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\title{Theory of Stochastic Space-Time. \\
1. Gravitation as Quantum Diffusion.}
\author{Zahid Zakir \\
%EndAName
Institute of Noosphere,\\
167a, B.Ipak Yuli, Tashkent 700187 Uzbekistan}

\maketitle

\begin{abstract}
Is is shown that a nonhomogeneous diffusion in flat space with a tensor of
diffusion can be described as a homogeneous diffusion in a Riemannian
manyfold with this tensor of diffusion as a metric tensor. The curvature and
the gravitation are represented as manifestations of structure of
stochastic space-time, i.e. as consequences of quantum fluctuations.
Einstein's equations for metric tensor represented as equations for tensor
of diffusion. The energy density of such diffusional gravitation is negative
defined and leads to renormalizable quantum gravity with vanishing
cosmological constant.
\end{abstract}

\section{Content}

1{\bf . Introduction}

{\bf \ 2. Stochastic mechanics of homogeneous diffusion.}

{\bf \ 3. Stochastic mechanics of nonhomogeneous diffusion.}

{\bf \ 4. Diffusion induced gravity.}

{\bf \ 5. The energy-momentum tensor of gravitational field.}

{\bf \ 6. Gravitational renormalization of fields and quantum gravity.}

{\bf \ 7. Conclusions.}

{\bf \ 8. References}

\section{Introduction}

Two basic phenomena of physics - the gravitation and the quantum
fluctuations - both depended only from the mass of objects and both have the
geometric nature. The geometrical origin of gravitation is well known, while
up to now the Nelson's discovery of the stochastic geometry of space-time 
\cite{Ne} not accepted as one of fundamental concepts of physics. The
stochastic mechanics is unique version of quantum mechanics, where quantum
fluctuations represented as manifestations of stochastic geometry of
space-time with constant diffusion coefficient $\nu _0=\hbar /2m$ (see also
review \cite{Bla}). Therefore, in stochastic mechanics we may naturally
consider more general properties of space-time in case of nonhomogeneous
diffusion with tensor of diffusion $\nu ^{ab}(x,t)$ .

In this paper will be shown that the stochastic structure of space-time with
tensor of diffusion induced a nontrivial metrics $g^{ab}(x,t)$ with nonzero
curvature and, therefore, it is more fundamental property of space-time then
the metrics. This means that the quantum fluctuations induced the
gravitation and that the gravitation can be treated as a quantum diffusional
effect.

The result, that gravitation and quantum fluctuations are not independent
phenomena, leads to the interesting solution of the problem of energy of
gravitational field. The energy-momentum density of gravitational field is a
tensor and it is negative. The full energy-momentum of the system matter +
gravitational field is vanished (nonnegative), since the energy-momentum
density tensor of diffusional gravitation fully compensated the
energy-momentum density tensor of the source of this field, as firstly was
pointed out for the Einstein's equations by H.Lorentz in 1916 \cite{Lo} and
T.Levi-Civita in 1917 \cite{LC}.

The nontrivial consequences of negative defined energy of gravitation for
field theory and cosmology described in parts 5-6 and in my preceeding paper 
\cite{ZZ2}.

\section{Stochastic mechanics of homogeneous diffusion}

Let we have the Nelson's diffusion \cite{Ne} of a nonrelativistic particle
with the mass $m$ in flat space-time $R^n$:%

\begin{equation}
dx_{\pm }^a(t)=b_{\pm }^a(x,t)dt+dw_{\pm }^a(t) 
\end{equation}

with:

\begin{equation}
\begin{array}{c}
\lim E[\Delta w_{\pm }^a(t)\mid x^a(t)]=0 \\  
\\ 
\lim E[\Delta w_{\pm }^a(t)\cdot \Delta w_{\pm }^b(t)\mid x^a(t)]=\pm 2\nu
_0\delta ^{ab}dt 
\end{array}
\end{equation}

