分形量子时空
Fractal Quantum Space-Time
摘要：
本文利用自旋网络态上面积算符谱的标度性质和高斯态上体积和长度算符的标度性质计算了圈量子引力（LQG）的谱维数。当探测标量场的能量在假想时间T内由高能量向低能量递减时，空间截面的谱维数从1.5变到3，在特定的假设下，在1.5相位上从2变到3。我们还计算了光谱维的空间，时间使用的自旋泡沫模型计算的面积谱运营商的标度。主要结果是普朗克尺度下的有效维数为2，低能下的有效维数为4。这一结果与非微扰量子引力的另外两种方法：“因果动力学三角剖分”和“渐近安全量子引力“是一致的。我们研究了所有可能的曲率不变量的标度性质，并且我们已经表明，奇异性问题似乎可以在自旋泡沫模型的量子引力协变公式中得到解决。对于一个特定形式的标度（或对于一个特定的面积算子谱），所有的曲率不变量也是正则的。
英文摘要：
In this paper we calculated the spectral dimension of loop quantum gravity (LQG) using the scaling property of the area operator spectrum on spin-network states and using the scaling property of the volume and length operators on Gaussian states. We obtained that the spectral dimension of the spatial section runs from 1.5 to 3, and under particular assumptions from 2 to 3 across a 1.5 phase when the energy of a probe scalar field decreases from high to low energy in a fictitious time T. We calculated also the spectral dimension of space-time using the scaling of the area spectrum operator calculated on spin-foam models. The main result is that the effective dimension is 2 at the Planck scale and 4 at low energy. This result is consistent with two other approaches to non perturbative quantum gravity: "causal dynamical triangulation" and "asymptotically safe quantum gravity". We studied the scaling properties of all the possible curvature invariants and we have shown that the singularity problem seems to be solved in the covariant formulation of quantum gravity in terms of spin-foam models. For a particular form of the scaling (or for a particular area operator spectrum) all the curvature invariants are regular also in the Trans-Planckian regime.
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