论文标题
重力的边缘模式。第三部分。转角简单约束
Edge modes of gravity. Part III. Corner simplicity constraints
论文作者
论文摘要
在重力的四四公式中，所谓的简单约束起着核心作用。它们出现在对理论的哈密顿分析中，以及从拓扑BF理论构建重力分区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符斑结构进行系统分析，并对简单约束进行彻底的分析。从Poincaré和Heisenberg对称性的前体相位空间开始，我们通过施加运动学约束来获得BF理论的角相空间。这相当于通过选择位置和旋转操作员的选择来固定海森堡框架。然后，简单的约束进一步将BF相空间的庞加莱对称性降低到Lorentz subgerbra。这张图提供了（量子）几何形状的类似粒子的描述：内部正常作用扮演了四弹，质量的barbero-immirzi参数，相对论位置的通量以及旋转和谐振荡器的框架。此外，我们表明角区元素对应于庞加莱旋转卡西米尔。我们通过在连续体中正确拆分角度的简单性约束成一等和二等零件来实现这一中心结果。我们构建了完整的Dirac可观察物，其中包括Poincaré的本地$ \ Mathfrak {Sl}（2，\ Mathbb {C}）$ subalgebra的生成器，以及切角角的组件，使$ \ m athfrak {slfrak {sl}（2，\ mathbbbbbbbb {rge {rge alge alge a al alge a}）然后，我们对无限维角代数的协方差且不可还原表示进行初步分析。此外，作为量化的替代途径，我们还引入了角代数的正则化，并根据扭曲几何形状的扩展概念来解释这种离散设置。
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.
