quantum field theory. The first issue deals with finding a suitable initial value formulation for the theory, taking into account the surface Hadamard condition and the complications arising from geodesic distances. The second issue discusses the combination of semiclassical gravity with theories of spontaneous quantum state collapse, proposing a jump rule for the metric and/or extrinsic curvature to ensure unique time evolution. The simpler cases of semiclassical scalars model and semiclassical electrodynamics are also studied.