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arXiv 0912.0220hep-thgr-qcmath-phmath.MP
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量子宇宙的谱维数

Spectral dimension of a quantum universe

Leonardo Modesto, Piero Nicolini

备注：
5 pages, 1 figure, references added, typos corrected, title changed, final version published in Physical Review D
发表信息：
Phys.Rev.D81:104040,2010
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摘要：

本文考虑在涨落流形上传播的扩散过程，用透明的方法计算了量子时空的谱维数。为了描述扩散的不稳定路径，我们通过平均平坦空间情况下量子流形的粒度来实现最小长度。作为结果，我们得到，对于大的扩散时间，量子时空的行为像一个光滑的离散维微分流形。另一方面，对于较小的扩散时间，时空看起来像一个有效维数减少的分形表面。对于扩散时间具有最小长度的特定情况，时空的谱维数等于2，这表明在这种情况下引力可能具有可重整化的特征。对于较小的扩散时间，光谱维数接近于零，使得任何物理解释在这种极端情况下都不太可靠。我们将我们的结果推广到背景场和曲率的情况。我们表明，在这种情况下，谱维数与扩散时间有一个更复杂的关系，和结论的引力重正化字符变得不那么直接，我们发现与平坦的空间分析。

英文摘要：

In this paper, we calculate in a transparent way the spectral dimension of a quantum spacetime, considering a diffusion process propagating on a fluctuating manifold. To describe the erratic path of the diffusion, we implement a minimal length by averaging the graininess of the quantum manifold in the flat space case. As a result we obtain that, for large diffusion times, the quantum spacetime behaves like a smooth differential manifold of discrete dimension. On the other hand, for smaller diffusion times, the spacetime looks like a fractal surface with a reduced effective dimension. For the specific case in which the diffusion time has the size of the minimal length, the spacetime turns out to have a spectral dimension equal to 2, suggesting a possible renormalizable character of gravity in this regime. For smaller diffusion times, the spectral dimension approaches zero, making any physical interpretation less reliable in this extreme regime. We extend our result to the presence of a background field and curvature. We show that in this case the spectral dimension has a more complicated relation with the diffusion time, and conclusions about the renormalizable character of gravity become less straightforward with respect to what we found with the flat space analysis.

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