# 1D Global Bosonization of Quantum Gravity

L. A. Glinka111E-mail to<EMAIL_ADDRESS><EMAIL_ADDRESS>
_Bogoliubov Laboratory of Theoretical Physics_ ,
_Joint Institute for Nuclear Research_ ,
_Joliot–Curie 6, 141980 Dubna, Moscow Region, Russia_

###### Abstract

Reduction of the Wheeler–DeWitt equation to the Klein–Gordon–Fock evolution
for bosonic field by using of global bosonization to one-dimensional is
proposed. The second quantization of the theory is carried out, and the
Quantum Gravity is constructed in terms of the Fock–Bogoliubov–Heisenberg
initial data operator basis. It is shown that this leads to understanding of
mass of the bosonic field as a scaled initial data mass by one-point
correlations of two bosonic fields.

##  1 Introduction: Unsolved Quantum Gravity

The Einstein–Hilbert field equations of General Relativity [1, 2] 222We use
the system of units $c=\hbar=k_{B}=8\pi G/3=1$.

$R_{\mu\nu}-\dfrac{R[g]}{2}g_{\mu\nu}+\Lambda
g_{\mu\nu}=3T_{\mu\nu},\leavevmode\nobreak\ \leavevmode\nobreak\
R[g]=g^{\kappa\lambda}R_{\kappa\lambda}$ (1)

where $g_{\mu\nu}$ is a non-degenerate and symmetric
$\left(\\!\\!\\!\begin{array}[]{c}0\vspace*{-4pt}\\\
2\end{array}\\!\\!\\!\right)$-tensor field, $R_{\mu\nu},\Lambda,T_{\mu\nu}$
are the metric-contracted Riemann curvature tensor, cosmological constant, and
stress-energy tensor, and $R[g]$ is the Ricci scalar curvature of a pseudo-
Riemannian manifold $(M,g)$ [3, 4], arise due to the Palatini principle [5]

$\dfrac{\delta S[g]}{\delta g_{\mu\nu}}=0,$ (2)

used to the Einstein–Hilbert action modified by a boundary term

$S[g]=-\dfrac{1}{3}\int_{\partial
M}d^{3}x\sqrt{h}K[h]+\int_{M}d^{4}x\sqrt{-g}\left\\{-\dfrac{1}{6}R[g]+\dfrac{\Lambda}{3}+\mathcal{L}\right\\},$
(3)

springs from allowing variations for which the normal derivatives on $\partial
M$ are non-zero, in order to cancel surface terms. Here $K[h]$ is the
extrinsic curvature of an induced three-dimensional spacelike boundary
$(\partial M,h)$, and $\mathcal{L}$ is the Matter fields Lagrangian provoking
the stress-energy tensor $T_{\mu\nu}$

$T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathcal{L}\right)}{\delta
g^{\mu\nu}}.$ (4)

Stationarity of the Matter fields results in existence of a global timelike
Killing vector field for a metric field $g_{\mu\nu}$. A coordinate system can
be chose such that the Killing vector field equals $\dfrac{\partial}{\partial
t}$ and the foliation $t=constant$ is spacelike. Then a metric field depends
at most on a spatial coordinates $x^{i}$, so the $t$ can be treated globally
[6], and $3+1$ decomposition of a metric

$\displaystyle g_{\mu\nu}=\left[\begin{array}[]{cc}-N^{2}+N_{i}N^{i}&N_{j}\\\
N_{i}&h_{ij}\end{array}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
g^{\mu\nu}=\left[\begin{array}[]{cc}-\dfrac{1}{N^{2}}&\dfrac{N^{j}}{N^{2}}\vspace*{5pt}\\\
\dfrac{N^{i}}{N^{2}}&h^{ij}-\dfrac{N^{i}N^{j}}{N^{2}}\end{array}\right],$ (9)
$\displaystyle h_{ik}h^{kj}=\delta_{i}^{j},\leavevmode\nobreak\
\leavevmode\nobreak\ N^{i}=h^{ij}N_{j},\leavevmode\nobreak\
\leavevmode\nobreak\ g=N^{2}h,$ (10)

