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arXiv 2103.14090gr-qcmath-phmath.MP
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圈量子宇宙学中高斯态的驯服涨落

Taming Fluctuations for Gaussian States in Loop Quantum Cosmology

Patrick Fraser

备注：
11 pages, 5 figures; To be published in Physical Review D
发表信息：
Phys. Rev. D 103, 086014 (2021)
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摘要：

我们没有观测到宇宙学尺度上的量子效应。因此，如果圈量子宇宙学（LQC）要提供对真实世界的精确描述，它必须允许时空几何的量子态在两个方面是半经典的：它们必须在单一的经典几何周围有尖锐的峰值，并且它们必须有小的量子涨落。一般认为高斯态具有这两种性质。毕竟，它们在普通的量子力学中是存在的。本文导出了LQC中高斯态涨落的精确闭合表达式及其由Robertson-Schrödinger不等式给出的下界。我们证明了，与普通的量子力学，波动高斯态在空间平坦，均匀和各向同性LQC发散的状态方差的增加（以及在相关的宇宙学模型具有相同的运动学希尔伯特空间和正则观测值）。然而，当完整长度与体积正则化参数成比例时，这些波动可以通过将基准体积取大来任意抑制，从而提供对其发散的分析控制。最后，我们表明，尽管如此，高斯状态在LQC一般不最小化的不确定性。此外，它是不存在这样的最小不确定性状态。在整个这项工作中，它变得很清楚如何重要的经常假设的条件的完整长度体积缩放，我们表明，当这个条件被违反，由此产生的理论表现出算子封闭的病理和其他奇异的代数功能。

英文摘要：

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-Schrödinger 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.

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