作为Bianchi恒等式的双曲四面体的闭包约束
The closure constraint for the hyperbolic tetrahedron as a Bianchi identity
摘要：
闭合约束是圈量子引力数学的核心部分。它编码了量子几何中自旋网络态的规范不变性，并提供了一种几何解释：自旋网络的每个装饰顶点都与$\mathbb{R}^{3}$中的量子化多面体对偶。例如，一个4价顶点被解释为一个四面体，由它的四个面的法向量决定。我们开发了一个框架，封闭约束被重新解释为一个比安奇身份，与法线定义为完整的多面体面周围的连接（从旋转的几何解释扭曲的几何构造）。这使我们能够定义双曲四面体（生活在$\mathbb{R}^{3,1}$中单位未来定向类空向量的3-双曲面中）的闭合约束，这些约束都是根据生活在$SU(2)$或$SB(2,\mathbb{C})$中的法线定义的。后者与为$q$ -变形圈量子引力发展的经典相空间完全吻合，该变形圈量子引力假设解释了非零宇宙学常数$Λ>0$。这是将$q$变形扭曲几何解释为实际离散双曲三角剖分的第一步。
英文摘要：
The closure constraint is a central piece of the mathematics of loop quantum gravity. It encodes the gauge invariance of the spin network states of quantum geometry and provides them with a geometrical interpretation: each decorated vertex of a spin network is dual to a quantized polyhedron in $\mathbb{R}^{3}$. For instance, a 4-valent vertex is interpreted as a tetrahedron determined by the four normal vectors of its faces. We develop a framework where the closure constraint is re-interpreted as a Bianchi identity, with the normals defined as holonomies around the polyhedron faces of a connection (constructed from the spinning geometry interpretation of twisted geometries). This allows us to define closure constraints for hyperbolic tetrahedra (living in the 3-hyperboloid of unit future-oriented spacelike vectors in $\mathbb{R}^{3,1}$) in terms of normals living all in $SU(2)$ or in $SB(2,\mathbb{C})$. The latter fits perfectly with the classical phase space developed for $q$-deformed loop quantum gravity supposed to account for a non-vanishing cosmological constant $Λ>0$. This is the first step towards interpreting $q$-deformed twisted geometries as actual discrete hyperbolic triangulations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