has also a global sense. In this case the action (3) becomes

$\displaystyle S[g]\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\int dt\leavevmode\nobreak\
L(\pi,\pi^{i},\pi^{ij},N,N_{i},h_{ij}),$ (11) $\displaystyle
L(\pi,\pi^{i},\pi^{ij},N,N_{i},h_{ij})\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\int_{\partial
M}d^{3}x\left\\{\pi\dot{N}+\pi^{i}\dot{N_{i}}+\pi^{ij}\dot{h}_{ij}-NH-
N_{i}H^{i}\right\\},$ (12)

where

$\displaystyle\dot{h}_{ij}$ $\displaystyle=$ $\displaystyle\dfrac{\partial
h_{ij}}{\partial t}=N_{i|j}+N_{j|i}-2NK_{ij},$ (13) $\displaystyle H$
$\displaystyle=$
$\displaystyle\sqrt{h}\left\\{K^{2}-K_{ij}K^{ij}+R[h]-2\Lambda-6T_{nn}\right\\},$
(14) $\displaystyle H^{i}$ $\displaystyle=$
$\displaystyle-2\pi^{ij}_{\leavevmode\nobreak\
;j}=-2\pi^{ij}_{\leavevmode\nobreak\
,j}-h^{il}\left(2h_{jl,k}-h_{jk,l}\right)\pi^{jk},$ (15)

where the second formula follows from the Gauss-Codazzi equations [7]. Here
$K_{ij}$ is the extrinsic-curvature tensor ($K=K^{i}_{i}$), and $\pi^{ij}$ is
the canonical conjugate momentum field to the field $h_{ij}$

$\pi^{ij}=\dfrac{\delta
L}{\delta\dot{h}_{ij}}=-\sqrt{h}\left(K^{ij}-h^{ij}K\right).$ (16)

Time-preservation requirement [8] of the primary constraints [9] for (11)

$\pi=\dfrac{\delta L}{\delta\dot{N}}\approx 0,\leavevmode\nobreak\
\leavevmode\nobreak\ \pi^{i}=\dfrac{\delta L}{\delta\dot{N_{i}}}\approx 0,$
(17)

leads to the secondary constraints

$H\approx 0,\leavevmode\nobreak\ \leavevmode\nobreak\ H^{i}\approx 0,$ (18)

called the Hamiltonian constraint and the diffeomorphism constraint,
respectively. The diffeomorphism constraint merely reflects spatial
diffeoinvariance, and the Hamiltonian constraint gives the dynamics. By (16)
the Hamiltonian constraint becomes the Einstein–Hamilton–Jacobi equation [10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]

$G_{ijkl}\pi^{ij}\pi^{kl}+\sqrt{h}\left(R[h]-2\Lambda-6T_{nn}\right)=0,$ (19)

where $G_{ijkl}$ is called the Wheeler superspace metric

$G_{ijkl}=\dfrac{1}{2\sqrt{h}}\left(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl}\right).$
(20)

Canonical quantization [45] of (19) by the commutation relations [46]

$\displaystyle i\left[\pi^{ij}(x),h_{kl}(y)\right]$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}+\delta_{l}^{i}\delta_{k}^{j}\right)\delta^{(3)}(x,y),$
(21) $\displaystyle i\left[\pi^{i}(x),N_{j}(y)\right]$ $\displaystyle=$
$\displaystyle\delta^{i}_{j}\delta^{(3)}(x,y),$ (22) $\displaystyle
i\left[\pi(x),N(y)\right]$ $\displaystyle=$ $\displaystyle\delta^{(3)}(x,y),$
(23)

leads to the Wheeler–DeWitt equation [47, 9]

$\left\\{-G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}+h^{1/2}\left(R[h]-2\Lambda-6T_{nn}\right)\right\\}\Psi[h]=0,$ (24)

and the other first class constraints are conditions on the wave function
$\Psi[h]$

$\pi\Psi[h]=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\pi^{i}\Psi[h]=0,\leavevmode\nobreak\ \leavevmode\nobreak\ H^{i}\Psi[h]=0.$
(25)

Furthermore, the canonical commutation relations hold

$\left[\pi(x),\pi^{i}(y)\right]=\left[\pi(x),H^{i}(y)\right]=\left[\pi^{i}(x),H^{j}(y)\right]=\left[\pi^{i}(x),H(y)\right]=0,$
(26)

and in consequence $H_{i}$ are generators of diffeomorphisms
$\widetilde{x}^{i}=x^{i}+\delta x^{i}$ [9]