Here the diffusion coefficient is taken time-symmetric, constant and equal
to $\nu _0=\hbar /2m$, where $\hbar $is Planck constant. We must take limit $%
\Delta t\rightarrow 0$ only after calculation of conditional expectations $%
E[...\mid x^a(t)]$. The drifts are:

\begin{equation}
b_{\pm }^a(x,t)=\lim E[\frac{\Delta x_{\pm }^a(t)}{\pm \Delta t}\mid x^a(t)] 
\end{equation}

and the acceleration defined as:%
\begin{equation}
\begin{array}{c}
a^a(x,t)=\frac 12(D_{+}D_{-}+D_{-}D_{+})x^a(t)=\frac
12(D_{+}b_{-}^a+D_{-}b_{+}^a)= \\  
\\ 
=\frac{\partial v^a}{\partial t}+({\bf v\cdot \nabla })v^a-({\bf u\cdot
\nabla })u^a-\nu \Delta u^a 
\end{array}
\end{equation}

where the stochastic time derivative is:%
\begin{equation}
D_{\pm }f=\lim E[\frac{f(x(t\pm \Delta t),t\pm \Delta t)-f(x,t)}{\pm \Delta t%
}\mid x^a(t)]=\frac{\partial f}{\partial t}+b_{\pm }^a\nabla ^af\pm \nu
\Delta f 
\end{equation}

and 
\begin{equation}
\begin{array}{c}
v^a=\frac 12(b_{+}^a+b_{-}^a), \\ 
u^a=\frac 12(b_{+}^a-b_{-}^a) 
\end{array}
\end{equation}

The equations of motion of classical particle is the Newton's equations:

\begin{equation}
ma^b=-\nabla ^bV 
\end{equation}

The mean value of acceleration is equal:

\begin{equation}
\begin{array}{c}
E[a^b(t)]=\int dx\cdot \rho (x,t)\cdot a^b(x,t)=\int dx\cdot \rho \cdot [%
\frac{\partial v^b}{\partial t}+({\bf v}\cdot {\bf \nabla })v^b] 
\end{array}
\end{equation}

and
\begin{equation}
E[\frac{\partial v^b}{\partial t}+({\bf v}\cdot {\bf \nabla })v^b]=-\nabla
^b\varphi 
\end{equation}

i.e. the acceleration of sample particle in static gravitational field don't
depended from the mass of this particle (the principle of equivalence).

The mean trajectories of free particles by drifts $b_{\pm }^a$ are
diffusional geodesic lines on $R^n$.

\section{Stochastic mechanics of nonhomogeneous diffusion.}

Let we consider in $R^n$ a general diffusion with the tensor of diffusion $%
\nu ^{ab}=\nu _0\gamma ^{ab}(x,t)$. Here $\gamma ^{ab}$ is the normalized
tensor of diffusion, which in case of Nelson's homogeneous diffusion is: $%
\gamma ^{ab}=\delta ^{ab}$. The mean trajectories of particles in general
case are not geodesic lines on $R^n$, but they have some deviations from
geodesics. For the description of general diffusion we are introduce {\it %
curvilinear coordinates }$x^i(x^a,t)${\it \ and vectors }$e_i^a${\it \ along
mean trajectories of drifts }of free particles. Then we have for coordinates
of the point $M$:%

\begin{equation}
\begin{array}{c}
x^a(M,t)=e_i^a(M,t)\cdot x^i(M,t) \\ 
dx_{\pm }^a(t)=b_{\pm }^a(x,t)dt+dw_{\pm }^a(t)
\end{array}
\end{equation}

where

\begin{equation}
\begin{array}{c}
e_i^ae_i^b=\delta ^{ab}, \\ 
e_i^ae_j^a=g_{ij} \\ 
g^{ab}=e_i^ae_j^bg^{ij} 
\end{array}
\end{equation}

Here $g^{ab}$ is the metrics of flat space-time $R^n$, $g^{ij}(x,t)$ is any
metric tensor of the manifold $M$ with curvilinear coordinates $x^i$,
forming by mean trajectories of drift $b_{\pm }^i(x,t)dt$ of the diffusion:

\begin{equation}
dx_{\pm }^i(t)=b_{\pm }^i(x,t)dt+d[e_a^i(x,t)\cdot w_{\pm }^a(t)] 
\end{equation}

The conditional probabilities in this curvilinear coordinates can be defined
only for small time intervals $\Delta t$ and along piece-wise smooth curve
approximating random curve near the point $M$.

