Position-Dependent Mass Quantum systems and ADM formalism

TL;DR

This paper explores the analogy between the Einstein-Hilbert action in general relativity and position-dependent mass models, developing a covariant Hamiltonian framework and proposing a new quantum cosmology wave equation.

Contribution

It introduces a novel analogy between GR and PDM models, constructs a covariant Hamiltonian, and extends quantum cosmology analysis beyond the standard ADM decomposition.

Findings

A covariant classical Hamiltonian for GR is formulated.

A new quantum wave equation for cosmology is proposed.

An equal-dimensional phase space is achieved using ADM decomposition.

Abstract

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.