$\displaystyle\left[h_{ij},i\int_{\partial M}H_{a}\delta x^{a}d^{3}x\right]$
$\displaystyle=$ $\displaystyle-h_{ij,k}\delta x^{k}-h_{kj}\delta
x^{k}_{\leavevmode\nobreak\ ,i}-h_{ik}\delta x^{k}_{\leavevmode\nobreak\
,j}\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (27)
$\displaystyle\left[\pi_{ij},i\int_{\partial M}H_{a}\delta x^{a}d^{3}x\right]$
$\displaystyle=$ $\displaystyle-\left(\pi_{ij}\delta
x^{k}\right)_{,k}+\pi_{kj}\delta x^{i}_{\leavevmode\nobreak\
,k}+\pi_{ik}\delta x^{j}_{\leavevmode\nobreak\ ,k}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (28)

or in more conventional form

$\displaystyle i\left[H_{i}(x),H_{j}(y)\right]\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int_{\partial M}H_{a}c^{a}_{ij}d^{3}z,$ (29)
$\displaystyle i\left[H(x),H_{i}(y)\right]\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!H\delta^{(3)}_{,i}(x,y),$ (30) $\displaystyle
i\left[\int_{\partial M}H\delta x_{1}d^{3}x,\int_{\partial M}H\delta
x_{2}d^{3}x\right]\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\int_{\partial
M}H^{a}\left(\delta x_{1,a}\delta x_{2}-\delta x_{1}\delta
x_{2,a}\right)d^{3}x,$ (31)

where $H_{i}=h_{ij}H^{j}$, and

$c^{a}_{ij}=\delta^{a}_{i}\delta^{b}_{j}\delta^{(3)}_{,b}(x,z)\delta^{(3)}(y,z)-\delta^{a}_{j}\delta^{b}_{i}\delta^{(3)}_{,b}(y,z)\delta^{(3)}(x,z)\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (32)

are structure constants of diffeomorphism group. Commutators (29-31) show the
first-class constrained system property.

The Wheeler–DeWitt equation (24) has been studied intensively since 30 years
[48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77]. In fact, this is an equation on
superspace [78], defined as a space of all equivalence class of metrics
related by the action of the diffeomorphism group of a compact, connected,
orientable, Hausdorff, $C^{\infty}$ 3-dimensional spacelike manifold without
boundary $\partial M$. If $Riem(\partial M)$ consists all $C^{\infty}$
Riemannian metrics on $\partial M$, and $Dif\\!f(\partial M)$ is a group of
all $C^{\infty}$ diffeomorphisms of $\partial M$ that preserve orientation,
then the Wheeler superspace $S(\partial M)$ is the space of all orbits of
$Dif\\!f(\partial M)$, _i.e._ $S(\partial M)=Riem(\partial M)/Dif\\!f(\partial
M)$. $S(\partial M)$ is a connected, second-countable, metrizeable space. All
geometries with the same kind of symmetry are manifold in $S(\partial M)$ \-
they have homeomorphic neighbouhoods. However, symmetric geometries
neighbourhoods are not homeomorphic to nonsymmetric geometries ones, and by
this $S(\partial M)$ is not manifold. Superspace can be decomposed by its
subspaces on a countable, partially-ordered,
$C^{\infty}$-Fr$\mathrm{\acute{e}}$chet manifold partition, that is an
inverted stratification indexed by the symmetry type - geometries with a given
symmetry are completely contained within the boundary of less symmetric
geometries. The minisuperspace models, _i.e._ Quantum Cosmology [79, 80, 81,
82, 83, 84, 85], study certain strata of superspace. Fischer [78] proved by
suitable choice of a subgroup of $Dif\\!f(\partial M)$ and by action of this
subgroup on $Riem(\partial M)$, for n-dimensional $\partial M$ the superspace
$S(\partial M)$ can be extended to a manifold $S_{e}(\partial M)$ such that
$\dim S_{e}(\partial M)/S(\partial M)=n(n+1)$.

