The classical Einstein-Hilbert (EH) action for general relativity (GR) is shown to be formally analogous to the classical system with position-dependent mass (PDM) models. The analogy is developed and used to build the covariant classical Hamiltonian as well as defining an alternative phase portrait for GR. The set of associated Hamilton's equations in the phase space is presented as a first-order system dual to the Einstein field equations. Following the principles of quantum mechanics, I build a canonical theory for the classical general. A fully consistent quantum Hamiltonian for GR is constructed based on adopting a high dimensional phase space. It is observed that the functional wave equation is timeless. As a direct application, I present an alternative wave equation for quantum cosmology. In comparison to the standard Arnowitt-Deser-Misner(ADM) decomposition and quantum gravity proposals, I extended my analysis beyond the covariant regime when the metric is decomposed into the 3+1 dimensional ADM decomposition. I showed that an equal dimensional phase space can be obtained if one applies ADM decomposed metric.

中文翻译：

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位置相关质量量子系统和 ADM 形式主义

广义相对论 (GR) 的经典爱因斯坦-希尔伯特 (EH) 作用被证明在形式上类似于具有位置相关质量 (PDM) 模型的经典系统。类比被开发并用于构建协变经典哈密顿量以及定义 GR 的替代相图。相空间中相关的哈密顿方程组被表示为爱因斯坦场方程的一阶系统对偶。遵循量子力学的原理，我为经典将军建立了一个正则理论。基于采用高维相空间构造了一个完全一致的 GR 量子哈密顿量。可以看出，泛函波动方程是永恒的​​。作为一个直接的应用，我提出了一个量子宇宙学的替代波动方程。与标准的 Arnowitt-Deser-Misner(ADM) 分解和量子引力建议相比，当度量被分解为 3+1 维 ADM 分解时，我将我的分析扩展到协变机制之外。我表明，如果应用 ADM 分解度量，可以获得等维相空间。