It is well known that Einstein’s equations assume a simple polynomial form in the Hamiltonian framework based on a Yang-Mills phase space. We re-examine the gravitational dynamics in this framework and show that time evolution of the gravitational field can be re-expressed as (a gauge covariant generalization of) the Lie derivative along a novel shift vector field in spatial directions. Thus, the canonical transformation generated by the Hamiltonian constraint acquires a geometrical interpretation on the Yang-Mills phase space, similar to that generated by the diffeomorphism constraint. In classical general relativity this geometrical interpretation significantly simplifies calculations and also illuminates the relation between dynamics in the ‘integrable’ (anti)self-dual sector and in the full theory. For quantum gravity, it provides a point of departure to complete the Dirac quantization program for general relativity in a more satisfactory fashion. This gauge theory perspective may also be helpful in extending the ‘double copy’ ideas relating the Einstein and Yang-Mills dynamics to a non-perturbative regime. Finally, the notion of generalized, gauge covariant Lie derivative may also be of interest to the mathematical physics community as it hints at some potentially rich structures that have not been explored.

中文翻译：

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引力动力学—哈密顿主义范式的新变化

众所周知，爱因斯坦方程式在基于Yang-Mills相空间的哈密顿量框架中采用简单的多项式形式。我们在此框架下重新检查了重力动力学，并显示出重力场的时间演化可以在空间上沿着新的位移矢量场重新表示为Lie导数（的规范协变泛化）。指示。因此，由哈密顿约束产生的规范变换获得了杨-米尔斯相空间上的几何解释，类似于由微分形约束产生的解释。在古典广义相对论中，这种几何解释大大简化了计算，也阐明了“可积”（反）自对偶扇区和整个理论中动力学之间的关系。对于量子引力，它为以更令人满意的方式完成广义相对论的狄拉克量化程序提供了起点。这种规范理论的观点也可能有助于将“爱因斯坦和杨米尔斯动力学与非扰动状态相关的”双重复制”思想扩展。最后，广义的概念