[math-ph]

# Wiener Process of Fractals and Path-Integrals II:
Emergent Einstein-Hilbert Action in Stochastic Process of Quantum Fields along
with the Ricci Flow of the Space Geometry

Amir Abbass Varshovi
ab.varshovi@sci.ui.ac.ir/amirabbassv<EMAIL_ADDRESS>Faculty of
Mathematics and Statistics, Department of Applied Mathematics and Computer
Science, University of Isfahan, Isfahan, IRAN.
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O. Box: 19395-5746, Tehran, IRAN.

###### Abstract

Abstract: This is the second paper of a series of researches (that commenced
with varshovi1  and is followed by varshovi2 ) that aims to interpret the
gravitational effects of nature within some consistent stochastic fractal-
based (intrinsically conformal) path-integral formulation. In the present
work, the Wiener stochastic process is employed to study the Brownian motion
of fractal functions on a closed Riemannian manifold with dynamical geometry
due to the Ricci flow. It has been shown that the Wiener measure automatically
leads to the Einstein-Hilbert action and the path-integral formulation of
scalar quantum field theory at the first local approximation. This would be
interpreted as the more fundamental formulation of quantum field theory in
presence of gravity. However, we establish that the emergence of Einstein-
Hilbert action is independent of the matter field interactions and is a merely
entropic effect stemming from the nature of the Ricci flow. We also extract an
explicit formula for the cosmological constant in terms of the Ricci flow and
Hamilton’s theorem for 3-manifold. Then, we discuss the cosmological features
of the FLRW solution in $\Lambda$CDM Model via the derived equations of the
Ricci flow. We also argue the correlation between our formulations and the
entropic aspects of gravity. Finally, we establish some theoretical evidence
that proves that the second law of thermodynamics is the basic source of and
more fundamental than gravity.

Keywords: Wiener Stochastic Process, Fractal Norm, Laplace-Beltrami Operator,
Closed Riemannian Manifold, Einstein-Hilbert Action, Hamilton’s Theorem, Ricci
Flow, Entropic Force, Cosmological Constant, $\Lambda$CDM Model, Second Law of
Thermodynamics.

##  I Introduction

Employing the Wiener stochastic process wiener  for Wierstrass-like functions
(the so called fractal functions) on a flat space manifold we have proved in
varshovi1  that the scalar quantum field theory stems from the Brownian motion
of fractal functions along the time direction.111See folland  for a nice
presentation of the Wiener stochastic process and its applications as a
probability measure. Actually, this consequence stands on a complex footing
which consists of i) _Fractals_ and ii) _Brownian motion_. In fact, fractal
functions are continuous but nowhere differentiable functions on the whole (or
a widespread subset of) of the underlying domain. We studied them according to
the properties of the Weierstrass functions weierstrass  and their asymptotic
behavior in Fourier coefficients. Following the Hardy’s results hardi  we
obtained a non-linear differential equation for such asymptotic behaviors of
fractal functions. By solving the differential equation we established that
$f:S\to\mathbb{R}$, with $S=[-L,L]^{D}$, is a fractal function if for each
$\sigma>0$, $\mu\geq 0$, and $\ell>0$ there exists some Fourier mode
$n=(n_{1},\cdots,n_{D})\in\mathbb{Z}^{D}$, so that

$\int_{0}^{|f_{n}|}e^{\sigma(\mu^{2}+n^{2})x^{2}/2}dx>\ell,$ (I.1)

wherein $f_{n}$ is the $n$-th Fourier coefficient of $f$, and
$n^{2}=\sum_{i=1}^{D}n_{i}^{2}$. In fact, if we set
$\Delta=-\sum_{i=1}^{D}\partial_{i}^{2}$ to be the Laplacian operator, then we
could readily adjust (I.1) for the spectrum of the Laplacian operators. Let
$\\{\psi_{\lambda,i}\\}_{\lambda\in\text{spec}\Delta}^{i\in D_{\lambda}}$ be
the complete set of orthonormal eigenfunctions of $\Delta$ (i.e.
$\int_{S}\psi_{\lambda,i}\psi_{\lambda^{\prime},i^{\prime}}d^{D}x=\delta_{\lambda\lambda^{\prime}}\delta_{ii^{\prime}}$)
for eigenvalues $0\leq\lambda$s and some possible degeneracy space
$D_{\lambda}$ labeled by $i$. Then $f:S\to\mathbb{R}$ is a fractal function if
for each $\sigma>0$, $\mu\geq 0$, and $\ell>0$ there exists some Fourier mode
$(\lambda,i)$ so that

$\int_{0}^{|f_{\lambda,i}|}e^{\sigma(\mu^{2}+\lambda)x^{2}/2}dx>\ell,$ (I.2)

wherein $f_{\lambda,i}=\int_{S}\psi_{\lambda}fd^{D}x$. In principle, the
mechanism of which we employed in varshovi1  for working out the fractality
condition (I.3) on the space set $S=[-L,L]^{D}$ (i.e. fractal periodic
functions with period $2L$ in each variable), could be generalized to any
compact Riemannian manifold $(M,g)$. Actually, here we may concentrate on
manifolds without boundary.222So the Neumann and Dirichlet boundary conditions
for extracting the eigenvalues and the eigenspaces of the Laplace-Beltrami
operator would be irrelevant. It is well-known that if $(M,g)$ is a closed
$D$-dimensional Riemannian manifold then the spectrum of the Laplacian
$\Delta$ is discrete and consists of an increasing sequence
$\\{\lambda_{k}\\}_{k=1}^{\infty}$ of eigenvalues (counted with degeneracies)
with $\lambda_{1}=0$ and $\lambda_{k}\to\infty$ as $k\to\infty$ yau .

