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Reduced Quantum General Relativity in Higher Dimensions

The higher dimensional Quantum General Relativity of a Riemannian manifold being an embedded space in a space-time being a Lorentzian manifold is investigated. The model of quantum geometrodynamics, based on the Wheeler-DeWitt equation reduced to a first order functional quantum evolution supplemented through an additional eigenequation for the scalar curvature, is formulated. Furthermore, making use of the objective quantum gravity and global one-dimensional conjecture, the general wave function beyond the Feynman path integral technique is derived. The resulting quantum gravity model creates the opportunity of potentially new theoretical and phenomenological applications for astrophysics, cosmology, and physics.


Introduction
Quantum geometrodynamics is the pioneering formulation of the quantum theory of gravitational field, for which the classical theory is the General Theory of Relativity, Cf. e.g. the Refs. [1]- [14], having straightforward applications in astrophysics, cosmology, and physics, Cf. e.g. the Refs. [15]- [31]. Its basic standpoint is the canonical primary quantization in the Dirac Hamiltonian formalism, Cf. e.g. the Refs. [32]- [33], applied to the Einstein-Hilbert action with possible inclusion of the York-Gibbons-Hawking boundary action in accordance with the Arnowitt-Deser-Misner decomposition of a space-time metric being a solution to the Einstein field equations. This strategy leads to the model of quantum gravity given through the Wheeler-DeWitt equation, the specific variant of the Schrödinger wave equation of quantum mechanics, for whose the dynamical object is geometry of a space embedded in a space-time and for whose the configuration space is the Wheeler superspace equipped with the DeWitt supermetric. In the present-day state of affairs, quantum geometrodynamics, as the core model for either critique or development, lays the foundations for a number models of quantum cosmology and quantum gravity and their applications, the best possible example are the professional books which were either published or republished in the recent fifteen years, Cf. e.g. the Refs. [34]- [80]. In this brief article, we shall consider the quantum geometrodynamics in the way which results in reduction of the Wheeler-DeWitt equation to the system of two equations: a first order quantum evolution and a supplementary eigenequation. Furthermore, we apply the objective quantum gravity and the global one-dimensional conjecture in order to construct the general solutions beyond the Feynman path integral technique.

Wheeler-DeWitt Equation
Let us consider a 1+D-dimensional space-time being a Lorentzian manifold characterized through (1+D)×(1+D) metric tensor g µν of signature (1, D), the Ricci curvature tensor (1+D) R µν and the Ricci scalar curvature (1+D) R = g µν (1+D) R µν , where a Greek index runs from 0 to D. Let a D-dimensional embedded space being a Riemannian manifold is characterized through D × D metric tensor h i j , the Ricci curvature tensor (D) R i j and the Ricci scalar curvature (D) R = h i j (1+D) R i j , where a Latin index runs from 1 to D. The foundational differential geometry formulas for 4-dimensional space-time curvatures remain unchanged in higher dimensions until application of the identity g µν g µν = D + 1. The Einstein tensor

Vacuum Gravitational Fields
The main object of quantum geometrodynamics is a wave function Ψ[h i j ] which is considered as a physical state. For vacuum gravitational fields, the Wheeler-DeWitt equation has the following form on the Wheeler superspace, and ℓ P = hG c 3 is the Planck length. We shall consider a special situation with the following definition of the first variation of a wave function where the constant multiplier has been chosen for convenience and ε i j are tensor coefficients, or equivalently is the canonically conjugate momentum operator in the Wheeler-Schrödinger representation. Calculating the second variation, one obtains or, equivalently, Whenever the relation (5) is rewritten in the form of a wave equation then it coincides with the Wheeler-DeWitt equation if which, through contractions with space metric, gives the results where the brackets refer to symmetrization. Calculating the first variation and making slightly tedious calculations with help of the Jacobi formula δ h = hh i j δ h i j , one obtains and, therefore, the aforementioned orthogonality condition can be expressed as h i j π i j Ψ[h i j ] = 0.