The tensor of diffusion is defined as:

\begin{equation}
\nu ^{ij}(x,t)=\frac 12\lim E_\gamma [\frac{\Delta x_{\pm }^i(t)\Delta
x_{\pm }^j(t)}{\pm \Delta t}\mid x^i(t)]=\nu _0\gamma ^{ij}(x,t) 
\end{equation}

If we introduce new type of parallel transport of tensors - {\it the
stochastic parallel transport on flat space-time along the mean trajectory
of drift:} 

\begin{equation}
de_a^i(x,t)=-\Gamma _{ml}^ie_a^ldx^m(t)-\frac 12d[\Gamma
_{ml}^ie_a^l]dx^m(t) 
\end{equation}

then such nonhomogeneous diffusion can be described as a homogeneous
diffusion on a Riemannian manyfold with constant diffusion coefficient $\nu
_0=const$ and the metric tensor $g^{ij}(x,t)$ with the correspondence $\nu
^{ij}(x,t)=\nu _0\cdot g^{ij}(x,t).$ Stochastic mechanics with tensor of
diffusion then can be treated as quantum mechanics on Riemannian manyfold.
Therefore, we may use the well known formulas of stochastic mechanics on
Riemannian manifold \cite{Gu1}.

Stochastic time derivatives then given by:

\begin{equation}
(D_{\pm }F)^i(x,t)=\frac{\partial F^i}{\partial t}+(b_{\pm }\cdot \nabla
)F^i\pm \nu _0(\Delta _{DR}F)^i 
\end{equation}

where $\nabla -$ is the Laplace-Beltrami operator in curved manifold, and

\begin{equation}
(\Delta _{DR}F)^i=\Delta F^i+R_j^i\cdot F^j 
\end{equation}

is the Laplace-de Rham operator, $\Delta =\nabla \cdot \nabla ,$ and $R_j^i$
is the Ricci tensor.

From the expression for the acceleration:

\begin{equation}
a^i(x,t)=\frac 12(D_{+}b_{-}+D_{-}b_{+})^i(x,t)=-\frac 1m\nabla ^iV 
\end{equation}

can be obtained equations:

\begin{equation}
\frac{\partial v^i}{\partial t}+(v\cdot \nabla )v^i-(u\cdot \nabla )u^i-\nu
_0\Delta u^i-\nu _0R_j^iF^j=-\frac 1m\nabla ^iV 
\end{equation}

The continuity equation for the probability density $\rho (x,t)$%
\begin{equation}
\frac{\partial \rho }{\partial t}+\nabla _i(\rho v^i)=0 
\end{equation}

gives $u^i=\nu _0(\nabla ^i\rho )/\rho .$ Then after Nelson's assumptions $%
mv^i=\nabla ^iS,$ $\nu _0=\hbar /2m,$ and

\begin{equation}
\psi (x,t)=\sqrt{\rho (x,t)}\cdot \exp [iS(x,t)/\hbar ] 
\end{equation}

where $S(x,t)$ is some function, $\hbar -$Planck's constant, $\psi (x,t)$ is
wave function, we obtain Schrodinger equation for particle motion on the
stochastic space with tensor of diffusion \cite{Gu1}:%

\begin{equation}
i\hbar \frac{\partial \psi }{\partial t}=-\frac{\hbar ^2}{2m}\Delta \psi
+V\psi  
\end{equation}

\section{Diffusion induced gravity.}

The mean value of acceleration not contain terms with $u^i$ and if not
external potential, we have intrinsic accelaration since the presence of
derivatives of metrics in Laplace-Beltrami operator:%
\begin{equation}
\begin{array}{c}
E[
\frac{\partial v_i}{\partial t}+(v\cdot \nabla )v_i]= \\ =E[\frac{\partial
v_i}{\partial t}+v^j\partial _jv_i-\Gamma _{ij}^kv^jv_k]=0
\end{array}
\end{equation}

This acceleration not contain dependence from the mass of particle, i.e. we
have analog of equivalence principle for gravitation. 

Formulas of stochastic mechanics are naturally extended to the case of
relativistic particle \cite{Gu2}. In this case we obtain nontrivial metrics
for space-time, which we can interprete as diffusion induced gravity. Then,
we can conclude, that nonhomogeneous diffusion leads to induced metrics and
curvature, since the tensor of diffusion $\nu ^{ij}(x,t)$ leads to the
acceleration of the particle and the mean value of this diffusion induced
acceleration not depended from the mass of the particle. 

