According to the Belinski-Khalatnikov-Lifshitz conjecture, close to a spacelike singularity different spatial points decouple, and the dynamics can be described in terms of the mixmaster (vacuum Bianchi IX) model. In order to understand the role played by quantum-gravity effects in this context, in the present work we consider the semiclassical behavior of this model. Classically, this system undergoes a series of transitions between Kasner epochs, which are described by a specific transition law. This law is derived based on the conservation of certain physical quantities and the rotational symmetry of the system. In a quantum scenario, however, fluctuations and higher-order moments modify these quantities, and consequently also the transition rule. In particular, we perform a canonical quantization of the model and then analytically obtain the modifications of this transition law for semiclassical states peaked around classical trajectories. The transition rules for the quantum moments are also obtained, and a number of interesting properties are derived concerning the coupling between the different degrees of freedom. More importantly, we show that, due to the presence of quantum-gravity effects and contrary to the classical model, there appear certain finite ranges of the Kasner parameters for which the system will not undergo further transitions and will follow a Kasner regime until the singularity. Even if a more detailed analysis is still needed, this feature points toward a possible resolution of the classical chaotic behavior of the model. Finally, a numerical integration of the equations of motion is also performed in order to verify the obtained analytical results.

中文翻译：

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混合大师模型的半经典研究：量子卡斯纳图

根据 Belinski-Khalatnikov-Lifshitz 猜想，接近类空间奇点的不同空间点解耦，动力学可以用 mixmaster（真空 Bianchi IX）模型来描述。为了理解量子引力效应在这种情况下所起的作用，在目前的工作中，我们考虑了这个模型的半经典行为。经典地，该系统在 Kasner 时期之间经历了一系列转变，这些转变由特定的转变定律描述。该定律是基于某些物理量的守恒和系统的旋转对称性推导出来的。然而，在量子场景中，涨落和高阶矩会改变这些量，因此也会改变过渡规则。尤其是，我们对模型进行规范量化，然后解析地获得这个过渡定律的修改，用于在经典轨迹周围达到峰值的半经典状态。还获得了量子矩的跃迁规则，并推导出了一些关于不同自由度之间耦合的有趣性质。更重要的是，我们表明，由于量子引力效应的存在并且与经典模型相反，出现了某些有限范围的 Kasner 参数，在这些范围内，系统不会经历进一步的跃迁，并且会遵循 Kasner 状态直到奇点. 即使仍需要更详细的分析，该特征也指向模型经典混沌行为的可能解决方案。最后，