Lukasz Andrzej Glinka*
Member of the American Association of International Researchers. New York, USA
Received Date: February 05, 2014; Accepted Date: February 25, 2014; Published Date: March 05, 2014
Citation: Glinka LA (2014) On The Residual Effective Potential within Global One-Dimensional Quantum Gravity. J Astrophys Aerospace Technol 2:103. doi: 10.4172/2329-6542.1000.103
Copyright: © 2014 Glinka LA. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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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 space-time. In 2012, it has already been presented by the author that this model can be extended to any Lorentzian D + 1-dimensional space-time, 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.
Global one-dimensionality conjecture; Quantum gravity model; Quantum geometrodynamics; Effective potential; Residual approximation
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. Actually, this conjecture leads to the non-trivial model of quantum gravity which differs from the standardly considered approaches [3-12].
The quantum-mechanical part of the global one-dimensional quantum gravity can be considered separately as a nice mathematical theory having possibly interesting physical ramifications. 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 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.
In this paper few selected elements of the globally one-dimensional quantum mechanics are discussed in some detail. 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. Few possible mathematical scenarios with respect to the form of the effective potential are suggested as possibly interesting from the theoretical physics point of view.
The content of this paper is as follows. In the Section 7 the global one-dimensional model of quantum gravity is briefly discussed, and its quantum mechanical part is presented in some detail. We start, in the Subsection 7.1, from the condensed presentation of the standard way resulting in the Wheeler–DeWitt equation of quantum Geometrodynamics which is the standardly considered model of quantum gravity and is the core fundament of all presently considered approaches. Then, in the Subsection 7.2, the idea of the global dimension, the resulting quantum mechanics, and the idea of the generalized functional expansion of the effective potential are presented. Next, in the Subsection 7.3, the idea of the invariant global dimension is digressed only, with no continuation in the further part of the paper. In the Section 8, the residual approximation of the effective potential, the role of maximally symmetric three-dimensional Einstein manifolds within the model of quantum gravity, and few conclusions having possibly interesting physical significance are discussed. The Section 7 is devoted to derivation of the ”geometric” wave functionals of the global quantum mechanics in terms of the special functions, with respect to the selected situations given by the three types of boundary conditions discussed one by one in the Subsections 7.1, 7.2, and 7.3. Finally, in the Section 8 we summarize briefly the paper.
Standard quantum geometrodynamics
In general relativity [13,14] a Lorentzian (pseudo–Riemannian)  manifold(M, g) with a metric and a distance and a distance , where , μ=0,1,2,3 is a coordinate system, characterized by the Christoffel symbols and curvatures: Riemann–Christoffel Ricci Ricci scalar R
is a space time being a solution of the Einstein field equations
Where κ is the Einstein constant, is the Einstein tensor, Λ is cosmological constant, and Tμv is stress-energy tensor of Matter fieldswhich arises through the variationprinciple  applied to the EinsteinHilbert action modified through the York–Gibbons– Hawking boundary term [17,18]
Where κ is the Gauss scalar curvature of a spacelike boundary L0 is the Lagrangian of Matter fields, and are the invariant measures of four-dimensional space-time and embedded three-dimensionalspace, respectively.