On the other hand, the eigenfunctions (resp. the eigenforms) of the Laplacian
operator provide a pure point spectrum, which means that
$L^{2}M=L^{2}(\Lambda^{0}M)$ (resp. $L^{2}(\Lambda^{n}M)$, $n>0$) admits a
complete orthonormal basis consisting of eigenfunctions $\psi_{\lambda,i}$ of
$\Delta$ with associated eigenvalues $\lambda$s, i.e.
$\Delta\psi_{\lambda,i}=\lambda\psi_{\lambda,i}$. This leads to a generalized
Fourier expansion of continuous functions (resp. forms) on compact Riemannian
manifolds. An extensive amount of investigations have been made concerning the
eigenvalues and their relationship to the background Riemannian
geometry.333See for example weyl ; weyl2 ; gallot ; besse ; donnelly ; gilkey1
; gilkey2 ; chavel ; yau ; li-yau ; yau0 ; deshmukh ; gilkey3 ; singer ;
berger ; buser ; cheeger ; ledoux ; lang ; egidi ; simon ; henrot  and the
references therein. These studies pertain to upper and lower bounds for
eigenvalues and their asymptotics which are of the most interest in our study.
There is a major conclusion that is valid for all compact Riemannian geometry;
The Weyl’s asymptotic law weyl ; weyl2  which asserts that $\lambda_{k}$
asymptotically behaves as $k^{2/D}$ as $k\to\infty$.

It is worth to remind that there are significant results for the first gap of
the Laplacian eigenvalues: a) The Yau’s theorem yau0 ; li-yau  which provides
a lower bound for non-vanishing Laplacian eigenvalues on Riemannian manifolds
with non-negative Ricci curvature as $\lambda_{1}\geq\frac{\pi^{2}}{d^{2}}$,
where $d$ is the diameter of $M$. b) The Lichnerowicz’s theorem gallot  for
positive definite Ricci curvature, which says that $\lambda_{1}\geq KD$
whenever $Ric_{\mu\nu}\geq(D-1)Kg_{\mu\nu}$. c) The Simon’s lower bound for an
Einstein’s manifold of $D\geq 3$ dimensions whose sectional curvature is
bounded from below by a constant $K_{0}$, which states that $M$ is either
isometric to $D$-sphere $S^{D}$ or else $\lambda_{1}>2DK_{0}$ simon .444See
deshmukh0 ; poor  for more detailed review about the harmonic theory in
Riemannian geometry and the lower bound of $\lambda_{1}$ in Einstein-like and
constant scalar curvature manifolds. d) And finally, Cheeger’s isoperimetric
constant puts an important geometric lower bound for $\lambda_{1}$, mostly
referred to as the Cheeger’s inequality cheeger . This inequality also gives
rise to an upper bound for $\lambda_{1}$ buser  which is essentially given in
terms of the Ricci curvature of $(M,g)$ ledoux .555See lang  for
generalization of Cheeger’s inequalities to the magnetic Laplacian
$\Delta^{\alpha}$ defined for $d\mapsto d+\alpha$, $\alpha\in\Omega^{1}M$, on
closed Riemannian manifold $M$. See also egidi  for more recent achievements
about the eigenvalues of the magnetic Laplacian $\Delta^{\alpha}$.

However, Weyl’s asymptotic result guarantees that the asymptotic behavior of
the Laplace-Beltrami operator on each $D$-dimensional closed Riemannian
manifold equals that of flat $D$-torus $\mathbb{T}^{D}$, which we essentially
considered in varshovi1 . Moreover, the fractality condition is in particular
a local property that can be studied by means of the harmonic analysis on
Riemannian manifolds. Indeed, according to the Sobolev’s theorem sobolev  any
element of the Sobolev’s space $\mathcal{H}_{t}$ with $t=[D/2]+1+s$, $0\leq
s<1$, will uniformly converge to a continuous function whose derivative fails
to converge on $M$, hence is a Weierstrass-like function.666See chapter 6 of
warner  for detailed applications of Sobolev’s spaces on Riemannian manifolds.
Here, the Sobolev class of $f:M\to\mathbb{R}$ is studied locally on $M$ and
the constant $s$ could be considered as the self-similarity scale of the
fractal function, as was pointed out by Hardy hardi .

