The Physical Relevance of the Fiducial Cell in Loop Quantum Cosmology

Fabio M. Mele; Johannes Münch

doi:10.1103/PhysRevD.108.106004

paper

The Physical Relevance of the Fiducial Cell in Loop Quantum Cosmology

arXiv:2109.10663 · doi:10.1103/PhysRevD.108.106004

Abstract

A common way to avoid divergent integrals in homogeneous spatially non-compact gravitational systems is to introduce a fiducial cell by cutting-off the spatial slice at a finite region $V_{o}$ . This is usually considered as an auxiliary regulator to be removed after computations by sending $V_{o} \to \infty$ . In this paper, we analyse the dependence of the classical and quantum theory of homogeneous, isotropic and spatially flat cosmology on $V_{o}$ . We show that each fixed $V_{o}$ regularisation leads to a different canonically independent theory. At the classical level, the dynamics of observables is not affected by the regularisation on-shell. For the quantum theory, however, this leads to a family of regulator dependent quantum representations and the limit $V_{o} \to \infty$ becomes then more subtle. First, we construct a novel isomorphism between different $V_{o}$ -regularisations, which allows us to identify states in the different $V_{o}$ -labelled Hilbert spaces to ensure equivalent dynamics for any value of $V_{o}$ . The $V_{o} \to \infty$ limit would then correspond to choosing a state for which the volume assigned to the fiducial cell becomes infinite as appropriate in the late-time regime. As second main result of our analysis, quantum fluctuations of observables smeared over subregions $V \subset V_{o}$ , unlike those smeared over the full $V_{o}$ , explicitly depend on the size of the fiducial cell through the ratio $V / V_{o}$ interpreted as the (inverse) number of subcells $V$ homogeneously patched together into $V_{o}$ . Physically relevant fluctuations for a finite region, as e.g. in the early-time regime, which would be unreasonably suppressed in a naïve $V_{o} \to \infty$ limit, become appreciable at small volumes. Our results suggest that the fiducial cell is not playing the role of a mere regularisation but is physically relevant at the quantum level and complement previous statements in the literature.

arXiv commentPublished version, 13 pages (REVTeX format), 1 figure. Minor clarifications added

The Physical Relevance of the Fiducial Cell in Loop Quantum Cosmology

Abstract

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