##  2 Global Bosonization to One Dimension

The superspace has no physical consequences [86] and is the main structural
problem of the theory. In this section we will construct linearization of the
Quantum Gravity, global bosonization to one dimension.

###  2.1 Reduction Problem

Let us consider the standard relation of General Relativity [87] between
variations of a metric field determinant and a metric field

$\delta g=gg^{\mu\nu}\delta g_{\mu\nu}=g\left(g^{00}\delta g_{00}+g^{ij}\delta
g_{ij}+g^{0j}\delta g_{0j}+g^{i0}\delta g_{i0}\right).$ (33)

The $3+1$ parametrization (9) allows determine the partial variations

$\displaystyle\delta g_{00}$ $\displaystyle=$ $\displaystyle-\delta
N^{2}+N^{i}N^{j}\delta h_{ij}+h_{ij}N^{i}\delta N^{j}+h_{ij}N^{j}\delta
N^{i},$ (34) $\displaystyle\delta g_{ij}$ $\displaystyle=$
$\displaystyle\delta h_{ij},$ (35) $\displaystyle\delta g_{0j}$
$\displaystyle=$ $\displaystyle h_{ij}\delta N^{i}+N^{i}\delta h_{ij},$ (36)
$\displaystyle\delta g_{i0}$ $\displaystyle=$ $\displaystyle h_{ij}\delta
N^{j}+N^{j}\delta h_{ij},$ (37)

as well as the total variation

$\displaystyle\delta g=N^{2}\delta h+h\delta N^{2}.$ (38)

Taking a contravariant metric field components of (9) we obtain from (33)

$\displaystyle N^{2}\delta h=N^{2}hh^{ij}\delta h_{ij},$ (39)

so that the global relation between first functional derivatives is
established

$\dfrac{\delta}{\delta h_{ij}}=hh^{ij}\dfrac{\delta}{\delta h}.$ (40)

The global reduction (40) has deep sense - the first functional derivative
operator $\dfrac{\delta}{\delta h_{ij}}$ is an object from a vector space
spanned by the contravariant 3-space metric $h^{ij}$. Therefore, as the
consequence of (40) one can determine the Wheeler–DeWitt second derivative
functional operator (24)

$\displaystyle-G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}=\dfrac{3}{2}h^{3/2}\dfrac{\delta^{2}}{\delta h^{2}},$ (41)

where was used the obvious identity

$\displaystyle\left(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl}\right)h^{ij}h^{kl}=\delta_{i}^{l}\delta^{i}_{l}+\delta^{j}_{l}\delta^{l}_{j}-\delta^{i}_{i}\delta^{k}_{k}=-3.$
(42)

Hence the Wheeler–DeWitt equation (24) becomes the one-dimensional
Klein–Gordon–Fock type evolution

$\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+m^{2}\right)\Psi[h]=0,$ (43)

where

$m^{2}\equiv m^{2}[h]=\dfrac{2}{3h}\left(R[h]-2\Lambda-6T_{nn}\right),$ (44)

is the square of mass of the bosonic field $\Psi[h]$. By using of the notation

$\Phi=\left[\begin{array}[]{c}\Psi\\\
\Pi_{\Psi}\end{array}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
\vec{\partial}=\left[\begin{array}[]{c}\dfrac{\delta}{\delta h}\\\
0\end{array}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{M}=\left[\begin{array}[]{cc}0&1\\\ -m^{2}&0\end{array}\right]\geq 0,$
(45)

the second order scalar equation (43) becomes the first order vector equation

$\left(i\mathbf{\Gamma}\vec{\partial}-\mathbb{M}\right)\Phi[h]=0,$ (46)

where $\Gamma$ matrices obey the relations

$\mathbf{\Gamma}=\left[-i\mathbf{1},\mathbf{0}\right],\leavevmode\nobreak\
\leavevmode\nobreak\
\left\\{\mathbf{\Gamma}^{a},\mathbf{\Gamma}^{b}\right\\}=2\eta^{ab}\mathbf{1},\leavevmode\nobreak\
\leavevmode\nobreak\ \eta^{ab}=\left[\begin{array}[]{cc}-1&0\\\
0&0\end{array}\right],$ (47)

where $\mathbf{1}$ and $\mathbf{0}$ are unit and null two-dimensional
matrices.