Since, the main criteria for studying the fractality in varshovi1  stemmed
from the asymptotic behavior of Fourier-Laplace coefficients (i.e. the
expansion coefficients in terms of the eigenfunctions of the Laplacian
operator on $S=[-L,L]^{D}\cong\mathbb{T}^{D}$) of Weierstrass-like functions
we readily obtain the same differential equations for such coefficients on
closed Riemannian manifold $M$. Therefore, we have already obtained the
generalized version of Theorem 2 of varshovi1 :777We omit the detailed
arguments of the issue here and refer the reader to sections II and III of
varshovi1  for a more comprehensive discussion.

Theorem 1; _Let $(M,g)$ be a closed Riemannian manifold with Laplacian
operator $\Delta=dd^{\dagger}+d^{\dagger}d$. Let
$\\{\psi_{\lambda,i}\\}_{\lambda\in\text{spec}\Delta}^{i\in D_{\lambda}}$ be
the complete set of orthonormal eigenfunctions of $\Delta$ for eigenvalues
$0\leq\lambda$s and some possible degeneracy space $D_{\lambda}$ labeled by
$i$. That is:
$\int_{M}\psi_{\lambda,i}\psi_{\lambda^{\prime},i^{\prime}}d\Omega_{g}=\delta_{\lambda\lambda^{\prime}}\delta_{ii^{\prime}}$,
for $d\Omega_{g}$ the Riemannian volume form on $M$. Then, $f:M\to\mathbb{R}$
is a fractal function if for each $\sigma>0$, $\mu\geq 0$, and $\ell>0$ there
exists some Laplacian Fourier mode $(\lambda,i)$ so that_

$\int_{0}^{|f_{\lambda,i}|}e^{\sigma(\mu^{2}+\lambda)x^{2}/2}dx>\ell,$ (I.3)

_wherein $f_{\lambda,i}=\int_{M}f\psi_{\lambda,i}d\Omega_{g}$._

Employing the above theorem we can accomplish the Wiener stochastic process of
fractal functions on general closed Riemannian manifold $(M,g)$. Through the
next section, we follow the strategy of varshovi1  on a given closed
Riemannian manifold (as the manifold of space) and work out the generalized
result of varshovi1 . In fact, we will establish that a plain generalization
of the Wiener Brownian process for propagating fractal functions on $M$ will
lead to the path-integral formulation of scalar quantum field theory on static
curved space-time.

However, the main objective of this paper is accomplished in section III.
Actually, in this section, the Wiener fractal measure is implemented for
dynamical space-time with some time-dependent metric $g_{ij}(t)$. In fact, we
will establish that the Einstein-Hilbert action can be rigorously worked out
from an entropic evolution of the space geometry due to the Ricci flow.
Moreover, we will establish that the emergence of gravity in this path-
integral formulation is independent of the quantum field theory and is a
merely entropic effect. Therefore, although the expectation value of the
stress-energy tensor of the involved quantum matter fields will affect the
metric of space $M$ the time evolution of the metric is imposed by an entropic
law comparable with the second law of thermodynamics. Hence, the Ricci flow
could be assumed as a natural gauge fixing term for Einstein’s field equation.

In section IV we study the cosmological results of the extracted stochastic-
entropic theory of gravity, the so called _Wiener fractal gravity_ , and work
out the cosmological constant due to Hamilton’s theorem for 3-manifolds
hamilton1 . Next, we will give a different interpretation for inflation and
the dark energy. Comparison with the $\Lambda$CDM Model and the admitted FLRW
solution in cosmology is concerned due in this section. Finally, in section V
we study the entropic features of the Ricci flow according to Perelman’s
seminal paper perelman , and we will conclude that the second law of
thermodynamics could be regarded as the basic source of gravitational effects
in nature. We then compare the Wiener fractal gravity with Verlinde’s entropic
gravity, stochastic gravity and Horava-Lifshitz gravity.

##  II Wiener Fractal Measure and Quantum Fields on Curved Spacetime

In fact, Theorem 1 lets us to define the _(massive) fractal norm_ on
Riemannian manifold $(M,g)$ as;

$\ell_{\sigma,\mu}(f_{\lambda,i})=\int_{0}^{|f_{\lambda,i}|}e^{\sigma(\mu^{2}+\lambda)x^{2}/2}dx.$
(II.1)

Let $\mathcal{N}\in\mathbb{N}$ and define $C_{\mathcal{N}}$ be the collection
of real functions $f:M\to\mathbb{R}$ which are finite linear combinations of
Laplacian eigenfunctions $\psi_{\lambda,i}$ with eigenvalues
$0\leq\lambda\leq\mathcal{N}$. Each individual element of $C_{\mathcal{N}}$ is
conventionally called an _$\mathcal{N}$ -bounded function (on $M$)_. Then, the
fractal norm (II.1) leads to a _Lebesgue fractal measure_ $d\mu_{LF}$ on
$C_{\mathcal{N}}$ as:

