On the semiclassical approach to quantum cosmology: relational particle model

Abstract

The emergent semiclassical time approach to resolving the problem of time in quantum gravity involves heavy slow degrees of freedom providing via an approximately Hamilton–Jacobi equation an approximate time standard with respect to which the quantum mechanics of light fast degrees of freedom can run. More concretely, this approach involves Born–Oppenheimer and WKB ansätze and some accompanying approximations. In this paper, I investigate this approach for concrete scaled relational particle mechanics models, i.e. models featuring only relative separations, relative angles and relative times. I consider the heavy–light interaction term in the light quantum equation—necessary for the semiclassical approach to work, first as an emergent-time-dependent perturbation of the emergent-time-dependent Schrödinger equation for the light subsystem. Secondly, I consider a scheme in which the backreaction is small but non-negligible, so that the l-subsystem also affects the form of the emergent time. I also suggest that the many terms involving expectation values of the light wavefunctions in both the (unapproximated) heavy and light equations might require treatment in parallel to the Hartree–Fock self-consistent approach rather than merely being discarded; for the moment this paper provides a counterexample to such terms being smaller than their unaveraged counterparts. Investigation of these ideas and methods will give us a more robust understanding of the suggested quantum-cosmological origin of microwave background inhomogeneities and galaxies.