论文标题
重力的边缘模式 - II：角度指标和洛伦兹电荷
Edge modes of gravity -- II: Corner metric and Lorentz charges
论文作者
论文摘要
在该系列的第二篇论文中，我们继续阐明了一个新的量子重力程序，该程序基于角落对称代数及其表示形式。在这里，我们专注于四个重力及其角度的符号势。我们首先对BF理论和Tetrad重力中出现的各种几何量进行详细分解。这提供了BF理论和简单性约束的合成潜力的新分解。然后，我们证明四型重力角相位空间的动力变量是时空叶片的内部态度，该变量与增压发电机和角coframe场相结合。这使我们能够得出几个关键结果。首先，我们构建角落洛伦兹的费用。除了所有重力配方共有的球体差异性外，这些电荷还增加了局部$ \ mathfrak {sl}（2，\ mathbb {c}）$ component $ component to tore tetrad Gravity的角对称代数。其次，我们还揭示了角度指标满足本地$ \ mathfrak {sl}（2，\ mathbb {r}）$ algebra，其casimir对应于角区域元素。由于角度指标的太空性质，该卡西米尔属于单一离散序列，因此对其频谱进行了量化。该结果将区域频谱的离散性与洛伦兹的不变性核对，在连续体中得到了证明，而无需诉诸于批量连接。第三，我们表明角相空间解释了为什么简单的约束在角落变得不合时宜。这一事实要求在经典连续性理论中已经存在批量和角象征结构之间的和解。理解这不可避免地导致边缘模式的引入。
In this second paper of the series we continue to spell out a new program for quantum gravity, grounded in the notion of corner symmetry algebra and its representations. Here we focus on tetrad gravity and its corner symplectic potential. We start by performing a detailed decomposition of the various geometrical quantities appearing in BF theory and tetrad gravity. This provides a new decomposition of the symplectic potential of BF theory and the simplicity constraints. We then show that the dynamical variables of the tetrad gravity corner phase space are the internal normal to the spacetime foliation, which is conjugated to the boost generator, and the corner coframe field. This allows us to derive several key results. First, we construct the corner Lorentz charges. In addition to sphere diffeomorphisms, common to all formulations of gravity, these charges add a local $\mathfrak{sl}(2,\mathbb{C})$ component to the corner symmetry algebra of tetrad gravity. Second, we also reveal that the corner metric satisfies a local $\mathfrak{sl}(2,\mathbb{R})$ algebra, whose Casimir corresponds to the corner area element. Due to the space-like nature of the corner metric, this Casimir belongs to the unitary discrete series, and its spectrum is therefore quantized. This result, which reconciles discreteness of the area spectrum with Lorentz invariance, is proven in the continuum and without resorting to a bulk connection. Third, we show that the corner phase space explains why the simplicity constraints become non-commutative on the corner. This fact requires a reconciliation between the bulk and corner symplectic structures, already in the classical continuum theory. Understanding this leads inevitably to the introduction of edge modes.