Where are respectively called the lapse function, the shift vector, and embedded space metric, is the intrinsic covariant shift vector. The 3+1 decomposition transforms the action functional (3) into the Hamiltonian form
Where φ symbolizes Matter fields, and nontrivial π’s, H,Hi are 
Where (3)R is the Ricci scalar of an embedding, is the Matter fields energy related to the normal vector field Extrinsic curvature , where is constrained by the equality
Where is symmetrized intrinsic covariant derivative of the shift. In accordance with DeWitt’s foundational considerations , Hi are the generators of the spatial diffeomorphisms
Where Time-preservation  of the primary constraints leads to secondary ones-scalar and vector constraints
Which create nontrivial first-class type constraints algebra 
are structure constants of the diffeomorphism group and all Lie’s bracketsof π’s and H’svanish. Scalar constraint determines dynamics, vector one merely reflects diffeoinvariance. Making use of the conjugate momenta, first formula in (12), and the scalar constraint transforms into the Hamilton–Jacobi type equation
is the DeWitt metric on the Wheeler super space, a factor space of all Riemannia metrics on , and a group of all diffeomorphisms of that preserve orientation [29-31]. The Dirac–Faddeev primary canonical quantization method [28,32] in the present case has the form
The solutions or rather representations of the momenta operators satisfying these canonical commutation relations is the question of choice. In quantum Geometrodynamics, the Wheeler metric representation is usually taken into account. In such a representation the momenta operators are analogous to the momentum operator in quantum mechanics
and other first class constraints
Merely reflect diffeoinvariance, and are not important for this model. Since about 60 years classical and quantum aspects of the Wheeler–DeWitt Geometrodynamics have studied widely (Cf. e.g. the books and collective volumes) in [3-12], and individual papers, [34-48].
The global dimension
The global one–dimensionality conjecture within quantum general relativity [1,2] assumes the effective physical role of the determinant of a three-dimensional space embedded into a four-dimensional spacetime
Where is the Levi-Civita density, and takes into account the following situation as physically interesting
In other words, it replaces the functional dependence on the space metric elements and Matter fields through the function dependence on the determinant of a space metric. Applying the transformation of variables to the Wheeler–DeWitt equation (16), one must change the functional differentiation with respect to by the classical differentiation with respect to h. It can be simply done through application of the Jacobi rule for differentiation of the determinant of four-dimensional metric which leads to theresult
More detailed explanation can be found in the .
Making few very elementary algebraic manipulations, one obtains from the equation (16) the globally one-dimensional quantum mechanics
Here is the effective potential
This is a simple algebraic sum of the three fundamental energetic constituents
Related to pure geometry of an embedded three-dimensional space (G), cosmological constant (C), and Matter fields (M).
On the one hand, identification of the effective potential with the square of mass of the boson expresses the model of quantum gravity (21) as the classical theory of massive bosonic field The construction of quantum field theory by the method of the static Fockreper of creators and annihilators, and related thermodynamics of quantum states can be also done elementary. This part of the model of quantum gravity was discussedin the 2012’ monograph by the author , and is not the main motive of this paper.
On the other hand, one can take into account the non-relativistic interpretation of the one-dimensional evolution (21), and treat the received global quantum gravity model as the effective onedimensional Schrödinger quantum mechanics with a certain selected effective potential being a functional of determinant of a threedimensional embedding. In the spirit of this approach the potential has intriguing meaning - (22) is the equality between “effective physics”, which can be constructed by other type considerations,and three basic constituents related to an embedding space – ”geometric”, ”cosmological”, and ”material”.
Let us assume that the concrete form of is fixed. Then one can express the Ricci scalar of a three-dimensional embedding as follows
and, therefore, the two last terms in the brackets can be treated as the dark energy density contribution
One can list several typical examples of the physical scenarios within the global one-dimensional model of quantum gravity, with respect to the concretely fixed form of the effective gravitational potential
1. The case of the constant non-vanishing effective gravitational potential In such a situation, the Ricci scalar curvature of the embedded space and the global one-dimensional quantum gravity aregiven by the equations
Where a wave is functional related to
2. The case of the trivial effective gravitational potential In such a situation, the three-dimensional Ricci scalar curvature and the global one-dimensional quantum gravity are
Where a “free” wave is functional related to
3. The case when the sum of the geometric and cosmological contributions is identically vanishing but the material contribution does not vanish identically In such a situation, the three-dimensional Ricci scalar curvature and the global onedimensional quantum gravity are
Where a “material” wave is functional related to the material contribution
4. The case when the sum of the geometric and material contributions is identically vanishing while in general the cosmologicalcontribution is non-trivial In such a situation, the Ricci scalar curvature of the embedded space and the global onedimensional quantum gravity are respectively
Here is the “cosmological” wave functional related to the cosmological contribution
5. The case when the sum of the cosmological and material contributions is identically vanishing whereas the geometric contribution is non-zero In such a situation, the energy density of Matter fields and the global one-dimensional quantum gravity are given by the equations
Where is the “geometric” wave functional related to the geometric contribution VG ≠ 0.