$d\mu_{LF}=\prod^{D_{\mathcal{N}}}dx_{\lambda,i},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}dx_{\lambda,i}=e^{\sigma(\mu^{2}+\lambda)f_{\lambda,i}^{2}/2}df_{\lambda,i},$
(II.2)

wherein $D_{\mathcal{N}}$ is the dimension of $C_{\mathcal{N}}$. Since $M$ is
compact $C_{\mathcal{N}}$ is finite-dimensional (i.e.
$D_{\mathcal{N}}<\infty$), hence $d\mu_{LF}$ is well-defined. Now, we are
ready to study the Brownian motion of $\mathcal{N}$-bounded functions by means
of the Wiener stochastic process and the background Lebesgue fractal measure
$d\mu_{LF}$. This Brownian motion is defined by means of an appropriate
_Wiener fractal measure_ on $C_{\mathcal{N}}$ for Riemannian manifold $(M,g)$.
Actually, the definition is straightforward and is given upon what we have
obtained in varshovi1 . We remember that the symmetric Wiener measure for a
stochastic process in target space $\mathbb{R}^{D}$ from the initial position
$x_{I}$ at $t_{0}=-T$ to the final position $x_{F}$ at $t_{N+1}=T$, within the
time slicing $-T=t_{0}<t_{1}<\cdots<t_{N}<t_{N+1}=T$ is:

$\begin{gathered}dW(t_{N},\cdots,t_{1})=\left(4\pi
T\right)^{D_{\mathcal{N}}/2}~{}\exp\left(|x_{F}-x_{I}|^{2}/4T\right)\\\
\times\frac{1}{\left(2\pi\tau_{N+1}\right)^{D_{\mathcal{N}}/2}}e^{-|x_{F}-x_{N}|^{2}/2\tau_{N+1}}d^{D}x_{N}\frac{1}{\left(2\pi\tau_{N}\right)^{D_{\mathcal{N}}/2}}e^{-|x_{N}-x_{N-1}|^{2}/2\tau_{N}}\times\cdots\\\
\times\cdots\times\frac{1}{\left(2\pi\tau_{2}\right)^{D_{\mathcal{N}}/2}}e^{-|x_{2}-x_{1}|^{2}/2\tau_{2}}d^{D}x_{1}\frac{1}{\left(2\pi\tau_{1}\right)^{D_{\mathcal{N}}/2}}e^{-|x_{1}-x_{I}|^{2}/2\tau_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{gathered}$
(II.3)

for $x_{i}=x(t_{i})$, $\tau_{i}=t_{i}-t_{i-1}$, and
$|x_{i}-x_{i-1}|^{2}=\sum_{j=1}^{D}(x^{j}_{i}-x^{j}_{i-1})^{2}$. Therefore,
the symmetric Wiener for Brownian motion of fractal functions on $M$
restricted to $C_{\mathcal{N}}$ would be (assuming $x_{I}=x_{F}=0$):

$\begin{gathered}dW(t_{N},\cdots,t_{1})=\left(4\pi
T\right)^{D_{\mathcal{N}}/2}\left(\prod_{k=1}^{N+1}\Xi_{k}\right)\times\left(\prod_{k=1}^{N}d^{D_{\mathcal{N}}}f_{\lambda,i}(t_{k})\right),\end{gathered}$
(II.4)

for

$\begin{gathered}\Xi_{k}=\frac{1}{\left(2\pi\tau_{k}\right)^{D_{\mathcal{N}}/2}}\times\exp\left(-\frac{\tau_{k}}{2}\sum_{\lambda\in\text{spec}\Delta}\sum_{i\in
D_{\lambda}}\left(\frac{1}{\tau_{k}}\int_{f_{\lambda,i}(t_{k-1})}^{f_{\lambda,i}(t_{k})}e^{\sigma(\lambda+\mu^{2})x^{2}/2}dx\right)^{2}\right)\\\
\times\exp\left(\frac{\sigma}{2}\int_{M}\left(g^{ij}\partial_{i}f(t_{k-1})\partial_{j}f(t_{k-1})+\mu^{2}f^{2}(t_{k-1})\right)d\Omega_{g}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{gathered}$
(II.5)

We refer to (II.4) as the _Wiener fractal measure_. Being of a nonlocal
formula, this probability measure could be turned to a local structure by
employing the following approximation as $N\to\infty$:

Strategy A: _When $\tau_{k}\to 0$ (i.e. $N\to\infty$) we can simply apply the
following replacement within the Wiener fractal measure:_

$\frac{1}{\tau_{k}}\int_{f_{\lambda,i}(t_{k-1})}^{f_{\lambda,i}(t_{k})}e^{\sigma(\lambda+\mu^{2})x^{2}/2}dx~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mapsto~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{\partial
f_{\lambda,i}}{\partial t}(t_{k-1}),$ (II.6)

_which leads to:_

$\begin{gathered}\Xi_{k}=\frac{1}{\left(2\pi\tau_{k}\right)^{D_{\mathcal{N}}/2}}\exp\left(-\tau_{k}\int_{M}\frac{1}{2}\Big{\\{}\left(\partial_{t}f(t_{k-1})\right)^{2}-\frac{\sigma}{\tau_{k}}\left(g^{ij}\partial_{i}f(t_{k-1})\partial_{j}f(t_{k-1})+\mu^{2}f^{2}(t_{k-1})\right)\Big{\\}}d\Omega_{g}\right).\end{gathered}$
(II.7)

