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 qunatum gravity proposals, I extended my analysis beyond the covariant regime when the metric is decomposed in to the $3+1$ dimensional ADM decomposition. I showed that an equal dimensional phase space can be obtained if one apply ADM decomposed metric.
Highlights
Starting from classical mechanics, there are at least three interesting ways to extend the theory each of which introduces a constant of nature that is absent in classical mechanics: (1) at large velocities with respect to the velocity of light c the theory extends to special relativity; (2) at small distances certain physical quantities get quantized in units of the reduced Planck’s constant h corresponding to quantum mechanics and (3) a gravitational force can be introduced via Newton’s constant G leading to Newtonian gravity
There are two well-known ways to combine two of these extensions: (1) extending classical mechanics with high velocities and gravity leads to general relativity and (2) extending classical mechanics to high velocities and small distances leads to quantum field theory
There is a third way, namely extending classical mechanics to small distances and gravity. This would lead to a theory of non-relativistic (NR) quantum gravity
Summary
Starting from classical mechanics, there are at least three interesting ways to extend the theory each of which introduces a constant of nature that is absent in classical mechanics: (1) at large velocities with respect to the velocity of light c the theory extends to special relativity; (2) at small distances certain physical quantities get quantized in units of the reduced Planck’s constant h corresponding to quantum mechanics and (3) a gravitational force can be introduced via Newton’s constant G leading to Newtonian gravity. There is a third way, namely extending classical mechanics to small distances and gravity This would lead to a theory of non-relativistic (NR) quantum gravity. The maximal extension to high velocities, small distances and gravity leads to the long sought for theory of quantum gravity. Of how essential relativity is in constructing a theory of quantum gravity or, put differently, whether one can take in a consistent way the NR limit of quantum gravity Motivated by this we wish to address the following intriguing question: can one define a consistent NR theory of quantum gravity?. We will show how the geometry corresponding to NR string theory can be viewed as a generalization of the well-known Newton-Cartan (NC) geometry that underlies NC gravity
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