A complete Lagrangian and Hamiltonian description of the theory of self-gravitating light-like matter shells is given in terms of gauge-independent geometric quantities. For this purpose the notion of an extrinsic curvature for a null-like hypersurface is discussed and the corresponding Gauss-Codazzi equations are proved. These equations imply Bianchi identities for spacetimes with null-like, singular curvature. Energy-momentum tensor-density of a light-like matter shell is unambiguously defined in terms of an invariant matter Lagrangian density. Noether identity and Belinfante-Rosenfeld theorem for such a tensor-density are proved. Finally, the Hamiltonian dynamics of the interacting system: "gravity + matter" is derived from the total Lagrangian, the latter being an invariant scalar density.

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.

We present a quantum version of the vacuum Bianchi IX model by implementing affine coherent state quantization combined with a Born-Oppenheimer-like adiabatic approximation. The analytical treatment is carried out on both quantum and semiclassical levels. The resolution of the classical singularity occurs by means of a repulsive potential generated by our quantization procedure. The quantization of the oscillatory degrees of freedom produces a radiation energy density term in the semiclassical constraint equation. The Friedmann-like lowest energy eigenstates of the system are found to be dynamically stable.

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Following our previous papers concerning the quantization of the vacuum Bianchi-IX model within or beyond the Born-Oppenheimer and adiabatic approximation, we develop a more elaborate analysis of the dynamical properties of the model based the vibronic approach utilized in molecular physics. As in the previous papers, we restrict our approach to the harmonic approximation of the anisotropy potential in order to obtain resoluble analytical expressions.