SL(2,C)相干不变张量的渐近行为
Asymptotics of $\mathrm{SL}(2,\mathbb{C})$ coherent invariant tensors
AI总结：
本文研究SL(2,C)主系列表示中n腿广义Clebsch-Gordan系数不变量的半经典极限，发现其临界构型对应可通过Lorentz变换互变的三维多边形，并利用鞍点近似给出首阶幂律衰减公式，结果可应用于EPRL模型及超越固定时间规范的自旋泡沫研究。
摘要：
我们研究了一类对应于具有n≥3条腿的广义Clebsch-Gordan系数的SL(2,C)主系列无限维酉表示的不变张量的半经典极限。我们发现了量子标签的临界构型，其不变量呈幂律衰减。它们描述了可以通过Lorentz变换相互变形的三维多边形。这是通过将多边形的边向量视为双矢量的电部分来定义的，这些双矢量满足其电部分与磁部分之间的（依赖于框架的）关系，该关系在圈量子引力文献中被称为γ-简单性。该框架依赖于标记不变量基元素的SU(2)自旋。我们利用临界点计算了鞍点近似，并给出了不变量的首阶近似。如果SU(2)自旋取最低值，则幂律是普适的；否则依赖于n。作为附带结果，我们给出了任意框架下γ-简单性的紧凑公式。这些结果可应用于当前的EPRL模型，也有助于未来旨在超越自旋泡沫模型中固定时间规范使用的研究。
英文摘要：
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.
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