投影Ponzano-Regge自旋网络及其对称性
Projective Ponzano-Regge spin networks and their symmetries
摘要：
我们提出了一种新的层次结构的投影自旋网络的Ponzano-Regge型从组装的五个四边形的组合4-单形兼容的几何实现在欧几里得4-空间。它的关键内容是，一方面是射影Desargues位形及其空间对偶给出的关联结构，另一方面是SU（2）的6 j符号的Biedenharn-Elliott恒等式。投影组合和代数特征之间的相互作用依赖于角动量的再耦合理论，这是一种在过去几十年中成功实现的离散量子引力模型的方法。Regge对称性的作用--$6j$的一种有趣的离散对称性，它超越了这个符号的标准四面体对称性--也将被简要地讨论，以强调它在提供投射自旋网络的自然正则化中的作用，该网络通过SU（2）的q变形以某种方式模拟标准正则化。
英文摘要：
We present a novel hierarchical construction of projective spin networks of the Ponzano-Regge type from an assembling of five quadrangles up to the combinatorial 4-simplex compatible with a geometrical realization in Euclidean 4-space. The key ingrendients are the projective Desargues configuration and the incidence structure given by its space-dual, on the one hand, and the Biedenharn--Elliott identity for the 6j symbol of SU(2), on the other. The interplay between projective-combinatorial and algebraic features relies on the recoupling theory of angular momenta, an approach to discrete quantum gravity models carried out successfully over the last few decades. The role of Regge symmetry --an intriguing discrete symmetry of the $6j$ which goes beyond the standard tetrahedral symmetry of this symbol-- will be also discussed in brief to highlight its role in providing a natural regularization of projective spin networks that somehow mimics the standard regularization through a q-deformation of SU(2).
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