Loop quantum gravity (LQG) is a non-perturbative attempt at quantization of a classical phase space description of gravity in terms of SU(2) connections and electric fields. As emphasized recently Ashtekar and Varadarajan (2020 arXiv:2012.12094 [gr-qc]), on this phase space, classical gravitational evolution in time can be understood in terms of certain gauge covariant generalizations of Lie derivatives with respect to a spatial SU(2) Lie algebra valued vector field called the electric shift. We present a derivation of a quantum dynamics for Euclidean LQG which is informed by this understanding. In addition to the physically motivated nature of the action of the Euclidean Hamiltonian constraint so derived, the derivation implies that the spin labels of regulating holonomies are determined by corresponding labels of the spin network state being acted upon thus eliminating the ‘spin j-ambiguity’ pointed out by Perez. By virtue of Thiemann’s seminal work, the Euclidean quantum dynamics plays a crucial role in the construction of the Lorentzian quantum dynamics so that our considerations also have application to Lorentzian LQG.

环量子引力 (LQG) 是根据SU (2) 连接和电场对引力的经典相空间描述进行量化的非微扰尝试。正如最近 Ashtekar 和 Varadarajan (2020 arXiv:2012.12094 [gr-qc]) 所强调的，在这个相空间上，经典引力随时间的演化可以通过李导数相对于空间 SU的某些规范协变概括来理解(2) 称为电位移的李代数向量场。我们提出了欧几里得 LQG 的量子动力学推导，这是由这种理解提供的。除了如此导出的欧几里得哈密顿约束的作用的物理动机性质之外，该推导意味着调节完整的自旋标签由所作用的自旋网络状态的相应标签确定，从而消除了“自旋j -歧义”佩雷斯指出。凭借 Thiemann 的开创性工作，欧几里得量子动力学在洛伦兹量子动力学的构建中起着至关重要的作用，因此我们的考虑也适用于洛伦兹 LQG。