Actually, upon the strategy A we readily find:

$\begin{gathered}\prod_{k=1}^{N+1}\Xi_{k}=\left(\frac{1}{\left(2\pi\theta\right)^{D_{\mathcal{N}}/2}}\right)^{N+1}\exp\left(-\int_{\mathcal{M}}\frac{1}{2}\Big{\\{}\left(\partial_{t}f\right)^{2}-\frac{\sigma}{\theta}\left(g^{ij}\partial_{i}f\partial_{j}f+\mu^{2}f^{2}\right)\Big{\\}}dt\wedge
d\Omega_{g}\right),\end{gathered}$ (II.8)

wherein $\mathcal{M}=[-T,T]\times M$ is the space-time continuum,888Indeed, as
we pointed out in varshovi1  we distinguish between terminologies _spacetime_
and _space-time_ , for which the former is realized to admit space and time
dimensions substantially as an identical essence, whereas the latter embraces
an inherent difference between them. and $\theta=dt=2T/N$ ($\to 0$) is the
unit of uniform time slicing. Set $c=\sqrt{\sigma/\theta}$ to be the speed of
light in vacuum, then we obtain:

$\begin{gathered}\prod_{k=1}^{N+1}\Xi_{k}=\left(\frac{1}{\left(2\pi\theta\right)^{D_{\mathcal{N}}/2}}\right)^{N+1}\exp\left(-c\int_{\mathcal{M}}\frac{1}{2}\Big{\\{}\mathbf{g}^{\mu\nu}\partial_{\mu}f\partial_{\nu}f-\mu^{2}f^{2}\Big{\\}}d\Omega\right),\end{gathered}$
(II.9)

where $d\Omega=dx^{0}\wedge d\Omega_{g}$ is the Riemannian volume form on the
space-time continuum $\mathcal{M}$ for the induced time-independent metric
$\mathbf{g}=\mathbf{g}_{\mu\nu}dx^{\mu}\otimes dx^{\nu}=dx^{0}\otimes
dx^{0}-g_{ij}dx^{i}\otimes dx^{j}$ with $x^{0}=ct$. Note that $\mathbf{g}$ is
a Lorentzian metric with signature $(+,-,-,-)$, hence the theory is locally
Lorentz invariant. If one defines $f=\gamma^{-1}\phi$ for normalization factor
$\gamma=\sqrt{\hbar c}$ we read:

$\begin{gathered}\prod_{k=1}^{N+1}\Xi_{k}=\left(\frac{1}{\left(2\pi\theta\right)^{D_{\mathcal{N}}/2}}\right)^{N+1}\exp\left(-\frac{1}{\hbar}\int_{\mathcal{M}}\frac{1}{2}\Big{\\{}\mathbf{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{m^{2}c^{2}}{\hbar^{2}}\phi^{2}\Big{\\}}d\Omega\right),\end{gathered}$
(II.10)

for $m=\hbar\mu/c$ to be the physical mass.999For more detailed discussion
about the relations between fractal parameters $\sigma$ and $\mu^{2}$ and the
physical constants $\hbar$ and $c$ see varshovi1 . Hence, for the Feynman
measure

$\mathfrak{D}\phi=\left(4\pi
T\right)^{D_{\mathcal{N}}/2}\left(\frac{1}{\left(2\pi\theta\right)^{D_{\mathcal{N}}/2}}\right)^{N+1}\gamma^{-ND_{\mathcal{N}}}\left(\prod_{k=1}^{N}d^{D_{\mathcal{N}}}\phi_{\lambda,i}(t_{k})\right),$
(II.11)

the Wiener fractal measure with strategy A becomes:

$dW_{\textbf{A}}(t_{N},\cdots,t_{1})=\exp\left(-\frac{1}{\hbar}\int_{\mathcal{M}}\frac{1}{2}\Big{\\{}\mathbf{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{m^{2}c^{2}}{\hbar^{2}}\phi^{2}\Big{\\}}d\Omega\right)\mathfrak{D}\phi.$
(II.12)

However, as we discussed in varshovi1  the replacement (II.6) causes the heat
kernel of the Gaussian term of the Wiener measure to be replaced with a
quadratic function which is not always dominating. Therefore, the Wiener
fractal measure (II.12) is not well-defined over $C_{\mathcal{N}}$. To
overcome this subtlety, we should transfer to complex analysis. As we pointed
out in varshovi1  incorporating an infinitesimal heat kernel term after
analytic continuation of the Wiener fractal measure would compensate some of
the pathologies emerging from replacement (II.6) via the equation:

$\lim_{\varepsilon\to
0}\Big{|}\int_{\mathbb{R}}e^{(ia-\varepsilon)x^{2}}dx\Big{|}=\int_{\mathbb{R}}e^{-|a|x^{2}}dx.$
(II.13)

Then, the next strategy is:

