  The Mixmaster solution to Einstein field equations was examined by C. Misner
in an effort to better understand the dynamics of the early universe. We
highlight the importance of the quantum version of this model for early
universe. This quantum version and its semi-classical portraits are yielded
through affine and standard coherent state quantizations and more generally
affine and Weyl-Heisenberg covariant integral quantizations. The adiabatic and
vibronic approximations widely used in molecular physics can be employed to
qualitatively study the dynamics of the model on both quantum and
semi-classical levels. Moreover, the semi-classical approach with the exact
anisotropy potential can be effective in numerical integration of some
solutions. Some promising physical features such as the singularity resolution,
smooth bouncing, the excitation of anisotropic oscillations and a substantial
amount of post-bounce inflation as the backreaction to the latter are pointed
out. Finally, a realistic cosmological scenario based on the quantum mixmaster
model, which includes the formation and evolution of local structures is
outlined.
