论文标题
量子空间的光谱尺寸和尺寸光谱
Spectral dimensions and dimension spectra of quantum spacetimes
论文作者
论文摘要
量子重力的不同方法通常可以预测，基本级别的时空维度不是4。另一方面，非共同几何的观点表明，量子空间应该以离散的复合体集合 - 维度频谱来表征。在这里，我们表明，这两个概念相互补充，维度光谱在揭示光谱维度的紫外线行为方面非常有用。我们进行了扩展的分析，突出了故障点，并用两个具体的例子说明了总体结果：量子球和$κ$ -Minkowski SpaceTime，用于一些不同的Laplacians。特别是，我们发现前者的光谱尺寸表现出对数周期振荡，随着变形参数趋向于经典值，其幅度迅速衰减。相比之下，在$κ$ -Minkowski时空上的三名laplacians中的任何一个中的任何一个都没有发生这种振荡。
Different approaches to quantum gravity generally predict that the dimension of spacetime at the fundamental level is not 4. The principal tool to measure how the dimension changes between the IR and UV scales of the theory is the spectral dimension. On the other hand, the noncommutative-geometric perspective suggests that quantum spacetimes ought to be characterised by a discrete complex set -- the dimension spectrum. Here we show that these two notions complement each other and the dimension spectrum is very useful in unravelling the UV behaviour of the spectral dimension. We perform an extended analysis highlighting the trouble spots and illustrate the general results with two concrete examples: the quantum sphere and the $κ$-Minkowski spacetime, for a few different Laplacians. In particular, we find out that the spectral dimensions of the former exhibit log-periodic oscillations, the amplitude of which decays rapidly as the deformation parameter tends to the classical value. In contrast, no such oscillations occur for either of the three considered Laplacians on the $κ$-Minkowski spacetime.
