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.
Highlights
It was recently proposed by the author, [1,2], to take into account the global one–dimensionality conjecture within quantum general relativity, where the global dimension is determinant of metric of a three-dimensional space embedded into four-dimensional space-time
The global quantum mechanics can be interpreted in terms of radial-type Schrödinger wave equation and, for this reason; it straightforwardly leads to the strict physical relation with atomic and nuclear physics
We have considered the analytical form of the effective potential, and concentrated an especial attention on the physical conclusions following from the residual effective potential, which on some wellestablished conventional level is directly identified with the attractive Newton’s gravitation or the repulsive Coulomb’s electrostatics
Summary
It was recently proposed by the author, [1,2], to take into account the global one–dimensionality conjecture within quantum general relativity, where the global dimension is determinant of metric of a three-dimensional space embedded into four-dimensional space-time This conjecture leads to the non-trivial model of quantum gravity which differs from the standardly considered approaches [312]. In and of itself this fragment of the model is globally one-dimensional quantum mechanics describing 3 + 1-decomposed solutions of the Einstein field equations of general relativity In this theory quantum gravity is given by the one-dimensional Schrödinger equation, where the single dimension is the global dimension. The generalized functional expansion of the effective potential and the residual approximation of the expansion, which corresponds to the embedding being the maximally symmetric three-dimensional Einstein manifolds, whose the physical meaning is reconstruction the Newton–Coulomb type potential within the model of quantum gravity, are considered. Extrinsic curvature Kij , where TrKij ≡ K , is constrained by the equality
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