迴圈量子重力

此时用卡当的几何法，用三足（triad）一次型来表示度规张量,

${\displaystyle e_{a}^{i}e_{b}^{j}\delta _{ij}=q_{ab}}$

假设与三足相容的联络为${\displaystyle \Gamma _{a}^{i}}$(称为自旋联络),三维空间的外部曲率张量为${\displaystyle K_{a}^{i}}$,${\displaystyle \gamma }$为任何实数，定义一个新的联络

${\displaystyle A_{a}^{i}=\Gamma _{a}^{i}+\gamma K_{a}^{i}}$

即阿希提卡-巴贝罗联络。其正则动量为${\displaystyle E_{i}^{a}=det(e)e_{i}^{a}}$。使用狄拉克约束处理方法可得三个第一类约束:

1. 高斯约束: ${\displaystyle D_{a}E_{i}^{a}=0}$
2. 微分同胚约束: ${\displaystyle F_{ab}^{i}E_{i}^{a}=0}$
3. 哈密顿约束: ${\displaystyle {\frac {1}{\sqrt {|det(E)|}}}F_{ab}^{i}E_{j}^{a}E_{k}^{b}\epsilon _{ijk}+...=0}$

${\displaystyle D_{a}}$为阿希提卡-巴贝罗联络定义的协变微商，${\displaystyle F_{ab}^{i}}$为阿希提卡-巴贝罗联络定义的曲率张量.由于有高斯约束的关系,所以迴圈量子引力是一种类似规范场论的理论.

• Rodolfo Gambini and Jorge Pullin, Loop Quantum Gravity for Everyone, World Scientific, 2020.
• Carlo Rovelli, "Reality is not what it seems", Penguin, 2016.
• Martin Bojowald, Once Before Time: A Whole Story of the Universe 2010.
• Carlo Rovelli, What is Time? What is space?, Di Renzo Editore, Roma, 2006.
• Lee Smolin, Three Roads to Quantum Gravity, 2001

• （英文）迴圈量子引力論中的引力子傳遞子(Graviton propagator in loop quantum gravity) （页面存档备份，存于）-- We compute some components of the graviton propagator in loop quantum gravity, using the spinfoam formalism, up to some second order terms in the expansion parameter.
• （英文）量子引力與標準模型(Quantum Gravity and the Standard Model) （页面存档备份，存于）-- Shows that a class of background independent models of quantum spacetime have local excitations that can be mapped to the first generation fermions of the standard model of particle physics.