We have seen that application of the global reduction (40) to the
Wheeler–DeWitt equation (24), that has also a global nature by a character of
the decomposition (9), results in the bosonic quantum mechanics (43). This
scalar-type second order functional evolution was reduced directly to the
vector-type first order functional equation (46) with some two-component field
$\Phi[h]$ as a solution. In the equation (43) as well as in its the reduced
form (46) the superspace metric is completely absent. By this reason the most
mysterious element of the Wheeler Quantum Gravity’s logics was formally
excluded from considerations – the notion of superspace as well as its
mathematical properties are not need to further analysis. In further
developments of this paper we will concentrate on canonical quantization in
the bosonic Fock space of the reduced equation (46).

###  2.2 Fock–Bogoliubov–Heisenberg initial data basis

Next step of the bosonization is the field quantization of the equation (46)

$\Phi[h]\rightarrow\mathbf{\Phi}[h]\Rightarrow\left(i\mathbf{\Gamma}\vec{\partial}-\mathbb{M}\right)\mathbf{\Phi}[h]=0,$
(48)

according to canonical commutation relations proper for the Bose statistics
[88, 89, 90]

$\displaystyle\left[\mathbf{\Pi}_{\Psi}[h^{\prime}],\mathbf{\Psi}[h]\right]$
$\displaystyle=$ $\displaystyle-i\delta(h^{\prime}-h),$ (49)
$\displaystyle\left[\mathbf{\Pi}_{\Psi}[h^{\prime}],\mathbf{\Pi}_{\Psi}[h]\right]$
$\displaystyle=$ $\displaystyle 0,$ (50)
$\displaystyle\left[\mathbf{\Psi}[h^{\prime}],\mathbf{\Psi}[h]\right]$
$\displaystyle=$ $\displaystyle 0.$ (51)

By using of the second quantization method [91, 92, 93], from the equation
(43) spring that the field operator $\mathbf{\Phi}[h]$ of the reduced equation
(46) can be represent in the Fock space of annihilation and creation
functional operators

$\mathbf{\Phi}[h]=\mathbb{Q}[h]\mathfrak{B}[h],$ (52)

where $\mathfrak{B}[h]$ is a dynamical basis in the Fock space

$\mathfrak{B}[h]=\left\\{\left[\begin{array}[]{c}\mathsf{G}[h]\\\
\mathsf{G}^{\dagger}[h]\end{array}\right]:\left[\mathsf{G}[h^{\prime}],\mathsf{G}^{\dagger}[h]\right]=\delta\left(h^{\prime}-h\right),\left[\mathsf{G}[h^{\prime}],\mathsf{G}[h]\right]=0\right\\},$
(53)

and $\mathbb{Q}[h]$ is the second quantization matrix

$\mathbb{Q}[h]=\left[\begin{array}[]{cc}\dfrac{1}{\sqrt{2|m[h]|}}&\dfrac{1}{\sqrt{2|m[h]|}}\\\
-i\sqrt{\dfrac{|m[h]|}{2}}&i\sqrt{\dfrac{|m[h]|}{2}}\end{array}\right].$ (54)

In this way the operator equation (48) becomes the equation for a basis
$\mathfrak{B}[h]$

$\dfrac{\delta\mathfrak{B}[h]}{\delta
h}=\left[\begin{array}[]{cc}-im[h]&\dfrac{1}{2m[h]}\dfrac{\delta m[h]}{\delta
h}\\\ \dfrac{1}{2m[h]}\dfrac{\delta m[h]}{\delta
h}&im[h]\end{array}\right]\mathfrak{B}[h].$ (55)

Actually, there is a nonlinearity given by coupling between annihilation and
creation operators present as nondiagonal terms in (55), so the equation (55)
can not be solved standardly. In order to solving, let us suppose that in the
Fock space exists a new basis $\mathfrak{B}^{\prime}[h]$

