On the Residual Effective Potential within Global One-Dimensional Quantum Gravity

TL;DR

This paper explores a simplified one-dimensional quantum gravity model derived from a higher-dimensional framework, focusing on the effective potential and residual approximation, which simplifies the mathematical complexity of quantum geometrodynamics.

Contribution

It introduces a residual approximation within the global one-dimensional quantum gravity model, leading to a Newton-Coulomb type potential and simplifying the effective potential analysis.

Findings

Derived a generalized functional expansion of the effective potential.

Developed a residual approximation for maximally symmetric Einstein manifolds.

Proposed physically interesting scenarios based on specific effective potential forms.

Abstract

The global one-dimensional quantum gravity is the model of quantum gravity which arises from the global one-dimensionality conjecture within quantum general relativity, first considered by the author in 2010 and then in 2012. In this model the global dimension is a determinant of a metric of three-dimensional space embedded into an enveloping Lorentizan four-dimensional spacetime. In 2012, it has already been presented by the author that this model can be extended to any Lorentzian D + 1-dimensional spacetime, where D is a dimension of space, and resulting in the global one-dimensional model of a higher dimensional quantum gravity. The purely quantum-mechanical part of this model is a minimal effective model within the quantum geometrodynamics, introduced by J.A. Wheeler and B.S. DeWitt in the 1960s, but the effective potential is manifestly different from the one considered by Wheeler & DeWitt. Moreover, in our model the wave functionals solving the quantum gravity are one-variable smooth functions and, therefore, the troublesome mathematical technique of the Feynman functional integration present in the Hawking formulation of quantum gravity, is absent is this model, what makes it a mathematically consistent theory of quantum gravitation. In this paper, we discuss in some detail a certain part of the global one-dimensional model already proposed in 2010, and then developed in 2012. The generalized functional expansion of the effective potential and the residual approximation, which describe the embedded spaces which are maximally symmetric three-dimensional Einstein's manifolds, whose lead to the Newton-Coulomb type potential in the quantum gravity model, are considered. Furthermore, scenarios related to few selected specific forms of the effective potential are suggested as physically interesting and discussed.