Wheeler-De Witt equations for fourth-order quantum cosmology

Abstract

The author compares the methods proposed by Boulware (1984), and Buchbinder and Lyachovich (1987), to establish a Hamiltonian formalism for fourth-order gravity and studies another possibility to circumvent second-order derivatives of the metric tensor components on the level of the Lagrangian for fourth-order theories of gravity, by introduction of a scalar field. This is done in a way different from the common procedure of doing a conformal transformation of the metric. It is demonstrated how the well known fourth-order field equations result. On the basis of the Hamiltonian formalism, the author considers the quantum cosmology for homogeneous and isotropic closed models. As a first application, the WKB approximation is discussed neglecting the spatial curvature. Chosen initial conditions have the consequence that the resulting wavefunction leads to the inflationary stage of the cosmic evolution.

Extended gravity theories and the Einstein--Hilbert action

“…The first of these can always be eliminated as part of a total divergence while the second can only be rewritten in terms of R 2 in homogeneous and isotropic metrics. It could be treated in a similar manner to the R 2 term in what follows, but this would require the introduction of a tensor field ϕ µν [14] and so cannot be described as scalar-tensor gravity. Considering the action quadratic only in the Ricci scalar, variation with respect to the metric g µν yields the Euler-Lagrange field equations…”

R+R2gravity asR+ back reaction

“…The prediction from classical analysis is indeed interesting, but near the Planck time a quantum analysis is needed. There has been some literature on quantizing the minisuperspace model for R + R2 gravity [4,5]. In the literature, because of the technical difficulties of quantizing higher derivative theories, only very simple cases (i.e., no coupling to matter) have been considered.…”

“…Because of the technical difficulties of solving such constraints, quantum cosmology for quadratic gravity has been solved for only *Electronic address: jkung@abacus.bates.edu 0556-2821/95/52( 12)/6922(7)/$06. 00 -52 simple systems such as a vacuum [4,5]. I n this paper, we explore the consequences of viewing R + R2 gravity as R + perturbation (higher derivative version of back reaction) [6].…”

$${f(\mathcal R)}$$ quantum cosmology

“…These are basically done in the classical regimes, although there are some works [10,11] dealing with quantum models.…”

“…Although this can be done in the standard framework of quantum mechanics, as it is discussed in the literature [10,11], using the causal interpretation of quantum mechanics introduced by de-Broglie and Bohm [12][13][14][15][16][17][18][19] is a good idea at least for gravity and cosmology. This is because Bohmian quantum mechanics does not suffer from some essential conceptual problems of the standard quantum mechanics which show themselves more transparently in gravity.…”