WKB Approaches to Restore Time in Quantum Cosmology: Predictions and Shortcomings

Abstract

In this review, we analyse different aspects concerning the possibility to separate a gravity-matter system into a part which lives close to a quasi-classical state and a “small” quantum subset. The considered approaches are all relying on a WKB expansion of the dynamics by an order parameter and the natural arena consists of the Bianchi universe minisuperspace. We first discuss how, limiting the WKB expansion to the first order of approximation, it is possible to recover for the quantum subsystem a Schrödinger equation, as written on the classical gravitational background. Then, after having tested the validity of the approximation scheme for the Bianchi I model, we give some applications for the quantum subsystem in the so-called “corner” configuration of the Bianchi IX model. We individualize the quantum variable in the small one of the two anisotropy degrees of freedom. The most surprising result is the possibility to obtain a non-singular Bianchi IX cosmology when the scenario is extrapolated backwards in time. In this respect, we provide some basic hints on the extension of this result to the generic cosmological solution. In the last part of the review, we consider the same scheme to the next order of approximation identifying the quantum subset as made of matter variables only. This way, we are considering the very fundamental problem of non-unitary morphology of the quantum gravity corrections to quantum field theory discussing some proposed reformulations. Instead of constructing the time dependence via that one of the classical gravitational variables on the label time as in previous works, we analyse a recent proposal to construct time by fixing a reference frame. This scheme can be reached both introducing the so-called “kinematical action”, as well as by the well-known Kuchar–Torre formulation. In both cases, the Schrödinger equation, amended for quantum gravity corrections, has the same morphology and we provide a cosmological implementation of the model, to elucidate its possible predictions.

Hamiltonian formulation of gravity as a spontaneously-broken gauge theory of the Lorentz group

“…Furthermore, it was shown that the cases (g + = 1, g − = 0) and (g + = 0, g − = 1) correspond to an extension of general relativity that includes solutions corresponding to an additional effective pressureless perfect fluid matter source. Interestingly, such a matter source has been of prior interest as a possible solution to the problem of time in quantum gravity [25,27,[30][31][32]. We find it encouraging that in the present models, the perfect fluid arises 'naturally' from the theory and does not have to be independently posited.…”

On the emergence of a classical isotropic universe from a quantum f(R) Bianchi cosmology in the Jordan frame

“…This is the so-called reduced phase space quantisation (RPSQ) that is the most straightforward procedure since it is an exact algorithm requiring no WKB approximation based on the wave function of the Universe, even if the Hamiltonian density can be non-local. The second one is achieved by implementing both the WKB and Born-Oppenhaimer approximations, namely, the Vilenkin approach, for details see [21,22].…”

“…a physical clock is constructed via an internal gravity or matter degree of freedom [18]. Some examples of how to construct a time variable for the quantum gravity dynamics via a specific field are [19][20][21]23,24].…”

Study of the Inflationary Spectrum in the Presence of Quantum Gravity Corrections

“…In fact, there are physical settings, both in early cosmology and for the gravitational collapse, in which the background metric cannot be regarded as a purely classical dynamics, but simply an essentially classical dynamics affected by quantum fluctuations of the geometry. This line of research has been till now developed by a relatively limited number of studies [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], in which the first problem to be addressed is the emergence of a time variable for QFT from the gravity-matter Wheeler-DeWitt equation.…”