Singularity avoidance in a quantum model of the Mixmaster universe

Abstract

We present a quantum model of the vacuum Bianchi-IX dynamics. It is based on four main elements. First, we use a compound quantization procedure: an affine coherent state quantization for isotropic variables and a Weyl quantization for anisotropic ones. Second, inspired by standard approaches in molecular physics, we make an adiabatic approximation (Born-Oppenheimer-like approximation). Third, we expand the anisotropy potential about its minimum in order to deal with its harmonic approximation. Fourth, we develop an analytical treatment on the semi-classical level. The resolution of the classical singularity occurs due to a repulsive potential generated by the affine quantization. This procedure shows that during contraction the quantum energy of anisotropic degrees of freedom grows much slower than the classical one. Furthermore, far from the quantum bounce, the classical re-collapse is reproduced. Our treatment is put in the general context of methods of molecular physics, which can include both adiabatic and non-adiabatic approximations.

Quantum Belinski–Khalatnikov–Lifshitz scenario

“…Our fully quantum results show that the preliminary results obtained for the diagonal BIX within the semiclassical affine coherent states approximation [29,30] are correct. Appendix B presents the affine coherent states applied in these papers, which define another parametrization of our coherent states.…”

Covariant affine integral quantization(s)

“…This work is a continuation of previous studies devoted to affine integral quantization of the half-plane [1,2,3,4] and its applications to early or quantum cosmology [5,6,7,8,9,10]. In the latter works, the method was based on the use of affine coherent states or wavelets.…”

Critical Evaluation of Common Claims in Loop Quantum Cosmology

“…A straightforward quantization of this Hamiltonian would be given in terms of the dilation generatorD = V H, which is well-defined and self-adjoint in a quantization of the positive real line, V > 0. It is therefore a basic operator in affine quantum cosmology [40,41,42,43,44], in which the volume is restricted to positive values. In loop quantum cosmology, both signs are allowed for the oriented volume v, taking into account the orientation of space.…”