论文标题
$ \ mathrm {sl}的渐近学
Asymptotics of $\mathrm{SL}(2,\mathbb{C})$ coherent invariant tensors
论文作者
论文摘要
我们研究了主要系列的$ \ mathrm {Slrm {sl}（2，\ Mathbb {c}）$ \ mthrm {Slrm {sl} $的无限维统一表示子的半经典限制，对应于$ n \ geq3 $ teg的普通clebsch-gordan系数。我们发现量子标签的关键配置具有不变性的幂律衰减。他们描述了3D多边形，这些多边形可以通过Lorentz转换彼此变形。这是定义的，查看多边形的边缘向量是双方的电部分，在环路量子重力文献中，其电动和磁性零件（称为$γ$ -simplicity）之间的电动零件（依赖帧依赖）关系。框架取决于SU（2）自旋标记不变的基础元素。我们使用临界点计算鞍点近似，并提供不变性的前阶近似。如果SU（2）旋转具有最低的值，否则$ n $依赖于否则，则幂律是通用的。作为综上的结果，我们为任意框架中的$γ$ -Simplicity提供了一个紧凑的公式。结果对当前的EPRL模型有应用，但也针对未来的研究，旨在超越自旋泡沫模型中固定时表的使用。
We study the semiclassical limit of a class of invariant tensors for infinite-dimensional unitary representations of $\mathrm{SL}(2,\mathbb{C})$ of the principal series, corresponding to generalized Clebsch-Gordan coefficients with $n\geq3$ legs. We find critical configurations of the quantum labels with a power-law decay of the invariants. They describe 3d polygons that can be deformed into one another via a Lorentz transformation. This is defined viewing the edge vectors of the polygons are the electric part of bivectors satisfying a (frame-dependent) relation between their electric and magnetic parts known as $γ$-simplicity in the loop quantum gravity literature. The frame depends on the SU(2) spin labelling the basis elements of the invariants. We compute a saddle point approximation using the critical points and provide a leading-order approximation of the invariants. The power-law is universal if the SU(2) spins have their lowest value, and $n$-dependent otherwise. As a side result, we provide a compact formula for $γ$-simplicity in arbitrary frames. The results have applications to the current EPRL model, but also to future research aiming at going beyond the use of fixed time gauge in spin foam models.
