On the Role of Fiducial Structures in Minisuperspace Reduction and Quantum Fluctuations in LQC

Abstract

In spatially non-compact 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 theory before homogeneous reduction, the physical significance of the fiducial cell has been largely debated, especially in the context of loop quantum cosmology. Understanding its role 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 reduction to spatially homogeneous and isotropic minisuperspaces within the canonical framework and apply it to a massive scalar field theory and gravity. Our strategy consists in implementing spatial homogeneity via second-class constraints for 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 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 homogenenity is imposed. Additionally, the procedure allows to track the information lost during reduction and how the error depends on $V_o$. We then move to the quantisation of the classically reduced theories, focusing on the relation between 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.