Strategy B: _To compensate the pathology of strategy A we should employ the
analytic continuation of the Wiener fractal measure with an augmented heat
kernel $\varepsilon$-term due to (II.13). Hence, we must obtain:_

$\begin{gathered}dW(t_{N},\cdots,t_{1})\approx
dW_{\textbf{A,B}}(t_{N},\cdots,t_{1})\\\
=\exp\left(\frac{i}{\hbar}\int_{\mathcal{M}}\frac{1}{2}\Big{\\{}\mathbf{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{(m^{2}-i\varepsilon)c^{2}}{\hbar^{2}}\phi^{2}\Big{\\}}d\Omega\right)\mathfrak{D}\phi,\\\
=\exp\left(\frac{i}{\hbar}\int_{\mathcal{M}}\frac{1}{2}\sqrt{|\det\mathbf{g}|}\Big{\\{}\mathbf{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{(m^{2}-i\varepsilon)c^{2}}{\hbar^{2}}\phi^{2}\Big{\\}}d^{D+1}x\right)\mathfrak{D}\phi,\end{gathered}$
(II.14)

_within the limit of $\varepsilon\to 0$._

In particular, after employing strategies A and B we are effectively worked
out the path-integral formulation of a scalar quantum field theory on a curved
space-time. In principle, a well-defined (renormalizable) quantum field theory
on curved space-time $\mathcal{M}$ is defined for a sequence of Wiener
integrable functions, say
$F^{(\mathcal{N})}(t_{N},\cdots,t_{1})=\prod_{k=1}^{N}\exp\left(\tau_{k}\int_{M}V^{(\mathcal{N})}(\phi(t_{k-1}))d\Omega\right)$,
the so-called interaction term, each of which being defined over
$C_{\mathcal{N}}$, so that the Wiener path-integrals

$\begin{gathered}I_{\mathcal{N}}=\int_{C_{\mathcal{N}}}F^{(\mathcal{N})}(t_{N},\cdots,t_{1})\big{\\{}\phi(t_{i_{1}})\cdots\phi(t_{i_{n}})\big{\\}}dW(t_{N},\cdots,t_{1})\\\
~{}~{}~{}~{}~{}~{}~{}\approx\int_{C_{\mathcal{N}}}F^{(\mathcal{N})}_{\textbf{B}}(t_{N},\cdots,t_{1})\big{\\{}\phi(t_{i_{1}})\cdots\phi(t_{i_{n}})\big{\\}}dW_{\textbf{A,B}}(t_{N},\cdots,t_{1})\end{gathered}$
(II.15)

result in finite amplitudes as $\mathcal{N}\to\infty$ (and $N\to\infty$ as an
appropriate monotone increasing function of $\mathcal{N}$ varshovi1 ) for each
$n$, wherein
$F_{\textbf{B}}^{(\mathcal{N})}(t_{N},\cdots,t_{1})=\prod_{k=1}^{N}\exp\left(-i\tau_{k}\int_{M}V^{(\mathcal{N})}(\phi(t_{k-1}))d\Omega\right)$.
As we discussed in varshovi1  employing negative renormalizable potential
terms $V^{(\mathcal{N})}$ always lead to finite and predictable expectation
values $I_{\mathcal{N}}$ as $\mathcal{N},N\to\infty$. Indeed, the former
property ensures that the energy levels are bounded from below, and the latter
one puts forward a reasonable correlation between $V^{(\mathcal{N})}$s (and
the corresponding coupling constants) at different regimes of energy labeled
with $\mathcal{N}$.101010For more details about the renormalizability of a
scalar quantum field theory within the Wiener fractal mechanism see varshovi1
. Also see holland ; dewitt ; wald1 ; wald2  for more discussions about
quantization, regularization, renormalization, Casimir effect, Bogoliubov
transformation and the $S$-matrix components of such theories. Moreover, we
refer the readers to hu  for an interesting review on the topic.

In particular, a similar mechanism to the above machinery could be employed to
work out the Maxwell and the Yang-Mills actions on curved space-time
$\mathcal{M}$ varshovi2 . Actually, for such gauge theories, one should
replace the Fourier expansion of the Laplacian operator with that of the
generalized Bochner Laplacian defined on principal bundles over the space
manifold $M$. Therefore, one may claim that the Standard Model, as the
admitted prescription for the fundamental forces of nature, including gauge
fields, fermions and the scalar Higgs boson, could be described and
interpreted fascinatingly as a Brownian process of geometric structures over
the space manifold, i.e. the universe. In the next section we show that the
gravitation effect, within the framework of general relativity, is
spectacularly worked out by evolving the cosmos geometry via an entropic
equation.

##  III Einstein-Hilbert Action and the Ricci Flow of the Space Metric

Let us revisit the symmetric Wiener measure (II.4) once again. Suppose that we
need to perform some change of variables
$F_{i}:\mathbb{R}^{D}\to\mathbb{R}^{D}$ at each intermediate time section
$t_{i}$, $1\leq i\leq N$.

