标题： 量子群交错空间来自量子弯曲四面体 显示英文标题

标题： Quantum Group Intertwiner Space From Quantum Curved Tetrahedron

摘要： 在本文中，我们通过将组合量子化应用于在arXiv:1506.03053中定义的四面体形状的相空间，发展了一种均匀曲率四面体几何的量子理论。 我们的方法基于这个相空间与SU(2)平直连接在四孔球面上的模空间之间的关系。 量子化结果得到物理Hilbert空间作为量子闭合约束的解，这量化了均匀曲率四面体的经典闭合条件$M_4M_3M_2M_1=1$, $M_\nu\in$ SU(2)。 量子群Uq(su(2))作为量子四面体的规范对称性出现。 量子四面体的物理Hilbert空间与Uq(su(2))的四价不变量的Hilbert空间一致。 此外，我们定义了量化四面体面面积的面积算符并计算了谱。 所得谱在大自旋区域与通常的环量子引力面积谱一致，但在小自旋情况下有所不同。 这项工作与存在宇宙常数的3+1维环量子引力密切相关，并为该理论中量子群的出现提供了依据。 显示英文摘要

摘要： In this paper, we develop a quantum theory of homogeneously curved tetrahedron geometry, by applying the combinatorial quantization to the phase space of tetrahedron shapes defined in arXiv:1506.03053. Our method is based on the relation between this phase space and the moduli space of SU(2) flat connections on a 4-punctured sphere. The quantization results in the physical Hilbert space as the solution of the quantum closure constraint, which quantizes the classical closure condition $M_4M_3M_2M_1=1$, $M_\nu\in$ SU(2), for the homogeneously curved tetrahedron. The quantum group Uq(su(2)) emerges as the gauge symmetry of a quantum tetrahedron. The physical Hilbert space of the quantum tetrahedron coincides with the Hilbert space of 4-valent intertwiners of Uq(su(2)). In addition, we define the area operators quantizing the face areas of the tetrahedron and compute the spectrum. The resulting spectrum is consistent with the usual Loop-Quantum-Gravity area spectrum in the large spin regime but is different for small spins. This work closely relates to 3+1 dimensional Loop Quantum Gravity in presence of cosmological constant and provides a justification for the emergence of quantum group in the theory.