The independence of the acceleration from the mass leads also to the
accelerations of macroscopic objects - bodies of basis of reference frame
and devices. Acceleration of the reference frame means the appearance of
nontrivial metrics and nonzero curvature of space-time. So, the existence of
the diffusion induced curvature means the existence of the diffusion induced
gravity. Einstein's equations for the curvature and the metric tensor then
can be represented as equations for the tensor of diffusion.

Identification of the metric structure of space-time with general diffusion
leads to interpretation of gravitation as secondary effect of quantum
fluctuations and we come to {\it diffusion induced gravity} and to {\it %
stochastic interpretation of gravitation (stochagravity?).}

In stochastic interpretation of gravitation we consider equations for the
metrics $g_{ij}(x,t)$ as consequence of diffusion equations. The gravitation
is the energy-momentum effect and we deal with some kind of thermodiffusion,
where diffusional flow directed opposite to differences of energies between
two points of space:%

\begin{equation}
j=-\alpha \cdot (E_1-E_0) 
\end{equation}

Let we have the particle at very large distance from the source of
gravitational field, where the energy density of the field is $G_0$. After
diffusion process the particle moved near the source, where the energy
density of the field is $G_1.$ For existence of the attractive diffusion we
must have negative difference of energy densities:

\begin{equation}
G_1-G_0=\Delta G\prec 0. 
\end{equation}

Therefore, if we interprete the gravitation as some diffusion process, the
energy density of gravitational field must be negative. We can suppose, that
if the energy-momentum tensor of matter $T_{ij}$ must be the source of
gravitational field with energy-momentum density $G_{ij},$ we have

\begin{equation}
G_{ij}\sim -T_{ij} 
\end{equation}

i.e. this tensors have opposite sign in equations. The Einstein's equations
for metrics%

\begin{equation}
R_{ij}-\frac 12g_{ij}\cdot R=-\kappa T_{ij} 
\end{equation}

we may treat as equations for tensor of diffusion $\nu _{ij}(x,t),$ i.e. as
basic equations of the theory, which connect $\nu _{ij}(x,t)$ by $T_{ij}$ .
The metric tensor $g_{ij}(x,t)$ then become secondary object and it is
expression of local diffusion $\nu _{ij}(x,t)$ only. Here the
energy-momentum density tensor $T_{ij}$ represented as source for
deformations of stochastic structure of space-time. 

As result, we have some {\it duality} between deformations of stochastic
structure and Riemannian structure of manyfold.

\section{The energy-momentum tensor of gravitational field.}

\subsection{The tensor of energy-momentum density}

The energy-momentum density tensor of gravitational field can be defined
from the Einstein's equations as left hand side of this equations:

\begin{equation}
G_{ik}=\frac 1\kappa (R_{ik}-\frac 12g_{ik}R) 
\end{equation}

Then the full energy of the system (gravitational field and other matter) is
cancelled:

\begin{equation}
G_{ik}+T_{ik}=0 
\end{equation}

This form of the tensor of energy-momentum for gravitation was proposed
firstly by H.A.Lorentz \cite{Lo} in 1916 and T.Levi-Chivita \cite{LC} in
1917.

The conservation of the energy-momentum here is automatically:

\begin{equation}
\nabla ^i(G_{ik}+T_{ik})=\partial ^i(G_{ik}+T_{ik})\equiv 0 
\end{equation}

The stability of the system guarantied only by intrinsic dynamics, described
by Einstein's equations. The creation of matter can be ''organized'', if we
can split the space-time into gravitation and other forms of matter. We can
annihilate the matter with gravitation, if we can pack the energy of
gravitational field of the system into small volume.

\subsection{ Cosmological constant is vanished.}

The appearance of a matter condensate with the nonzero cosmological constant 
$\lambda _m$ always induced the cosmological constant for the gravitational
field $\lambda _g,$ which exactly compensated $\lambda _m$, and as result, 
{\it the full cosmological constant of the system is vanished }$\lambda
=-\lambda _g+${\it \ }$\lambda _m=0.$

\subsection{ Singularities and Black Holes are annihilated.}

The particle and it's gravitational field annihilated near the singularity
point inside of the black hole and {\it therefore the singularities vanished
in the classical theory.}

The process of annihilation of all particles inside of the black hole leads
to the decreasing of an area of the horizon and radius with speed of light.
The external gravitational field of the black hole partly released from the
source and will be emitted as gravitational waves and particles (gravitons).
Therefore, the black holes annihilated during the process of their formation
by emission of strongly gravitational waves and high energy gravitons.