$\begin{gathered}dW(t_{N},\cdots,t_{1})=\mathfrak{C}_{N}\frac{1}{\left(2\pi\tau_{N}\right)^{D/2}}e^{-|\Delta
x_{N}|^{2}/2\tau_{N}}d^{D}\Delta
x_{N}\times\cdots\times\frac{1}{\left(2\pi\tau_{1}\right)^{D/2}}e^{-|\Delta
x_{1}|^{2}/2\tau_{1}}d^{D}\Delta x_{1}\\\
=\mathfrak{C}_{N}\frac{1}{\left(2\pi\tau_{N}\right)^{D/2}}e^{-|F_{N}(\Delta
x_{N})|^{2}/2\tau_{N}}d^{D}F_{N}(\Delta
x_{N})\times\cdots\times\frac{1}{\left(2\pi\tau_{1}\right)^{D/2}}e^{-|F_{1}(\Delta
x_{1})|^{2}/2\tau_{1}}d^{D}F_{1}(\Delta x_{1})\\\
=\mathfrak{C}_{N}\frac{1}{\left(2\pi\tau_{N}\right)^{D/2}}e^{-|F_{N}(\Delta
x_{N})|^{2}/2\tau_{N}}J_{N}d^{D}\Delta
x_{N}\times\cdots\times\frac{1}{\left(2\pi\tau_{1}\right)^{D/2}}e^{-|F_{1}(\Delta
x_{1})|^{2}/2\tau_{1}}J_{1}d^{D}\Delta x_{1},~{}~{}~{}~{}\end{gathered}$
(III.1)

for $\Delta x_{i}=x_{i}-x_{i-1}$, and
$\mathfrak{C}_{N}=\left(\frac{2T}{\tau_{N+1}}\right)^{D/2}~{}\exp\left(|x_{F}-x_{I}|^{2}/4T-|x_{F}-x_{N}|^{2}/2\tau_{N+1}\right)$,111111The
contribution of $x_{N}$ in $\mathfrak{C}_{N}$ would provide an integration on
the last time section within the total action, hence it is essentially a null
set integral and can be easily removed from the calculations. where $J_{i}$ is
the Jacobian determinant of transformation $F_{i}$. We can simply employ
(III.1) for a dynamical geometry of space Riemannian manifold $M$. This
geometric evolution is absolutely independent of the topology of $M$, but
rather it would have been related to the differential geometric structures
due; hence the Riemannian metric $g$. Therefore, since the Wiener fractal
measure was worked out for the Fourier expansion in terms of the Laplacian
eigenfunctions on $(M,g)$, the time evolution of $g$ would cause some change
of variables within the measure as we considered in (III.1). The next theorem
provides a consistent framework for such transformations versus infinitesimal
variation of the metric.121212For the variation of eigenvalues and
eigenfunctions of the Laplacian (and its generalized versions such as
$(p,q)$-Laplacian and the Witten Laplacian) versus the metric evolution see
for example azami ; besse ; berger2 ; hwang ; dicerbo ; berger et al ; lic ;
hou ; fang ; cao ; elsoufi  and the references therein

Theorem 2 (ref. berger2 ); _Let $g(t)=g+th$, $|t|\ll 1$, be a small variation
of the metric on Riemannian manifold $(M,g)$, for $h$ a symmetric
$(0,2)$-tensor field on $M$. Set $\Delta^{g(t)}$ and $d\Omega_{g(t)}$ to be
respectively the Laplacian operator and the Riemann volume form due to the
metric $g(t)$. Let $\lambda$ be an eigenvalue of $\Delta=\Delta^{g(0)}$ with
degeneracy space $D_{\lambda}$ labeled by $i$. Then, there exist
$\psi_{\lambda,i}(t)\in C^{\infty}(M)$, for each
$\lambda\in\text{spec}\Delta$, and $i\in D_{\lambda}$, such that;_

a) _$\Delta^{g(t)}\psi_{\lambda,i}(t)=\lambda_{i}(t)\psi_{i}(t)$ , for each
$\lambda\in\text{spec}\Delta$ and $i\in D_{\lambda}$, with $t$, $|t|\ll 1$._

b) _$\lambda_{i}(t)$ and $\psi_{\lambda,i}(t)$ depend analytically on $t$,
$|t|\ll 1$, with $\lambda_{i}(0)=\lambda$, for each $i\in D_{\lambda}$._

c) _$\\{\psi_{\lambda,i}(t)\\}^{i\in
D_{\lambda}}_{\lambda\in\emph{spec}\Delta}$ is orthonormal with respect to
integration on $M$ with $d\Omega_{g(t)}$._

Actually, we assume that the space metric would vary with the Ricci flow:

$\frac{\partial}{\partial t}g_{ij}(t)=-2Ric_{ij}(t),$ (III.2)

for $Ric(t)$ the Ricci curvature of evolving metric $g(t)$. In fact, as we
will argue in the following, we regard the Ricci flow as an imposed natural
entropic gauge fixing term for Einstein’s field equation. One should note that
although the Einstein equation is scale invariant, i.e. is symmetric with
respect to transformation $g_{ij}\to cg_{ij}$ for some $c>0$, the Ricci flow
disobeys this property. Hence, once the scale of time dimension $t$ is fixed
(see Eq. (III.3) in below where the time scale $\zeta$ is introduced) the
Ricci flow fixes the scale of $g_{ij}$ and consequently the space volume.
Recently, the Ricci flow has attracted a lot of attention in building a well-
defined model of (topological) quantum gravity and several research have shown
some theoretical evidence for an intimate correlation between gravity and the
Ricci flow.131313See for example top1 ; top2 ; top3 ; isi ; graf ; lulli ;
woolgar ; dz ; dz2 ; dz3 ; dowker ; morison .