6. A more general explicit form of the effective gravitational potential can be constructed in the spirit of complex analysis. Let us consider ad hoc functional generalization of the Laurent series expansion in the global dimension h of the effective gravitational potential Veff [h]
in an infinitesimal neighborhood, i.e. in a one-sphere (circle) of radius h∈, of any ad hoc fixed initial value h0 of the global dimension
The numbers an are the series coefficients determined by the classical functional integral
which is straightforward functional generalization of the Cauchy integral formula with the Lebesgue–Stieltjes measure–the classical functional Radon measure δ h
Let us take into considerations the most general choice of h0 which similarly to h is assumed to be a complex number. In such a situation, the Ricci scalar curvature of a three-dimensional embedded space takes the following form
where bn is the combined series coefficient
and the global one-dimensional model of quantum gravity is then defined by the equation
Making use of the triangle inequality one can write
so that it can be deduced straightforwardly that
Applying the well-known inequality for any Riemann integral
where f is considered as the Riemann-integrable function and the integral is considered as defined, to the coefficients an and bn one obtains the inequality
where a−1 is the residue of the effective gravitational potential at the fixed point h = h0 . This residue is determined by the straightforward functional generalization of the Cauchy integral formula
where C(h∈) traces out circle around the fixed point h0 in counterclockwise manner on the punctured disk . If the center of the circle h−h0 is a pole of order n, then the residue is defined by the simple limiting procedure
Making use of the following reasoning
one sees that for any n the inequality holds
and hence the inequality (43) can be rewritten as
In the light of the triangle inequality one can write
and because of the relation
one receives the following relation
Finally, application of the inequality (50) in the equivalent form
lead us to the following upper bound
Another bound for can be obtained as follows. In the light of the definition (43) one can write
and, consequently, one can deduce the recursive relation for the coefficients bn
which after simple algebraic manipulations leads to the relation
This equation can be rewritten as
which after taking into account the triangle inequality gives the relation
which leads to the conclusion
Once again, making use of the triangle inequality one has
This inequality can be rewritten in the following form
or more conveniently
In the light of the inequality (54) and the relation
one obtains finally the following bound
which taken together with the previous result (55) allows to deduce the inequality for the fixed point
which can be resolved immediately and gives the consistency condition for the fixed point
and in itself is a non-trivial solution of the initial data problem.
Naturally, there is plenty of other possibilities for the choice of a concrete form of the effective gravitational potential . However, in the next section we shall discuss only a particular situation.
The invariant global dimension
Let us note that in general the global one-dimensional quantum mechanics (21) cane be transformed by the second change of variables
where is any functional in the global dimension h. In this case one can rewrite the global one-dimensional wave equation (21) in the form
and if the coefficient does not vanish identically (or the transformation (70) is non-singular) then the equation (72) can be rewritten as ξ
where the new potential V[ξ] is scaled effective potential Veff expressed by the new dimension
One sees easily that the following choice of the “gauge”
Transforms the quantum mechanics (21) into itself. The choice of the transformation of variables in the form (74) is the simplest transformation of the kind within the Wheeler– DeWitt theory. Other, more advanced propositions, can be generated directly from this basic case, and should be justified by some rational arguments. Let us choose the transformation of variables in the form
It is clear that this selection can be justified by the fact that is the invariant volume element on an embedding with assumption that 1. . In this manner the dimension has an invariant nature. The choice (75) yields the equation (72) with the following modified effective potential
Moreover, the singularity evidently vanishes, but actually causes that must be studied with respect to the new“invariant” dimension ξ .
