We consider the loop quantum theory of the spherically symmetric model of gravity coupled to Gaussian dust fields, where the Gaussian dust fields provide a material reference frame of the space and time to deparametrize gravity. This theory, used to study the quantum features of the spherically symmetric BH, is constructed based on a 1-dimensional lattice $\gamma \subset \mathbb{R}$. Taking advantage of the path integral formulation, we investigate the quantum dynamics and obtain an effective action. With this action, we get an effective continuous description of this quantum lattice system which is not the same as the one described by the effective Hamiltonian used in [M. Han and H. Liu, Improved effective dynamics of loop-quantum-gravity black hole and Nariai limit], i.e., the classical Hamiltonian with the holonomy correction. It turns out that, the Hamiltonian derived in this paper can return to that used in [M. Han and H. Liu] only for macro black holes, since the lattice $\gamma$ is required to be sufficiently fine. Indeed, it is necessary to propose this fine-grained lattice structure in order to well describe the underlying lattice theory by the continuous description.

中文翻译：

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黑洞的缩减相空间量化：路径积分和有效动力学

我们考虑了与高斯尘埃场耦合的重力球对称模型的圈量子理论，其中高斯尘埃场提供了空间和时间的物质参考系来去参数化重力。该理论用于研究球对称 BH 的量子特征，是基于一维晶格构建的$C\subset \mathbb{电阻}$. 利用路径积分公式，我们研究了量子动力学并获得了有效的作用。通过这个动作，我们得到了这个量子晶格系统的有效连续描述，这与 [M. Han 和 H. Liu，改进的环量子引力黑洞和 Nariai 极限的有效动力学]，即具有完整校正的经典哈密顿量。事实证明，本文导出的哈密顿量可以回归到[M. Han and H. Liu] 只针对宏观黑洞，自格$C$要求足够精细。事实上，有必要提出这种细粒度的晶格结构，以便通过连续描述很好地描述基础晶格理论。