A coarse graining technique akin to block spin transformations that groups together fiducial cells in a homogeneous and isotropic universe has been recently developed in the context of loop quantum cosmology. The key technical ingredient was an SU(1,1) group and Lie algebra structure of the physical observables as well as the use of Perelomov coherent states for SU(1,1). It was shown that the coarse graining operation is completely captured by changing group representations. Based on this result, it was subsequently shown that one can extract an explicit renormalization group flow of the loop quantum cosmology Hamiltonian operator in a simple model with dust-clock. In this paper, we continue this line of investigation and derive a coherent state path integral formulation of this quantum theory and extract an explicit expression for the renormalization-scale dependent classical Hamiltonian entering the path integral for a coarse grained description at that scale. We find corrections to the nonrenormalized Hamiltonian that are qualitatively similar to those previously investigated via canonical quantization. In particular, they are again most sensitive to small quantum numbers, showing that the large quantum number (spin) description captured by so called “effective equations” in loop quantum cosmology does not reproduce the physics of many small quantum numbers (spins). Our results have direct impact on path integral quantization in loop quantum gravity, showing that the usually taken large spin limit should be expected not to capture (without renormalization, as mostly done) the physics of many small spins that is usually assumed to be present in physically reasonable quantum states.

中文翻译：

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环量子宇宙学中的路径积分重整化

最近在环量子宇宙学的背景下开发了一种类似于阻止自旋变换的粗粒度技术，该技术将均匀和各向同性的宇宙中的基准单元组合在一起。关键的技术成分是物理可观测量的 SU(1,1) 群和李代数结构，以及对 SU(1,1) 使用 Perelomov 相干态。结果表明，通过更改组表示可以完全捕获粗粒度操作。基于这一结果，随后证明可以在一个简单的带有尘埃时钟的模型中提取循环量子宇宙学哈密顿算符的显式重整化群流。在本文中，我们继续这条研究路线并推导出该量子理论的相干状态路径积分公式，并提取重整化尺度相关经典哈密顿量的明确表达式，该公式进入路径积分以在该尺度上进行粗粒度描述。我们发现对非重整化哈密顿量的修正与先前通过规范量化研究的修正在性质上相似。特别是，它们再次对小量子数最敏感，这表明循环量子宇宙学中所谓的“有效方程”所捕获的大量子数（自旋）描述并不能再现许多小量子数（自旋）的物理学。我们的结果对环量子引力中的路径积分量化有直接影响，