The very good point of reference in searching for the dimension ξ is the normalization condition of the Schrödinger quantum mechanics, which for the considered situation takes the form of the Lebesgue– Stieltjes/Radon integral
Where Ω(hI, h) some region of inerrability in a space of all threedimensional embedding is’ s with metric hij and a determinant h=det hij. In fact this is the main condition for possible solutions of the studied model:
Proposition. Inerrability of the wave functional in the sense of functional integration in (77) determines the new dimension
The generalized dimension can be established in the region of inerrability as by using of the formula
In this paper we will study few consequences of the simplest choice (74). We will use standard argument which states that the normalization condition (77) establishes inerrability constants of an arbitrary solution of the Schrödinger theory. The model in the invariant global dimension was discussed in detail in the .
Let us consider the situation in which the series coefficients of the effective gravitational potential are
We shall call such a case the residual approximation. In this approximation the effective gravitational potential (22) takes the form
which is formally the Newton–Coulomb potential, i.e. has the behavior like 1/ h where h is interpreted as a kind of radial quantity. The value of the coefficient a−1 is unknown, but it is assumed that this coefficient exists. It can be verified by straightforward easy calculation that in such a situation the combined series coefficients bn are
and, consequently, the Ricci scalar curvature of the three-dimensional space becomes
Where as the equation (21), defining the global one-dimensional model of quantum gravity. Takes the form
The Ricci scalar curvature (88) defines certain states of the geometry of the three-dimensional embedded space. However, even when one considers the case of vacuum, i.e. when both the energy density of Matter fields and cosmological constant are identically vanishing e = 0, Λ = 0 , it is rather difficult to establish a three dimensional metric tensor for which the Ricci scalar curvature behaves like
Interestingly, in the most general situation the residue of the threedimensional Ricci scalar curvature calculated at a fixed point h0 is
i.e. it can be taken ad hoc identical to zero if and only if the residue of the energy density of Matter fields is
If one takes ad hoc the following relation
Then the Ricci scalar curvature of an induced three-dimensional geometry of embedded space takes the form
and its residue at a fixed point h0
Identically vanishes if and only if the residue of the energy density of Matter fields at a fixed point h0 has the value
In such a situation, also the geometry of an embedded threedimensional space manifold is automatically Ricci-flat if and only if the energy density of Matter fields takes the following form
It is easy to see that another possible Ricci-flat three-dimensional manifold is obtained for the identically vanishing cosmological constant Λ = 0 and the following value of the energy density of Matter fields
Newton–Coulomb quantum gravity
In the most general situation three-dimensional spaces having induced metrics characterized by the Ricci scalar curvature of the form (84) are not yet known explicitly in literature. However, it is evident that in the particular situation h0 = 0 , which is fully consistent with the general condition (69), the state of affairs is determined by much more simplified equations
Let us consider this particular case as the basic situation. We shall call the global one-dimensional quantum gravity described by the system of equations (93)-(94) the Newton–Coulomb quantum gravity.
As an example, we shall consider first the case of constant Ricci curvature. It is not difficult to see that in the most general view such a situationcorresponds to the identically vanishing energy density of Matter fields
We shall call such a case the Newton–Coulomb stationary quantum gravity. In such particular situation, the Ricci scalar curvature of threedimensional embedded space becomes
and, consequently, the Ricci curvature tensor, which characterizes the intrinsic geometry of the manifolds, describes the three-dimensional Einstein manifolds 
where λ the sign of the Einstein manifolds, in these specific conditions, is completely defined by parameters of the Newton–Coulomb stationary quantum gravity–the cosmological constant and the residue of the effective gravitational potential–as follows
Interestingly, the crucial consequence of identical vanishing of the energy density of Matter fields is the property of maximal symmetry of the Einstein manifolds described by the sign (98). For this reason, the Newton–Coulomb stationary quantum gravity possesses highly nontrivial geometrical interpretation: such a situation describes embedded three-dimensional manifolds which are maximally symmetric Einstein manifolds.
