arXiv:1605.02546v1 [physics.gen-ph] 26 Apr 2016
Reduced Quantum General Relativity in Higher Dimensions Lukasz Andrzej Glinka∗ and Patrick Linker† May 10, 2016
Abstract 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.
1
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 Schr¨ odinger 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 ∗ Independent Science Editor & Author, Non-fiction & Philosophy Writer, Poland; Member, American Association of International Researchers, USA; E-mail:
[email protected] † M.Sc. Student, Fachgebiet Kontinuumsmechanik, Technische Universit¨ at Darmstadt, 64287 Darmstadt, Germany;E-mail:
[email protected]
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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.
2
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)Rµν and the Ricci scalar curvature (1+D)R = g µν (1+D)Rµν , 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 hi j , the Ricci curvature tensor (D)Ri j and the Ricci scalar curvature (D)R = hi j (1+D)Ri 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 gµν = D + 1. The Einstein tensor 1 (1+D)G (1+D)R (1+D)R, in presence of a cosmological constant Λ D µν = µν − g µν 2 and matter fields described through a stress-energy tensor Tµν , gives the field 8π G equations (1+D)Gµν + ΛD gµν = κD Tµν where κD = κ ℓD−3 and κ = 4 is the P c Einstein constant. By virtue of the intrinsic geometry, the field equations for a Lorentzian manifold transform into classical geometrodynamics of a Riemannian manifold obeying the Gauss-Codazzi-Ricci equations.
2.1
Vacuum Gravitational Fields
The main object of quantum geometrodynamics is a wave function Ψ[hi j ] which is considered as a physical state. For vacuum gravitational fields, the WheelerDeWitt equation has the following form √ δ2 + ℓ2P h (D)R Ψ[hi j ] = 0, (1) Gi jkl δ hi j δ hkl 1 where h = det hi j , Gi jkl = √ hik h jl + hil h jk − hi j hkl is the DeWitt supermetric 2 h r h¯ G is the Planck length. We shall on the Wheeler superspace, and ℓP = c3 consider a special situation ℓ2P (D)R Ψ[hi j ] = δ 2 Ψ[hi j ] , (2) with the following definition of the first variation of a wave function δ δ Ψ[hi j ] = εi j Ψ[hi j ] , δ hi j 2
(3)
where the constant multiplier has been chosen for convenience and εi j are tensor coefficients, or equivalently (4) δ Ψ[hi j ] = 2iκD ℓP εi j π i j Ψ[hi j ] ,
δ δ 1 ℏc = −i is the canonically conjugate momenD 2κD ℓP δ hi j δ 16π ℓP hi j tum operator in the Wheeler-Schr¨odinger representation. Calculating the second variation, one obtains
where π i j = −i
δ δ 2 δ 2 Ψ[hi j ] = ℓ2P (D)R Ψ[hi j ] = δ εi j Ψ[hi j ] + εi j εkl Ψ[hi j ] , δ hi j δ hi j δ hkl
(5)
or, equivalently, δ 2 Ψ[hi j ] = ℓ2P (D)R Ψ[hi j ] = 2iκD ℓP εi j π i j Ψ[hi j ] − 4κD2 ℓ2P εi j εkl π i j π kl Ψ[hi j ] . (6) Whenever the relation (5) is rewritten in the form of a wave equation √ √ √ δ2 δ (7) + ℓ2P h (D)R Ψ[hi j ] = 0, − hδ εi j − hεi j εkl δ hi j δ hkl δ hi j then it coincides with the Wheeler-DeWitt equation if
δ εi j
δ Ψ[hi j ] = 0, δ hi j
(8)
what expresses the orthogonality condition δ εi j π i j Ψ[hi j ] = 0 and leads to the conclusion (D)R Ψ[hi j ] = 0. Moreover, one this compatibility also requires √ − hεi j εkl = Gi jkl what gives the equation
εi j εkl =
1 hi j hkl − hik h jl − hil h jk , 2h
which, through contractions with space metric, gives the results r D−2 εi j = hi j , 2Dh D−2 D−2 Gi jkl = − √ hi j hkl = − √ hi(k h jl) , 2D h 2 h
(9)
(10) (11)
where the brackets refer to symmetrization. Calculating the first variation
δ εi j = εkl
δ εi j , δ hkl
(12)
and making slightly tedious calculations with help of the Jacobi formula δ h = hhi j δ hi j , one obtains
δ εi j
=
δ Gi jkl
=
(D − 2)2 hi j , 4Dh p D(D − 4) 2D(D − 2) hδ εi j δ εkl , 2 (D − 2)3
−
(13) (14)
and, therefore, the aforementioned orthogonality condition can be expressed as hi j π i j Ψ[hi j ] = 0. 3
2.2
Non-Vacuum Gravitational Fields
Non-vacuum gravitational fields are described the Wheeler-DeWitt equation in the form √ δ2 Ψ[hi j , φ ] = 0, (15) + ℓ2P h (D)R − 2ΛD − 2κD ρ Gi jkl δ hi j δ hkl
where the symbol φ denotes dependence on matter fields, and ρ = Tµν nµ nν represents matter fields energy density related to a vector field nµ normal to an embedded space. Inclusion of the strategy of the previous subsection gives √ δ2 δ δ2 Ψ[hi j , φ ] = 0, Gi jkl + εi j εkl + h δ εi j − 2ℓ2P(ΛD + κD ρ ) δ hi j δ hkl δ hi j δ hi j δ hkl (16) and, through compatibility with the case of vacuum gravitational fields, consistency holds for hi j or, equivalently,
δ 8Dℓ2P h Ψ[hi j , φ ] = − (ΛD + κD ρ ) Ψ[hi j , φ ] , 2 δ hi j (D − 2)
ihi j π i j Ψ[hi j , φ ] = −
4DℓP h (ΛD + κD ρ ) Ψ[hi j , φ ] , 2 κD (D − 2)
(17)
(18)
and this is the reduced wave equation of quantum geometrodynamics. In this state of affairs, one can derive immediately δ 2 (19) Ψ[hi j , φ ] = δ hi j δ hkl 8D2 ℓ2P h2 κD δρ 8ℓ2P + Λ + (ΛD + κD ρ )2 Ψ[hi j , φ ] , h = − κ ρ − D D i j 2 2 (D − 2) Dh δ hi j (D − 2) hi j hkl
where we have employed the Jacobi formula, and, for this reason, one can construct the equation (D) R Ψ[hi j ] = (20) 2 D−2 κD δ ρ 8ℓP 4Dh (ΛD + κD ρ )2 − − 1 (ΛD + κD ρ ) + hi j Ψ[hi j , φ ] , 2 D−2 2Dh (D − 2) Dh δ hi j which together with the Eq. (17) forms the system of equations.
