The Born–Oppenheimer method, quantum gravity and matter

Abstract

Abstract. We illustrate and examine diverse approaches to the quantum matter-gravity system which refer to the Born-Oppenheimer (BO) method. In particular we first examine a quantum geometrodynamical approach introduced by other authors in a manner analogous to that previously employed by us, so as to include back reaction and non-adiabatic contributions. On including such effects it is seen that the unitarity violating effects previously found disappear. A quantum loop space formulation (based on a hybrid quantisation, polymer for gravitation and canonical for matter) also refers to the BO method. It does not involve the classical limit for gravitation and has a highly peaked initial scalar field state. We point out that it does not resemble in any way to our traditional BO approach. Instead it does resemble an alternative, canonically quantised, non BO approach which we have also previously discussed.

Construction of quantum Dirac observables and the emergence of WKB time

“…We have also analysed the connection between the relational view adopted in this paper with another popular approach to the "problem of time": the semiclassical emergence of "WKB time" [24,25,39,39,40,40,49]. This approach is relevant because many important phenomenological applications of quantum cosmology have been developed in this "semiclassical" framework [41][42][43][44][45][46][47][48]. Thus, it is worthwhile to understand its relation to the more complete quantum theory based on the physical Hilbert space and associated quantum Dirac observables.…”

“…Eq. (56) together with (48) implies that sgn(ṫ) = −sgn(p t ) = σ = const., which leads to |t(b) − t(a)| = σ(t(b) − t(a)). Using (56) and (57), we can now insert the relational solution (50) in the integrand of (47) to obtain the on-shell action:…”

“…Moreover, the explicit connection between this relational view and other approaches to the "problem of time" has remained unclear. In particular, a popular "solution" is the "semiclassical emergence of time": time only exists when the wave function(al) of the gravitational field is in a semiclassical regime (see [24,25,39,40] for a review and [41][42][43][44][45][46][47][48] for phenomenological applications). For this reason, this emergent semiclassical time is often referred to as "WKB time" [39,40,49].…”

Gauge Fixing and the Semiclassical Interpretation of Quantum Cosmology

“…The semiclassical interpretation can be generalised to what is often referred to as the Born-Oppenheimer (BO) approach to quantum gravity and cosmology [17][18][19][20][37][38][39][40][41][42][43][44][45][46], as it was inspired by the BO approximation in molecular physics [47,48]. We will use the terms 'semiclassical interpretation', 'semiclassical approach' and 'BO approach' interchangeably.…”

Semiclassical approximation of the Wheeler–DeWitt equation: arbitrary orders and the question of unitarity

“…One is a direct expansion with respect to the parameter M in (1) [3], the other is a more or less direct application of the molecular Born-Oppenheimer scheme [4,5]; its main difference lies in the treatment of back reaction on the gravitational sector and the preservation or violation of unitarity in the matter sector. A recent comparison can be found in [6] and in the Appendix of [7].…”