圈量子引力中次简单性约束的注记
A note on the secondary simplicity constraints in loop quantum gravity
摘要：
关于圈量子引力和自旋泡沫的文献中出现了一场争论，争论的焦点是，将连接简化为列维-奇维塔的次级简单性约束是否应该隐含形状匹配条件，将扭曲几何简化为雷吉几何。我们解决这个问题，使用一个简单的模型与平坦的动力学，其中次要的简单性约束产生的动力学保存的主要。我们发现，形状匹配的条件出现，从而提供支持，一个肯定的问题。起源这些额外的条件是要找到不同的图形本地化的哈密顿量和主要的简单性约束。我们的结果与Dittrich和Ryan先前的结论一致，并将其有效性扩展到Lorentzian签名和任意细胞分解。我们特别展示了如何（规范不变版本的）扭曲角Xi扭曲的几何形状等于上壳的Regge二面角乘以Immirzi参数，从而恢复离散的外在几何从Ashtekar-Barbero holonomy。最后，我们确认，平坦性意味着使用扭曲的几何形状和4维版本的顶点条件出现在t Hooft的多边形模型中的Levi-Civita和形状匹配条件。
英文摘要：
A debate has appeared in the literature on loop quantum gravity and spin foams, over whether the secondary simplicity constraints, reducing the connection to be Levi-Civita, should imply the shape matching conditions, reducing twisted geometries to Regge geometries. We address the question using a simple model with a flat dynamics, in which secondary simplicity constraints arise from a dynamical preservation of the primary ones. We find that shape matching conditions arise, thus providing support to an affirmative question. The origin of these extra conditions is to be found in the different graph localisation of the Hamiltonian and primary simplicity constraints. Our results are consistent with previous claims by Dittrich and Ryan, and extend their validity to Lorentzian signature and arbitrary cellular decompositions. We show in particular how the (gauge-invariant version of the) twist angle xi featuring in twisted geometries equals on-shell the Regge dihedral angle multiplied by the Immirzi parameter, thus recovering the discrete extrinsic geometry from the Ashtekar-Barbero holonomy. Finally, we confirm that flatness implies both the Levi-Civita and the shape-matching conditions using twisted geometries and a 4-dimensional version of the vertex condition appearing in 't Hooft's polygon model.
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