Consequently, one can deduce straightforwardly the classification of the three-dimensional embedded spaces, which are maximally
symmetric three-dimensional Einstein manifolds (97), with respect to the value of the sign λ (98) of a manifold in dependence on the cosmological constant and the residue a−1:
The Newton–Coulomb stationary quantum gravity, defined by the effective gravitational potential determines the threedimensional embedded spaces which are the maximally symmetric three-dimensional Einstein manifolds, characterized by the sign of the form (98). There is the classification of such manifolds with respect to the cosmological constant Λ and the value a−1 of the residue of the effective gravitational potential
1. If the sign of manifold is non-zero λ ≠ 0 and the residue of the effective gravitational potential is a negative real , then the effective gravitational potential corresponds to the Newtonian attractive potential energy
a) If the cosmological constant is a positive real Λ = + | Λ | then the maximally symmetric Einstein three-manifolds are characterized by the positive Ricci scalar curvature
b) If the cosmological constant is a negative real Λ = − | Λ | then the maximally symmetric Einstein three-manifolds are characterized by the Ricci scalar curvature
2. If the sign of manifold is non-zero λ ≠ 0 and the residue of the effective gravitational potential is a positive real , then the effective potential becomes the Coulomb repulsive potential energy
a) If the cosmological constant is a negative real Λ = − | Λ | then the maximally symmetric Einstein three-manifolds are characterized by negative Ricci scalar curvature
b) If the cosmological constant is a positive real Λ = + | Λ | then the maximally symmetric Einstein three-manifolds are characterized by the Ricci scalar curvature
3. If the sign of manifold is identically vanishing¸ λ = 0 , i.e. the maximally symmetric Einstein three-manifolds are Ricci-flat manifolds, then one can determine uniquely the value of the residue of the effective gravitational potential as follows
In such a situation, one obtains the values of the cosmological constant
where m is the mass of a body generating Newtonian gravitational field in vacuum and is its gravitational radius, q is the charge generating Columbic electrical field in vacuum and is its electrical radius.
Note that, in fact, by taking into account ad hoc the relation for the series coefficients (38), the residue a−1 of the effective gravitational potential is the Cauchy integral of Veff at a fixed point h0 = 0
and its value can be straightforwardly established as
where subscript “0” means the value of a quantity calculated in h = 0 .
Let us note that, when one shall to associate the residual effective potential with any realistic quantized Kepler problem, i.e. with employing the Newtonian or the Columbic potentials, one should to identify the global dimension with a spatial distance
In this case, with the formal identification of the functional derivative of h and the classical derivative of r, i.e. in fact the equality between the functional and classical integral measures δ h = dr well– known in classical mechanics , the wave functional Ψ[h] becomes the radial wave function Ψ[r], and the evolution (83) becomes familiar radial type Schrödinger equation
Where the number can be taken straightforwardly from the Newton law of gravitation or from the Coulomb law of electricity. The received wave equation (112) possibly describes an atomic system.
Note that there are many possible choices of the metrics hij with the same value of the determinant h = r . For instance one can take the simple variant
However, more generally, one can parameterize the relation (111) by SO(3) group rotation matrix which allows use the Eulerian angles (θ ,?,φ ) as follows
Where matrices are rotation matrices around the selected p-axis
This point of view was discussed in much advanced detail in the .
In this section we shall consider certain solutions of the global one-dimensional model of quantum gravity (21) for the case of the residual approximation of the effective gravitational potential Veff implemented in the previous section. In the most general situation, the considered quantum mechanical evolution
is solved by two types of wave functions where the attractive wave functions are associated with the Newton-like effective gravitational potential, and the repulsive ones are associated with the Coulomb-like effective gravitational potential. Because of the manifest one-dimensionality of the functional evolutionary equation (116), one can solve this equation in the framework of the theory of ordinary differential equations by interpretation of the functional derivative as the ordinary one, i.e. and the wave functional as a wave function Ψ[h] = Ψ(h) with no loss of generality.
