重力的边缘模式。部件III。角点简单性约束
Edge modes of gravity. Part III. Corner simplicity constraints
摘要：
在引力的四分体公式中，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理论构造引力配分函数时的拉格朗日路径积分中。在这里，我们开发了一个系统的角辛结构编码的引力对称代数的分析，并进行了彻底的分析的简单性约束。从具有庞加莱和海森堡对称性的先驱相空间出发，通过施加运动学约束，得到了BF理论的角相空间。这相当于用位置和自旋算符的选择来固定海森堡框架。简单性约束则进一步将BF相空间的庞加莱对称性降低到洛伦兹子代数。这幅图提供了（量子）几何的类似粒子的描述：内法线扮演着四动量的角色，巴贝罗-伊米尔齐参数扮演着质量的角色，通量扮演着相对论位置的角色，而框架扮演着自旋谐振子的角色。此外，我们还证明了角面积元对应于Poincaré自旋Casimir。我们实现了这一中心的结果，通过适当的分裂，在连续，角简单性约束成第一和第二类部分。我们构造了狄拉克观测量的完备集，其中包括庞加莱的局部$\mathfrak{sl}(2,\mathbb{C})$子代数的生成元，以及满足$\mathfrak{sl}(2,\mathbb{R})$代数的切角度量的分量.然后，我们提出了一个初步的分析的协变和连续的不可约表示的无限维角代数。此外，作为量子化的另一种途径，我们还引入了正则化的角代数和解释这个离散设置的扭曲几何的扩展概念。
英文摘要：
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 Poincaré 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 Poincaré 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 Poincaré 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 Poincaré, 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.
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