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According to the Belinski-Khalatnikov-Lifshitz conjecture, close to aspacelike singularity different spatial points decouple, and the dynamics canbe described in terms of the Mixmaster (vacuum Bianchi IX) model. In order tounderstand the role played by quantum-gravity effects in this context, in thepresent work we consider the semiclassical behavior of this model. Classically,this system undergoes a series of transitions between Kasner epochs, which aredescribed by a specific transition law. This law is derived based on theconservation of certain physical quantities and the rotational symmetry of thesystem. In a quantum scenario, however, fluctuations and higher-order momentsmodify these quantities, and consequently also the transition rule. Inparticular, we perform a canonical quantization of the model and thenanalytically obtain the modifications of this transition law for semiclassicalstates peaked around classical trajectories. The transition rules for thequantum moments are also obtained, and a number of interesting properties arederived concerning the coupling between the different degrees of freedom. Moreimportantly, we show that, due to the presence of quantum-gravity effects andcontrary to the classical model, there appear certain finite ranges of theKasner parameters for which the system will not undergo further transitions andwill follow a Kasner regime until the singularity. Even if a more detailedanalysis is still needed, this feature points toward a possible resolution ofthe classical chaotic behavior of the model. Finally, a numerical integrationof the equations of motion is also performed in order to verify the obtainedanalytical results.

Semiclassical study of the Mixmaster model: the quantum Kasner map