\section{Gravitational Renormalization of Fields and Quantum Gravity}

In calculations of the energy levels in quantum field theory appeared some
infinities, which usually can be excluded be renormalization of masses and
charges of particles. For example, in calculation of the energy of scalar
field on curved background the renormalized cosmological constant defined as 
\cite{Bi}:
\begin{equation}
\Lambda =\Lambda _B+[\frac A{n-4}+B(\mu )] 
\end{equation}

where $\Lambda _B$ is initial value of cosmological constant, $A,B$- some
functions of mass and cut off factors, $n=4$ is the dimension of space-time.

In our version of gravitational energy, each infinity contribution to the
energy of the field or particle induced infinite contribution to the energy
of background gravitational field with opposite sign, which exactly
compensated contributions to the energy of the matter. So, we rewrite the
formula as:%
\begin{equation}
\Lambda =-\Lambda _g+[\frac A{n-4}+B(\mu )]=-\Lambda _g+\Lambda _m(\mu
)_{}=0 
\end{equation}

where $-\Lambda _g$ is the energy density of background gravitational field, 
$\Lambda _m(\mu )$ is the energy of vacuum of scalar field

Therefore, we can conclude, that negative defined energy of gravitation is
main renormalizational factor for the quantum field theory, and the
procedures of renormalization can be obtained from the first principles of
the field theory, if we include into Lagrangian gravitational part.

In quantum gravity we have surprising cancellations of positive energy
infinities in all orders of perturbation theory because of unduced
infinities of the energy of the background gravitational field in the left
hand side of Einstein equations.

\section{Conclusions.}

The main result of the paper is the fact, that the quantum fluctuations
induced the gravitation and that the gravitation can be treated as a quantum
effect. This result is very important not only for understanding of physical
nature of gravitation, but for the definition of the strategy of
construction of unified theories also.

If in stochastic treatment the gravitation and quantum fluctuations are not
independent phenomena, and if from this point of view the problems of energy
of gravitational field, vanishing of cosmological constant, excluding of
singularities, observability of black holes and quantization of gravity can
be solved naturally, then we may hope, that the unified theories can be
based partly on this kind of treatments. One attempt in this direction is
the method of thermostring quantization and thermostring treatment of
strings \cite{ZZ1} \cite{ZZ2}.

In the forthcoming paper we will show that the stochastic treatments of the
quantum theory and the gravitation are not only some hypotheses, but as in
general relativity, they can be considered as the consequences of some
general principles of invariance.

\begin{thebibliography}{99}
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\bibitem{Bla} {Blanchard Ph., Combe Ph., Zheng W. (1987) {\it Mathematical
and Physical Aspects of Stochastic Mechanics, }Lect.Not. in Phys., v.281,
171 p.}

\bibitem{ZZ1} {hep-th/9809170 Zakir Z. (1998) {\it Thermostring
Quantization. Interpretation of Strings as Particles at Finite Temperature; }\\
hep-th/9810247 Zakir.Z. (1998) {\it Are Strings Thermostrings?}}

\bibitem{ZZ2} {hep-th/9812246 Zakir Z. (1998) {\it The Negative Energy of
Gravitation as Stabilization Factor in Field Theory and Cosmology.}}

\bibitem{Gu1} {Dohrn D., Guerra F. (1978) Lett. Nuovo Cim., v.22, N 4,
p.121-127.}

\bibitem{Gu2} {Guerra F., Ruggiero P. (1978) Lett.Nuovo Cim., v.23, N 15,
p.529-534.} 

\bibitem{Sm} {Smolin L. (1986) Phys.Lett., 113A, N 8, p.408-412.}

\bibitem{Lo}  {Lorentz H.A. (1916) Versl.Akad.Amsterdam, v.25, p.468;\\
Lorentz H.A. (1937) Collected {Papers.} The Hague: M.Nijhoff, v.5, p.276-313.%
}

\bibitem{LC}  {Levi-Civita T. (1917) Rend.Acc.Lincei, ser.5a, v.XXVI,p.381.}

\bibitem{Bi}  {Birrell N.D., Davies P. (1982) {\it Quantum Fields in Curved
Space.}, Camb.U.Pr.}
\end{thebibliography}

\end{document}
