Semiclassical theories as initial value problems

Abstract

Motivated by the initial value problem in semiclassical gravity, we study the initial value problem of a system consisting of a quantum scalar field weakly interacting with a classical one. The quantum field obeys a Klein-Gordon equation with a potential proportional to the classical field. The classical field obeys an inhomogeneous Klein-Gordon equation sourced by the renormalised expectation value of the squared quantum field in a Hadamard state, Ψ|Φ 2 Ψ . Thus, the system of equations for the scalar fields reminisces of the semi-classical Einstein field equations with a Klein-Gordon field, where classical geometry is sourced by the renormalised stress-energy tensor of the quantum field, and the Klein-Gordon equation depends on the metric explicitly. We show that a unique asymptotic solution for the system can be obtained perturbatively at any fixed finite order in the weak coupling from initial data provided that the interaction is switched on and off smoothly in a spacetime region to the future of the initial data surface. This allows one to provide "free" initial data for the decoupled system that guarantees that the Wightman function of the quantum field be of Hadamard form, and hence that the renormalised Ψ|Φ 2 Ψ exist (in a perturbative sense) and be smooth. We comment on how to relax the switching of the interaction, which might be relevant for the corresponding problem in semiclassical gravity.

Existence and Uniqueness of Solutions of the Semiclassical Einstein Equation in Cosmological Models

“…where we have used the relation |χ 0 k | = 0. Recalling the estimates (29), having shown the integrability of f 3 at any time in (37) and knowing how to control χ k and its functional derivatives, the statements of the Proposition follow.…”

“…The price to pay is that the dimension of that dynamical system is infinite and furthermore, for the case of a generic state, it is not clear if all these moments correspond to those obtained for a meaningful quantum state. Another recent study of the initial value problem associated with semiclassical equations can be found in [37]. Furthermore, numerical analyses have been performed in the past by Anderson in the study of effects of particle creation in the early universe [3][4][5][6].…”

Generation of inflationary perturbations in the continuous spontaneous localization model: The second order power spectrum

“…Before addressing in full detail the main equations of our model, we present the framework that underlies our description of the space-time metric and that of the inflaton [6,[53][54][55][56][57]. The proposed model is based on the semiclassical gravity framework, in which gravity is treated classically and the matter fields are treated quantum mechanically.…”

“…The interested reader can consult Refs. [54][55][56][57]; in those works the formalism of the collapse proposal within the semiclassical gravity framework has been developed. In the present paper, we will only make use of the most relevant equations.…”

The Cosmological Semiclassical Einstein Equation as an Infinite-Dimensional Dynamical System

“…This has been significantly extended by Pinamonti and one of us [48] to full solutions of the SCE on FLRW spacetimes, including the energy constraint with initial values specified on a Cauchy surface, but still limited to the conformally coupled scalar field, and a particular choice of renormalization parameters and quantum state. Other recent work related to the SCE includes: [26], where numerical solutions to the equations presented here are analyzed; [40], where yet another approach to solving the SCE is developed; [36], where a simplified semiclassical problem is studied as an initial value problem; [51] and [35], where solutions in static spacetimes are analyzed.…”