3
Global One-Dimensional Objective Quantum Gravity
Let us apply the strategy proposed in the Ref. [54]. Through some general justification on the nature of a geometrodynamical wave function which particularly involves the DeWitt supposition, one can consider the solutions to the reduced quantum geometrodynamics which are objective wave functions, i.e. are dependent on at most the matrix invariants cn , 1 6 n 6 D, of 4
a space metric tensor hi j . Then, a geometrodynamical wave function becomes D X δ cn δ Ψ[hi j ] = |Ψ(c1 , . . . , cD )i, and, moreover, one can write δ = . The δ hi j δ h i j δ cn n=1 next step is taking into account the global one-dimensional conjecture which δ δh δ stands that |Ψ(c1 , . . . , cD )i = |Ψ[h]i, and gives = . Involving once δ cn δ cn δ h D X δ cn δ h i j = D, this stratagain the Jacobi formula and the obvious identity δ h i j δ cn n=1 δ δ δ δ egy leads to the relation = Dhhi j , and, consequently, hi j = D2 h δ hi j δh δ hi j δh δ2 δ2 = D4 h2 2 . In this state of affairs, the reduced wave equation and hi j hkl δ hi j δ hkl δh derived in the previous section transforms into the objective quantum gravity δ 8ℓ2P Ψ[hi j , φ ] , Ψ[hi j , φ ] = − (Λ + κ ρ ) D D δh D(D − 2)2
(21)
δ |Ψ[h]i = 0. Since the which also applies to vacuum gravitational fields as δh 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[hI ]i, where hI is an initial value of h. The general solution is an objective global one-dimensional wave function ( !) Z h 8ℓ2P ′ ′ |Ψ[h]i = N exp − |Ψ[hI ]i , (22) ρ [h ]δ h ΛD (h − hI ) + κD D(D − 2)2 hI where N stands for a normalization factor. Remarkably, an initial state |Ψ[hI ]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 Z hF
Ψ[h′ ] Ψ[h′ ] δ h′ = 1 with hF being the standard form of the Born condition hI
a final value of h and hΨ[hI ]|Ψ[hI ]i = 1, what leads to the conclusion N=
(Z
hF
hI
"
16ℓ2P exp − D(D − 2)2
ΛD (h − hI ) + κD ′
Z
h′
hI
ρ [h ]δ h ′′
′′
!#
δh
′
)−1/2
. (23)
Furthermore, one can determine D(D − 2)2 δ D3 (D − 2) δ 2 δ 2 |Ψ[h]i = ℓ2P (D)R Ψ[hi j ] = − |Ψ[h]i+ h 2 |Ψ[h]i , (24) 4 δh 2 δh
and calculating explicitly # " 2 8ℓ2P 8ℓ2P δ2 δ ρ [h] 2 |Ψ[h]i , |Ψ[h]i = (ΛD + κD ρ ) − κD δ h2 D(D − 2)2 D(D − 2)2 δh 5
(25)
one obtains δ ρ [h] 32D 4D2 2 2 R |Ψ[h]i = 2 (ΛD + κD ρ ) + |Ψ[h]i . hℓ (ΛD + κD ρ ) − hκD (D − 2)3 P D−2 δh (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 8ℓ2P (Λ + ) (h − h ) |Ψ[hI ]i , (27) κ ρ |Ψ[h]i = N exp − D D 0 I D(D − 2)2 1/2 1 2 4 ℓP (ΛD + κD ρ0 ) D−2 D N = 1/2 , (28) 16ℓ2P (ΛD + κD ρ0 ) (hF − hI ) 1 − exp − D(D − 2)2 32D 2 (D) 2 R |Ψ[h]i = 2 (ΛD + κD ρ0 ) + hℓ (ΛD + κD ρ0 ) |Ψ[h]i , (29) (D − 2)3 P (D)
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 δ ρ [h] 32D 4D2 2 2 2 (ΛD + κD ρ ) + |Ψ[h]i = 0. (30) hℓ (Λ + h κ ρ ) − κ D D D (D − 2)3 P D−2 δh
4
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, 6
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.
Acknowledgements L.A. Glinka thanks to P. Linker for private communication which led to this joined work.
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