Building quantum space-time: spin foams and the renormalization group

为引力和量子物质定义一个一致的框架仍然是理论物理学中的一个基本难题。我们希望这个重要问题的解决方案在于量子引力理论，它描述了最小长度的时空，即普朗克尺度。这样的理论将为长期存在的问题提供新的见解，例如宇宙起源于大爆炸奇点。然而，通往量子引力理论的道路还不清楚。一个有希望的候选者是自旋泡沫重力。简而言之，自旋泡沫是时空的路径积分：不是只考虑单个几何，而是考虑所有几何并通过振幅加权。为了定义这个积分，时空被分成离散的“积木”，其中自旋泡沫在这些积木的所有形状和大小上求和。重要的是，这些模型包含了广义相对论的基本原则，背景独立性：在任何时候都不需要参考固定的背景几何。然而，为了释放自旋泡沫的全部潜力并将其转化为预测理论，我将在这个项目中解决三个相互关联的关键挑战：通过重整化识别一致模型，可计算性和可观测量的提取，以揭示量子时空的性质。时空细分为离散的构建块不是唯一的，例如，可以将空间-时间划分成几个粗略的或许多精细的构建块。一般来说，这种基准选择会显著影响结果，但哪一个（如果有的话）会给出正确的结果？我将通过背景独立重整化来解决这个难题，在这里我将自旋泡沫振幅与离散化联系起来，以便结果是一致的。为了实现这种方法，我将系统地实现自旋泡沫理论空间的截断。为了成功地重整化，我必须研究由许多构建块组成的自旋泡沫，这使得数值技术不可或缺。一方面，我将开发有效的算法来计算深量子体系中的自旋泡沫振幅。另一方面，我将带头使用蒙特卡罗方法来解开自旋泡沫模型的重整化群流。最后但并非最不重要的是，我将定义和研究可观测量，以揭示量子时空的性质。我将通过计算它们的曲率和光谱维数来研究自旋泡沫本身。后者是一种有效的尺度度量方法。此外，我还将解决将物质与量子时空耦合的关键问题。最终，我的目标是发现自旋泡沫和物质的相互重整化群。这个项目的这一部分特别重要，因为它将使我能够将自旋泡沫引力与量子引力的其他方法进行比较。此外，它可能会打开通往现象学的大门。

Defining a consistent framework for gravity and quantum matter remains a fundamental puzzle in theoretical physics. We expect the solution to this vital question to lie in a theory of quantum gravity that describes space-time at smallest lengths, the Planck scale. Such a theory would give new insights into long standing questions, e.g. the origin of the universe in a Big Bang singularity.However the road towards a theory of quantum gravity is not clear. One promising candidate is spin foam gravity. In a nutshell, a spin foam is a path integral of space-time: instead of only considering a single geometry, all geometries are considered and weighted by an amplitude. To define this integral, space-time is divided into discrete “building blocks”, where the spin foam sums over all shapes and sizes of these blocks. Crucially, these models embrace a fundamental principle of general relativity, background independence: at no point is a reference to a fixed background geometry necessary.However, to unlock the full potential of spin foams and to transform them into a predictive theory I will address three interconnected key challenges in this project: identification of consistent models through renormalization, computability and extraction of observables to uncover the properties of quantum space-time.The subdivision of space-time into discrete building blocks is not unique, e.g. one can divide space-time into a few coarse or many fine building blocks. In general this fiducial choice significantly influences the results, yet which one, if any, gives the right result? I will tackle this riddle by background independent renormalization, where I relate spin foam amplitudes across discretizations such that the results are consistent. To turn this method into reality I will systematically implement truncations of spin foam theory space.To succeed at renormalization, I must study spin foams consisting of many building blocks, which makes numerical techniques indispensable. On the one hand, I will develop efficient algorithms to compute spin foam amplitudes in the deep quantum regime. On the other hand, I will spearhead the usage of Monte Carlo methods to unravel the renormalization group flow of spin foam models.Last but not least I will define and study observables to reveal the properties of quantum space-time. I will examine spin foams themselves by computing their curvature and their spectral dimension. The latter is an effective dimension measure dependent on scale. Moreover, I will tackle the vital question of coupling matter to quantum space-time. Ultimately I aim for discovering the mutual renormalization group of spin foams and matter. This part of the project is particularly crucial since it will allow me to compare spin foam gravity to other approaches of quantum gravity. Furthermore, it might open the door towards phenomenology.