标题： 环圈自旋网络的福克空间在量子引力中的应用 显示英文标题

标题： The Fock Space of Loopy Spin Networks for Quantum Gravity

摘要： 在圈量子引力的粗粒化背景下，我们引入了环状和标记的自旋网络，这些结构将标准的自旋网络状态推广，以明确地考虑非平凡曲率和挠率。 这两种结构都放松了作用于自旋网络顶点处的闭合约束。 标记的自旋网络仅仅在每个顶点携带一个额外的自旋来编码总的闭合缺陷，而环状自旋网络允许每个顶点附带任意数量的环。 这些小环可以被解释为量子引力场的局部激发，并且我们讨论了赋予它们的统计性质。 环状自旋网络形成的Fock空间实现了圈量子引力的新截断，使得能够在固定背景图及其节点上附加的局域自由度上表述其图变化动力学。 这提供了一个框架，用于重新引入非平凡的背景量子几何，在该几何周围我们将研究扰动的有效动力学。 我们研究如何在这一框架中实现拓扑BF理论的动力学。 我们通过平行传输约束来实现平坦联络的投影，并特别关注那些通常被忽视的由$\delta$-分布的高阶导数定义的非平凡平坦解。 显示英文摘要

摘要： In the context of the coarse-graining of loop quantum gravity, we introduce loopy and tagged spin networks, which generalize the standard spin network states to account explicitly for non-trivial curvature and torsion. Both structures relax the closure constraints imposed at the spin network vertices. While tagged spin networks merely carry an extra spin at every vertex encoding the overall closure defect, loopy spin networks allow for an arbitrary number of loops attached to each vertex. These little loops can be interpreted as local excitations of the quantum gravitational field and we discuss the statistics to endow them with. The resulting Fock space of loopy spin networks realizes new truncation of loop quantum gravity, allowing to formulate its graph-changing dynamics on a fixed background graph plus local degrees of freedom attached to the graph nodes. This provides a framework for re-introducing a non-trivial background quantum geometry around which we would study the effective dynamics of perturbations. We study how to implement the dynamics of topological BF theory in this framework. We realize the projection on flat connections through holonomy constraints and we pay special attention to their often overlooked non-trivial flat solutions defined by higher derivatives of the $\delta$-distribution.