Non-Vacuum Gravitational Fields
Non-vacuum gravitational fields are described the Wheeler-DeWitt equation in the form where the symbol φ denotes dependence on matter fields, and ρ = T µν n µ n ν represents matter fields energy density related to a vector field n µ normal to an embedded space. Inclusion of the strategy of the previous subsection gives and, through compatibility with the case of vacuum gravitational fields, consistency holds for or, equivalently, and this is the reduced wave equation of quantum geometrodynamics. In this state of affairs, one can derive immediately where we have employed the Jacobi formula, and, for this reason, one can construct the equation which together with the Eq. (17) forms the system of equations.

Global One-Dimensional Objective Quantum Gravity
Let us apply the strategy proposed in the Ref. [54].
In this state of affairs, the reduced wave equation derived in the previous section transforms into the objective quantum gravity which also applies to vacuum gravitational fields as δ δ h |Ψ[h] = 0. Since the received model of objective quantum gravity formally is a first-order ordinary differential equation, its solution can be straightforwardly constructed avoiding the Feynman path integral method, which for the Wheeler superspace formalism is a slightly troublesome technique, in favour of a standard Cauchy problem for an initial wave function |Ψ[h I ] , where h I is an initial value of h. The general solution is an objective global one-dimensional wave function where N stands for a normalization factor. Remarkably, an initial state |Ψ[h I ] is a solution to the vacuum theory, and in itself is undetermined. Through the global one-dimensional nature of the model, normalization can be adopted in the standard form of the Born condition Furthermore, one can determine and calculating explicitly one obtains (26) For the aforementioned objective quantum gravity one can derive two simplest situations. First of all, we can consider classical gravitational fields in presence of a cosmological constant and a constant energy density of matter fields, for which and two particular situations, lambda-vacuum gravitational fields and vanishing cosmological constant, can be immediately derived from this case. More general case is a flat embedded space, for which one has the objective simplex-projected state and the normalization factor in the general forms, and

Discussion
We have shown that inclusion of the reduced quantum geometrodynamics creates the model of quantum gravity based on the Wheeler-DeWitt equation.
Looking through the prism of the Wheeler superspace, the aforementioned orthogonality condition related to a vacuum gravitational field has the physical interpretation as the analogue of the Lorentz/Lorenz gauge from the Maxwell electrodynamics, whereas for a general case of a non-vacuum gravitational field one obtains a considerably simplified model of quantum geometrodynamics. For this reason, the applied method has the most natural physical interpretation as the choice of a superspatial gauge. Equivalently, this technique can be regarded as the choice of the specific coordinate system in the configuration space of the General Theory of Relativity. Making use of the objective quantum gravity and the global one-dimensional conjecture for a geometrodynamical wave function, we have generated the global one-dimensional objective quantum gravity of an embedded Riemannian manifold. Remarkably, in the reduced quantum geometrodynamics all physical paths are compatible with the Wheeler-DeWitt equation, and, for this reason, the emerging model can be approached through the Feynman path integral formulation of quantum mechanics according to the idea of the Hartle-Hawking wave function. Prior to inclusion of the objective quantum gravity and the global one-dimensional conjecture, still the Feynman path integral type solutions to the Wheeler-DeWitt equation can be constructed, but inclusion of these two ideas leads to the solutions which are beyond the Feynman path integral formulation and are received through a solution to the Cauchy problem. Naturally, the most intriguing consequences of the reduced quantum general relativity, including the global one-dimensional objective quantum gravity, would be the particular phenomenological applications of the resulting model of quantum gravity. In our opinion, the scope of possible quantum gravitational effects could effect the phenomena of astrophysics, cosmology, and physics. Furthermore, pure theoretical applications of the proposed method in the context of loop quantization and string theory in themselves would be the great challenges for further development of the model.