$\mathfrak{B}^{\prime}[h]=\left\\{\left[\begin{array}[]{c}\mathsf{G}^{\prime}[h]\\\
\mathsf{G}^{\prime\dagger}[h]\end{array}\right]:\left[\mathsf{G}^{\prime}[h^{\prime}],\mathsf{G}^{\prime\dagger}[h]\right]=\delta\left(h^{\prime}-h\right),\left[\mathsf{G}^{\prime}[h^{\prime}],\mathsf{G}^{\prime}[h]\right]=0\right\\},$
(56)

for which the the Bogoliubov transformation

$\mathfrak{B}^{\prime}[h]=\left[\begin{array}[]{cc}u[h]&v[h]\\\
v^{\ast}[h]&u^{\ast}[h]\end{array}\right]\mathfrak{B}[h],\leavevmode\nobreak\
\leavevmode\nobreak\ |u[h]|^{2}-|v[h]|^{2}=1,$ (57)

and the Heisenberg evolution

$\dfrac{\delta\mathfrak{B}^{\prime}[h]}{\delta
h}=\left[\begin{array}[]{cc}-i\lambda[h]&0\\\
0&i\lambda[h]\end{array}\right]\mathfrak{B}^{\prime}[h],$ (58)

are supposed to hold together. We will call briefly this special basis as the
Fock–Bogoliubov–Heisenberg (FBH) operator basis. The diagonalization procedure
(56)-(58) converts the operator basis evolution (55) onto the Bogoliubov
coefficients one

$\dfrac{\delta}{\delta h}\left[\begin{array}[]{c}u[h]\\\
v[h]\end{array}\right]=\left[\begin{array}[]{cc}-im[h]&\dfrac{1}{2m[h]}\dfrac{\delta
m[h]}{\delta h}\\\ \dfrac{1}{2m[h]}\dfrac{\delta m[h]}{\delta
h}&im[h]\end{array}\right]\left[\begin{array}[]{c}u[h]\\\
v[h]\end{array}\right],$ (59)

and the basis $\mathfrak{B}^{\prime}[h]$ takes a meaning of static operator
basis associated with initial data

$\mathfrak{B}^{\prime}[h]\equiv\mathfrak{B}_{I}=\left\\{\left[\begin{array}[]{c}\mathsf{G}_{I}\\\
\mathsf{G}^{\dagger}_{I}\end{array}\right]:\left[\mathsf{G}_{I},\mathsf{G}^{\dagger}_{I}\right]=1,\left[\mathsf{G}_{I},\mathsf{G}_{I}\right]=0\right\\},$
(60)

within the vacuum state can be correctly defined

$|0\rangle_{I}=\left\\{|0\rangle_{I}:\mathsf{G}_{I}|0\rangle_{I}=0,\leavevmode\nobreak\
0={{}_{I}}\langle 0|\mathsf{G}_{I}^{\dagger}\right\\}.$ (61)

In the other words, the integrability problem consists in the equations (59).
However, the Bogoliubov coefficients are additionally constrained by the
hyperbolic rotation condition (56). By this it is useful to apply the
superfluid parametrization, for which the solutions are

$\displaystyle u[h]$ $\displaystyle=$
$\displaystyle\dfrac{1+\mu[h]}{2\sqrt{\mu[h]}}\exp\left\\{im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\mu[h^{\prime}]}\right\\},$ (62) $\displaystyle v[h]$
$\displaystyle=$
$\displaystyle\dfrac{1-\mu[h]}{2\sqrt{\mu[h]}}\exp\left\\{-im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\mu[h^{\prime}]}\right\\},$ (63)

where $\mu[h]$ is a mass scale

$\mu[h]=\dfrac{m_{I}}{m[h]}.$ (64)

This establish the relation between a dynamical basis $\mathfrak{B}[h]$ and
the initial data FBH basis $\mathfrak{B}_{I}$ as follows

$\mathfrak{B}[h]=\mathbb{G}[h]\mathfrak{B}_{I},$ (65)

where the transformation matrix $\mathbb{G}[h]$ is

$\displaystyle\mathbb{G}[h]=\left[\begin{array}[]{cc}\dfrac{\mu[h]+1}{2\sqrt{\mu[h]}}e^{-i\theta[h]}\vspace*{10pt}&\dfrac{\mu[h]-1}{2\sqrt{\mu[h]}}e^{i\theta[h]}\\\
\dfrac{\mu[h]-1}{2\sqrt{\mu[h]}}e^{-i\theta[h]}&\dfrac{\mu[h]+1}{2\sqrt{\mu[h]}}e^{i\theta[h]}\end{array}\right],$
(68)

where $i\theta[h]$ is given by a phase of (62). By this reason, the solution
of the equation (48) can be expressed in the initial data basis as

