位置相关的质量量子系统和ADM形式
Position-Dependent Mass Quantum systems and ADM formalism
摘要：
证明了广义相对论（GR）的经典Einstein-Hilbert（EH）作用量形式上类似于具有位置依赖质量（PDM）模型的经典系统。类比的开发和使用建立协变经典的哈密顿量，以及定义一个替代的相画像GR。相空间中的一组相关的哈密顿方程作为一个一阶系统的双爱因斯坦场方程。遵循量子力学的基本原理，我建立了一个经典一般的正则理论。在采用高维相空间的基础上，构造了广义相对论的完全自洽量子哈密顿量。观察到泛函波动方程是无时间的。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）分解和量子引力方案相比，当度规被分解为$3+1$维ADM分解时，我将我的分析扩展到协变区域之外。我证明了，如果应用ADM分解度量，可以得到一个等维相空间。
英文摘要：
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.
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