标题： 从哈密顿约束出发的图雷夫-维罗振幅 显示英文标题

标题： Towards the Turaev-Viro amplitudes from a Hamiltonian constraint

摘要： 3D环量子引力在宇宙学常数为零的情况下可以与在格点上离散化的$\textrm{SU}(2)$BF理论的量子化相关联。 在经典层面上，这个离散模型描述了离散的平坦几何，其相空间由$T^\ast \textrm{SU}(2)$构建而成。 在最近的一篇论文\cite{HyperbolicPhaseSpace}中，这个离散模型使用泊松-李群形式进行了变形，并被证明在保持拓扑性的同时描述了离散的双曲几何。 因此，它是描述带有（负）宇宙学常数的$\textrm{SU}(2)$BF理论离散化的良好候选者。 我们在此对这个模型进行量子化。 在运动学层面上，希尔伯特空间由基于$\mathcal{U}_{q}(\mathfrak{su}(2))$构建的自旋网络张开（其中$q$为实数）。 特别是，离散高斯约束的量子化自然导致$\mathcal{U}_{q}(\mathfrak{su}(2))$纹理。 我们还对一个三度面的哈密顿约束进行量子化，并表明物理态与量子6j符号成比例。 这表明带有$q$实数的图拉夫-维罗幅度是量子哈密顿量的一个解。 因此，该模型是描述具有（负）宇宙学常数的三维环状量子引力的一个自然候选。 显示英文摘要

摘要： 3D Loop Quantum Gravity with a vanishing cosmological constant can be related to the quantization of the $\textrm{SU}(2)$ BF theory discretized on a lattice. At the classical level, this discrete model characterizes discrete flat geometries and its phase space is built from $T^\ast \textrm{SU}(2)$. In a recent paper \cite{HyperbolicPhaseSpace}, this discrete model was deformed using the Poisson-Lie group formalism and was shown to characterize discrete hyperbolic geometries while being still topological. Hence, it is a good candidate to describe the discretization of $\textrm{SU}(2)$ BF theory with a (negative) cosmological constant. We proceed here to the quantization of this model. At the kinematical level, the Hilbert space is spanned by spin networks built on $\mathcal{U}_{q}(\mathfrak{su}(2))$ (with $q$ real). In particular, the quantization of the discretized Gauss constraint leads naturally to $\mathcal{U}_{q}(\mathfrak{su}(2))$ intertwiners. We also quantize the Hamiltonian constraint on a face of degree 3 and show that physical states are proportional to the quantum 6j-symbol. This suggests that the Turaev-Viro amplitude with $q$ real is a solution of the quantum Hamiltonian. This model is therefore a natural candidate to describe 3D loop quantum gravity with a (negative) cosmological constant.