$\mathbf{\Phi}[h]=\mathbb{Q}[h]\mathbb{G}[h]\mathfrak{B}_{I}.$ (69)

###  2.3 One-point correlations

The second quantized equation (43), _i.e._

$\left(\mu^{2}[h]\dfrac{\delta^{2}}{\delta
h^{2}}+m^{2}_{I}\right)\mathbf{\Psi}[h]=0,$ (70)

has a solution

$\displaystyle\mathbf{\Psi}[h]=\frac{\mu[h]}{2\sqrt{2m_{I}}}\left(\exp\left\\{-im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\mu[h^{\prime}]}\right\\}\mathsf{G}_{I}+\exp\left\\{im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\mu[h^{\prime}]}\right\\}\mathsf{G}_{I}^{\dagger}\right),$ (71)

that is a direct conclusion of the relation (69). This field acts on the
initial data vacuum state as follows

$\displaystyle\mathbf{\Psi}[h]|0\rangle_{I}$ $\displaystyle=$
$\displaystyle\frac{\mu[h]}{2\sqrt{2m_{I}}}e^{i\theta[h]}\mathsf{G}^{\dagger}_{I}|0\rangle_{I},$
(72) $\displaystyle{{}_{I}}\langle 0|\mathbf{\Psi}^{\dagger}[h]$
$\displaystyle=$ $\displaystyle{{}_{I}}\langle
0|\mathsf{G}_{I}\frac{\mu[h]}{2\sqrt{2m_{I}}}e^{-i\theta[h]}.$ (73)

By this reason, one can consider the following many-field quantum states

$\displaystyle|h,n\rangle$ $\displaystyle\equiv$
$\displaystyle\left(\mathbf{\Psi}[h]\right)^{n}|0\rangle_{I}=\left(\frac{\mu[h]}{2\sqrt{2m_{I}}}e^{i\theta[h]}\right)^{n}\mathsf{G}^{\dagger
n}_{I}|0\rangle_{I},$ (74) $\displaystyle\langle n^{\prime},h^{\prime}|$
$\displaystyle\equiv$ $\displaystyle{{}_{I}}\langle
0|\left(\mathbf{\Psi}^{\dagger}[h^{\prime}]\right)^{n^{\prime}}={{}_{I}}\langle
0|\mathsf{G}_{I}^{n^{\prime}}\left(\frac{\mu[h^{\prime}]}{2\sqrt{2m_{I}}}e^{-i\theta[h^{\prime}]}\right)^{n^{\prime}},$
(75)

and determine the two-point quantum correlator of two many-field states

$\displaystyle\langle
n^{\prime},h^{\prime}|h,n\rangle=\dfrac{\mu^{n^{\prime}}[h^{\prime}]\mu^{n}[h]}{\left(8m_{I}\right)^{(n^{\prime}+n)/2}}e^{-im_{I}\theta_{n^{\prime},n}[h^{\prime},h]}\langle
0|\mathsf{G}_{I}^{n^{\prime}}\mathsf{G}^{\dagger n}_{I}|0\rangle_{I},$ (76)

where

$\theta_{n^{\prime},n}[h^{\prime},h]=n^{\prime}\int_{h_{I}}^{h^{\prime}}\dfrac{\delta
h^{\prime\prime}}{\mu[h^{\prime\prime}]}-n\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime\prime}}{\mu[h^{\prime\prime}]}.$ (77)

Application of the normalization

$\langle 1,h_{I}|h_{I},1\rangle=\dfrac{1}{8m_{I}}{{}_{I}}\langle
0|0\rangle_{I}\equiv 1\Longrightarrow{{}_{I}}\langle 0|0\rangle_{I}=8m_{I},$
(78)

