Taming Fluctuations for Gaussian States in Loop Quantum Cosmology

TL;DR

This paper analyzes Gaussian states in loop quantum cosmology, revealing their divergent quantum fluctuations and lack of minimal uncertainty, and emphasizes the importance of holonomy length volume-scaling for consistent theory.

Contribution

It derives exact expressions for Gaussian state fluctuations in LQC and explores their divergence and uncertainty properties, highlighting the significance of holonomy length volume-scaling.

Findings

Gaussian fluctuations diverge with increasing state variance

Fluctuations can be suppressed by volume scaling of holonomy length

Gaussian states generally do not minimize quantum uncertainty

Abstract

We do not observe quantum effects on cosmological scales. Thus, if loop quantum cosmology (LQC) is to provide an accurate depiction of the real world, it must allow for quantum states of spacetime geometry which are semi-classical in two respects: they must be sharply peaked around a single, classical geometry, and they must have small quantum fluctuations. It is generally assumed that Gaussian states exhibit both of these properties. After all, they do in ordinary quantum mechanics. In this paper, we derive exact closed-form expressions for the fluctuations of Gaussian states in LQC and their lower bound given by the Robertson-Schr\"odinger inequality. We demonstrate that, contrary to ordinary quantum mechanics, fluctuations for Gaussian states in spatially flat, homogeneous and isotropic LQC diverge as the state variance increases (as well as in related cosmological models with the same kinematic Hilbert space and canonical observables). However, when the holonomy length is made to scale with a volume regularization parameter, these fluctuations may be arbitrarily suppressed by taking the fiducial volume to be large, providing analytic control over their divergence. Finally, we show that, despite this, Gaussian states in LQC generally do not minimize uncertainty. Moreover, it is conjectured that no such minimal-uncertainty states exist. Throughout this work, it becomes clear how important the often-assumed condition of holonomy length volume-scaling is; we show that when this condition is violated, the resulting theory exhibits operator closure pathologies and other exotic algebraic features.