In this manner, the problem to solve is given by the second order ordinary differential equation
This is well-known in the mathematical physics literature. The general solution of such a differential equation can be constructed straightforwardly by making use of the Bessel functions Jn and Yn for the case of the attractive potential
And in terms of the modified Bessel functions In and Kn for the case of repulsive potential
Where and are constants of integration, one takes standard definitions  of the Bessel functions of first and second kind,
And the modified Bessel functions of first and second kind,
Standardly, values of the second kind Bessel functions and modified ones for any integer’s n can be received by application of the limiting procedure
In further parts of this section we shall to present solutions of the quantum mechanics (116) with respecting of few selected boundary conditions for the general solutions (118) and (119).
Boundary conditions I
Let us consider the global one-dimensional quantum mechanics (21) with the boundary conditions for some selected initial value of the dimension
With using of the regularized hyper geometric functions
One can write out the general solutions (118) and (119) with respect to the boundary conditions (124)
Where the sign - in the hyper geometric function notation means that all , and the constants
for Newtonian case, and
For Columbic case.
Boundary conditions II
The second case which we want to present in this paper, are the boundary conditions for 1st and 2nd functional derivatives
By using of the hypergeometric functions, one can express the solution for attractive case as follows
Where and are constants defined as
Similarly for the repulsive case one obtains easily
Boundary conditions III
The last possible case of boundary conditions for the considered problem is
These conditions are formally improper for the problem; because of the lead to singular solutions. In this case, however, one can present the solutions in the form with formally singular constants. For the attractive potential one has
With constants (∈→0)
And similarly for the repulsive potential one obtains
With constants (∈→0)
However, when the following relation for the boundary conditions holds
Where are some (now unknown and arbitrary) nonsingular functional of hI and |α | , the sign + is related to the Newtonian case, and the sign - to the Columbic one, then the singularity of the solutions (142) and (145) is canceling. In this case the initial value ΨI for the attractive case is
And similarly for the repulsive one
Where are nonsingular constants of integration. The functional can be established by application of the condition (148) within the general solutions (142) and (145). It yields the results
Employing straightforwardly these results into the equalities (149) and (150) one obtains the integral equations for the functional f ± . For the columbic situation one receives the following equation
And similarly for the Newtonian case one derives the relation
In both the cases the integral operators acting on the functional f ± are nonsingular. By this reason one can put straightforwardly the formal limit (∈→0) in the integral equations (153) and (154), and by doing few elementary algebraic manipulations one can extract the searched functionals. The final results are as follows
In this manner the initial data conditions for the considered boundary conditions (141) cannot be chosen arbitrary, but according to the rules
The supposed equation for boundary values (148) is not unique, and can be replaced by other conditions. The discussed case, however, reflects the typical questions arising within the problem.
In this paper we have discussed the quantum mechanical model of quantum gravity arising from the global one–dimensional conjecture within quantum general relativity considered recently by the author . This model straightforwardly bases on the effective potential (22) being a simple algebraic sum of three fundamental energetic constituents - ”geometric”, ”cosmological”, and ”material”, with nontrivial change in potential behavior with respect to the initial model that was the Wheeler–DeWitt quantum Geometrodynamics (16). The relation between the models is established by change of both the differential operator and the potential.
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. Studying of this special case allowed concluding that in the case of Matter fields’s energy absence, in the global one–dimensional model of quantum gravity, the maximally symmetric three-dimensional Einstein manifolds are the characteristic embedding for the residual effective potential. Finally, we have found some solutions of the model of quantum gravity in the residual approximation.
We hope to discuss further conclusions of the global-one dimensional model of quantum gravity in the next contributions.
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