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

Abstract

We explore how the initial value problem may be formulated for globally hyperbolic, Hadamard, solutions of the semiclassical Einstein-Klein-Gordon equations. Given a set of data on an initial 3-surface, consisting of the values on the surface of a spacetime metric and its first 3 time derivatives off the surface, we introduce a notion of 'surface Hadamard' state on the CCR algebra of the surface. We conjecture that, for a given such set of classical Cauchy data with a surface Hadamard state satisfying the semiclassical constraint equations, the initial value problem will be well posed. We present similar conjectures for a semiclassical scalars model and semiclassical electrodynamics. Moreover, partly inspired by work of Parker and Simon in 1993, we define semiclassical gravity `physical solutions' to be those that are (jointly smooth) functions of $\hslash$ and of coordinates continuous in $\hslash$ at $\hslash =0$. We conjecture that for such solutions the second and third time derivatives of the metric off the surface need not be specified, but rather will be determined by that continuity condition. Assuming the initial value conjecture for such physical solutions holds, and that a stochastic rule were available which leads to quantum state collapses occurring on (non-intersecting) random Cauchy surfaces, we discuss the well-posedness of semiclassical gravity with stochastic quantum state collapses. We also discuss two notions of approximate physical semiclassical solutions (both with and without collapses): Namely solutions to order $\hslash$ (first discussed by Parker and Simon in 1993) and solutions to order ${\hslash }^0$. We point out that the latter do not require higher derivative terms or Hadamard subtractions, but that nevertheless order ${\hslash }^0$ semiclassical gravity is a distinct theory from classical general relativity capable of incorporating quantum interference phenomena.

47 pages. Corrections implemented, notably Def 11 (preliminary surface Hadamard condition) and certain equations on p. 33. Explanations improved and typos fixed. This version corresponds largely to the published version in JCAP

References in corpus (14)

• Lectures on the Cosmological Constant Problem
• Measurement Analysis and Quantum Gravity
• A Relativistic Dynamical Collapse Model
• Stable cosmological models driven by a free quantum scalar field
• Lectures on quantum energy inequalities
• Quantum-corrected black holes and naked singularities in (2+1)-dimensions
• Dynamical Backreaction in Robertson-Walker Spacetime
• Evaporation of four-dimensional dynamical black holes sourced by the quantum trace anomaly
• Towards a novel approach to semi-classical gravity
• Special cosmological models derived from the semiclassical Einstein equation on flat FLRW space-times
• Semiclassical gravity with a conformally covariant field in globally hyperbolic spacetimes
• Linear stability of semiclassical theories of gravity
• Can a particle detector cross a Cauchy horizon?
• Hadamard states on spherically symmetric characteristic surfaces, the semi-classical Einstein equations and the Hawking effect

Cited by in corpus (6)

• Quantum Field Theory in Curved Spacetime (2nd Edition)
• Cosmological de Sitter Solutions of the Semiclassical Einstein Equation
• Semiclassical Dynamics of Hawking Radiation
• What happens once an accelerating observer has detected a Rindler particle?
• The Hadamard condition on a Cauchy surface and the renormalized stress-energy tensor
• Semiclassical Gravity Beyond General Relativity: Insights from Torsion