Numerical study of the Lorentzian Engle-Pereira-Rovelli-Livine spin foam amplitude
Numerical study of the Lorentzian Engle-Pereira-Rovelli-Livine spin foam amplitude
复制标题
DOI：
10.1103/physrevd.100.106003
复制
发表时间：
2019-03
影响因子：
5
通讯作者：
P. Donà;Marco Fanizza;Giorgio Sarno;Simone Speziale
中科院分区：
文献类型：
--
作者：
P. Donà;Marco Fanizza;Giorgio Sarno;Simone Speziale
圈量子引力的洛伦兹EPRL自旋泡沫振幅是一个高振荡函数的多维非紧积分。利用SL（2，C）的无限维酉表示的Clebsch-Gordan系数根据SU（2）的系数分解的方法，我们能够首次提供顶点振幅的数值计算。所获得的值支持巴雷特和合作者获得的渐近公式与鞍点近似，显示，特别是，幂律衰减和振荡有关的Regge行动。比较提供了一个测试的效率的方法。截断分解到前几项提供了幂律衰减和振荡的定性匹配。对于矢量和欧氏Regge边界数据，仅用第一项就获得了定性匹配，这对应于简化的EPRL模型。我们评论的数值和扩展到更高的顶点的未来发展。我们完成了我们的工作与一些分析结果：我们提供了一个算法和明确的配置不同的几何形状，可以出现的边界数据，并解释所使用的分解的几何后果。
The Lorentzian EPRL spin foam amplitude for loop quantum gravity is a multi-dimensional non-compact integral of highly oscillating functions. Using a method based on the decomposition of Clebsch-Gordan coefficients for the unitary infinite-dimensional representations of SL(2,C) in terms of those of SU(2), we are able to provide for the first time numerical evaluations of the vertex amplitude. The values obtained support the asymptotic formula obtained by Barrett and collaborators with a saddle point approximation, showing, in particular, a power-law decay and oscillations related to the Regge action. The comparison offers a test of the efficiency of the method. Truncating the decomposition to the first few terms provides a qualitative matching of the power-law decay and oscillations. For vector and Euclidean Regge boundary data, a qualitative matching is obtained with just the first term, which corresponds to the simplified EPRL model. We comment on future developments for the numerics and extension to higher vertices. We complete our work with some analytic results: We provide an algorithm and explicit configurations for the different geometries that can arise as boundary data, and explain the geometric consequences of the decomposition used.