In spatially non-compact homogeneous minisuperpace models, spatial integrals in the Hamiltonian and symplectic form must be regularised by confining them to a finite volume V_o, known as the fiducial cell. As this restriction is unnecessary in the complete field theory before homogeneous reduction, the physical significance of the fiducial cell has been largely debated, especially in the context of (loop) quantum cosmology. Understanding the role of V_o is in turn essential for assessing the minisuperspace description’s validity and its connection to the full theory. In this work we present a systematic procedure for the field theory reduction to spatially homogeneous and isotropic minisuperspaces within the canonical framework and apply it to both a massive scalar field theory and gravity. Our strategy consists in implementing spatial homogeneity via second-class constraints for the discrete field modes over a partitioning of the spatial slice into countably many disjoint cells. The reduced theory’s canonical structure is then given by the corresponding Dirac bracket. Importantly, the latter can only be defined on a finite number of cells homogeneously patched together. This identifies a finite region, the fiducial cell, whose physical size acquires then a precise meaning already at the classical level as the scale over which homogeneity is imposed. Additionally, the procedure allows us to track the information lost during homogeneous reduction and how the error depends on V_o. We then move to the quantisation of the classically reduced theories, focusing in particular on the relation between the theories for different V_o, and study the implications for statistical moments, quantum fluctuations, and semiclassical states. In the case of a quantum scalar field, a subsector of the full quantum field theory where the results from the ‘first reduced, then quantised’ approach can be reproduced is identified and the conditions for this to be a good approximation are also determined.

中文翻译：

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关于基准结构在 LQC 中小超空间约简和量子涨落中的作用


在空间非紧齐次 minisuperpace 模型中，哈密顿量和对称形式的空间积分必须通过将它们限制在有限体积 V_o（称为基准单元）来正则化。由于在齐次还原之前的完全场论中，这种限制是不必要的，因此基准单元的物理意义一直存在很大争议，尤其是在（环）量子宇宙学的背景下。反过来，理解 V_o 的作用对于评估微超空间描述的有效性及其与完整理论的联系至关重要。在这项工作中，我们提出了一个系统的过程，用于将场论简化为规范框架内的空间均匀和各向同性微型超空间，并将其应用于大质量标量场论和引力。我们的策略包括通过离散场模式的二等约束来实现空间均匀性，将空间切片划分为可数的许多不相交单元。然后，简化理论的规范结构由相应的 Dirac 括号给出。重要的是，后者只能在有限数量的均匀拼接在一起的细胞上定义。这确定了一个有限区域，即基准单元，其物理大小在经典级别上已经获得了精确的含义，即施加同质性的尺度。此外，该程序还允许我们跟踪齐次还原过程中丢失的信息以及误差如何取决于 V_o。然后，我们转向经典简化理论的量化，特别关注不同 V_o 的理论之间的关系，并研究对统计矩、量子涨落和半经典状态的影响。 在量子标量场的情况下，确定了全量子场论的一个子部门，其中可以再现“先简化，然后量化”方法的结果，并且还确定了使其成为良好近似的条件。