allows define the following correlators

$\displaystyle\langle n^{\prime},h|h,n\rangle$ $\displaystyle=$
$\displaystyle\left(\dfrac{\langle 1,h|h,1\rangle}{{{}_{I}}\langle
0|0\rangle_{I}}\right)^{(n^{\prime}+n)/2}e^{-i(n^{\prime}-n)\theta[h]}{{}_{I}}\langle
0|\mathsf{G}_{I}^{n^{\prime}}\mathsf{G}^{\dagger n}_{I}|0\rangle_{I},$ (79)
$\displaystyle\dfrac{\langle n,h^{\prime}|h,n\rangle}{{{}_{I}}\langle
0|0\rangle_{I}}$ $\displaystyle=$ $\displaystyle\left(\dfrac{\langle
1,h^{\prime}|h,1\rangle}{{{}_{I}}\langle 0|0\rangle_{I}}\right)^{n},$ (80)

where

$\displaystyle\langle 1,h^{\prime}|h,1\rangle$ $\displaystyle=$
$\displaystyle\mu[h^{\prime}]\mu[h]\exp\left\\{im_{I}\int_{h^{\prime}}^{h}\dfrac{\delta
h^{\prime\prime}}{\mu[h^{\prime\prime}]}\right\\},$ (81) $\displaystyle\langle
1,h|h,1\rangle$ $\displaystyle=$ $\displaystyle\mu^{2}[h].$ (82)

The last formula (82) together with the definition (64) leads to the relation
between the mass of the bosonic field $\mathbf{\Psi}[h]$ and the initial data
mass $m_{I}$

$m[h]=\lambda[h]m_{I},\leavevmode\nobreak\ \leavevmode\nobreak\
\lambda[h]=\dfrac{1}{\sqrt{\langle 1,h|h,1\rangle}},$ (83)

that means the arbitrary mass $m[h]$ is only rescaled the initial data mass
$m_{I}$, and the scale $\lambda$ is directly related to one-point correlations
of the quantum bosonic field $\mathbf{\Psi}[h]$. Therefore, actually the mass
$m[h]$ for arbitrary $h$ is given by correlations of two bosonic fields
$\mathbf{\Psi}$ in the point $h$. Finally note that the two-point correlator
(81), that can be rewritten in the power series form

$\langle
1,h^{\prime}|h,1\rangle=\mu[h^{\prime}]\mu[h]\prod_{n=0}^{\infty}\sum_{p=0}^{\infty}a_{pn}[h,h^{\prime}|h_{I}]\left(\dfrac{\delta^{n}}{\delta
h^{n}}\mu^{2}[h]\Biggr{|}_{h_{I}}\right)^{p},$ (84)

with a coefficients

$a_{pn}[h,h^{\prime}|h_{I}]=\dfrac{1}{p!}\left[im_{I}\dfrac{(2n-3)!}{2^{2n-1}(n-1)!}\sum_{k=0}^{n+1}\dfrac{(-1)^{k}}{k!(n-k+1)!}(h_{I})^{n-k+1}\left(h^{k}-h^{\prime
k}\right)\right]^{p}.$ (85)

The series gives an opportunity to study perturbationally the two-point
correlations around the initial data point $h=h_{I}$.

##  3 Summary

In spite of a work in the Hamiltonian approach to General Relativity and the
primary quantization, the method of global bosonization to one $h$-dimension
of the Wheeler–DeWitt Quantum Gravity and its second quantization in the
Fock–Bogoliubov–Heisenberg initial data basis, which was presented in details
in this paper differs seriously from the previous authors considerations. The
main difference is a quantum field theory formulation of the Quantum Gravity,
that leads to the FBH initial data basis and considering the theory in terms
of the quantum bosonic field $\mathbf{\Psi}[h]$ associated with a
3-dimensional induced spacelike geometry $(\partial M,h)$. The proposed
approach is not the so called third quantization [94, 95, 96, 97, 98, 99,
100], where the Fock operator bases formalism is not applied. The main goal of
the presented linearization is a canceling of the Wheeler’s superspace notion
from considerations, and formulation of the Quantum Gravity in terms of the
Klein–Gordon–Fock operator evolution and the one-point correlations, that
results in the mass scale of the system.

The author benefited discussions from A.B. Arbuzov, B.M. Barbashov, V.N.
Pervushin, V.B. Priezzhev, D.V. Shirkov, and A.N. Sissakian. Special thanks
are directed to Profs. I.Ya. Aref’eva, G. ’t Hooft, and B.G. Sidharth for
interesting and critical remarks.

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