In the tetrad formulation of gravity, the so-called simplicity constraints play a central role. They appear in the Hamiltonian analysis of the theory, and in the Lagrangian path integral when constructing the gravity partition function from topological BF theory. We develop here a systematic analysis of the corner symplectic structure encoding the symmetry algebra of gravity, and perform a thorough analysis of the simplicity constraints. Starting from a precursor phase space with Poincar\'e and Heisenberg symmetry, we obtain the corner phase space of BF theory by imposing kinematical constraints. This amounts to fixing the Heisenberg frame with a choice of position and spin operators. The simplicity constraints then further reduce the Poincar\'e symmetry of the BF phase space to a Lorentz subalgebra. This picture provides a particle-like description of (quantum) geometry: The internal normal plays the role of the four-momentum, the Barbero-Immirzi parameter that of the mass, the flux that of a relativistic position, and the frame that of a spin harmonic oscillator. Moreover, we show that the corner area element corresponds to the Poincar\'e spin Casimir. We achieve this central result by properly splitting, in the continuum, the corner simplicity constraints into first and second class parts. We construct the complete set of Dirac observables, which includes the generators of the local $\mathfrak{sl}(2,\mathbb{C})$ subalgebra of Poincar\'e, and the components of the tangential corner metric satisfying an $\mathfrak{sl}(2,\mathbb{R})$ algebra. We then present a preliminary analysis of the covariant and continuous irreducible representations of the infinite-dimensional corner algebra. Moreover, as an alternative path to quantization, we also introduce a regularization of the corner algebra and interpret this discrete setting in terms of an extended notion of twisted geometries.

中文翻译：

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边缘重力模式。第三部分。拐角简单性约束

在引力的四重公式中，所谓的简单性约束起着核心作用。它们出现在理论的哈密顿分析中，以及从拓扑 BF 理论构造引力分配函数时的拉格朗日路径积分中。我们在这里对编码重力对称代数的角辛结构进行系统分析，并对简单性约束进行彻底分析。从具有庞加莱对称性和海森堡对称性的前驱相空间开始，我们通过施加运动学约束获得了BF 理论的角相空间。这相当于通过选择位置和自旋算子来固定海森堡框架。然后，简单性约束进一步将 BF 相空间的 Poincar\'e 对称性降低到洛伦兹子代数。这张图片提供了对（量子）几何的粒子状描述：内部法线扮演四动量的角色，巴贝罗-伊米尔齐参数扮演质量的角色，相对论位置的通量和框架的角色自旋谐振子。此外，我们表明角区域元素对应于 Poincar\'e 自旋 Casimir。我们通过在连续体中将拐角简单性约束正确拆分为第一类和第二类部分来实现这一中心结果。我们构建了完整的 Dirac 可观测值集，其中包括 Poincar\'e 的局部 $\mathfrak{sl}(2,\mathbb{C})$ 子代数的生成器，以及满足 $\'e 的切角度量的分量\mathfrak{sl}(2,\mathbb{R})$ 代数。然后，我们对无限维角代数的协变和连续不可约表示进行了初步分析。此外，作为量化的替代途径，我们还引入了角代数的正则化，并根据扭曲几何的扩展概念来解释这种离散设置。