Actually, the Ricci flow has been established to be a very fruitful mechanism
to improve metrics in Riemannian geometry whenever $M$ is compact. Indeed, it
has been shown that in some specific conditions the Ricci flow converges to a
canonical metric hamilton1 ; hamilton2 ; hamilton3 ; huisken . For instance,
in his seminal work, Hamilton has established that on each closed 3-manifold
with positive Ricci curvature the Ricci flow converges to a canonical metric
of positive constant sectional curvature hamilton1 . In this sense, the Ricci
flow may be regarded as a natural homotopy between a given metric of positive
Ricci curvature and a canonical metric of constant sectional curvature.
Moreover, Hamilton demonstrated that in this case the Ricci flow could be
resolved within a normalized version of which the volume of the manifold is
preserved along with the metric evolution. In principle, whenever the Ricci
flow holds and admits a unique solution on $(a,b)\subset\mathbb{R}$, it
converges to the canonical metric $g_{c}$ as $t\to b$, where $g_{c}$ is
necessarily an Einstein metric, i.e. $\lim_{t\to b}Ric_{ij}(t)=K{g_{c}}_{ij}$,
for some constant $K$.141414For: $K=\frac{1}{D}\int_{M}\lim_{t\to
b}R(t)d\Omega_{g_{c}}/\int_{M}d\Omega_{g_{c}}$.

Without loss of generality we may assume that the Ricci flow exists on
$(-a,a)\subset\mathbb{R}$, for some $a>0$. Hence, by rescaling $t\mapsto t/n$,
$n\in\mathbb{N}$, the domain of the solution changes to $(-na,na)$, while on
the right hand side of (III.3) one must replace $-2$ with $-2/n$. In
principle, since we concern about the stochastic process in $(-T,T)$ for
$T\to\infty$, the integer $n$ must be chosen as large as possible. Generally,
this Ricci flow could be rescaled as

$\frac{\partial}{\partial t}g_{ij}(t)=-2\zeta Ric_{ij}(t),$ (III.3)

for some small positive constant $\zeta$. In the following we will see that as
a very tiny amplitude $\zeta$ couples to negligible terms inserted in the
Einstein-Hilbert Lagrangian density. More precisely, the Ricci flow scaling
factor could be compared with the cosmological constant $\Lambda$.

However, it can be easily seen that the variation of the Riemannian volume
form $d\Omega_{g(t)}$ with respect to $t$ is given in terms of the scalar
curvature $R(t)$:

$\frac{\partial}{\partial t}d\Omega_{g(t)}=-\zeta R(t)d\Omega_{g(t)}.$ (III.4)

The above equation lets us to work out the explicit formulation of the Wiener
fractal measure (III.1) along the Ricci flow time evolution. Initially, one
should note that we have already considered a normalization condition as:

$\int_{M}\psi_{\lambda,i}(t)\psi_{\lambda,j}(t)d\Omega_{g(t)}=\delta_{ij}.$
(III.5)

Hence, by variation of (III.5) with respect to $t=\delta t$ we obtain:

$\begin{gathered}\epsilon_{\lambda,ii}(t)=1-\frac{t\zeta}{2}\int_{M}R(t)\psi_{\lambda,i}^{2}(t)d\Omega_{g(t)},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\epsilon_{\lambda,ij}(t)+\epsilon_{\lambda,ji}(t)=-t\zeta\int_{M}R(t)\psi_{\lambda,i}(t)\psi_{\lambda,j}(t)d\Omega_{g(t)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(i\neq
j)\end{gathered}$ (III.6)

wherein
$\epsilon_{\lambda,ij}(t)=\int_{M}\psi_{\lambda,i}(0)\psi_{\lambda,j}(t)d\Omega_{g(t)}$.
Hence, we would readily set;

$\epsilon_{\lambda,ij}(t)=\delta_{ij}-\frac{t\zeta}{2}\int_{M}R(t)\psi_{\lambda,i}(t)\psi_{\lambda,j}(t)d\Omega_{g(t)}.$
(III.7)

Consequently, along the variation of $g$ the basis of $C_{\mathcal{N}}$ move
linearly by the matrix $\epsilon=\oplus_{\lambda}\epsilon_{\lambda}$,
$\lambda\in\text{spec}\Delta$, which means that the Laplace-Fourier
coefficients $f_{\lambda,i}$s must be transformed via $\epsilon^{-1}$. In
fact, in (III.1) we must consider $F_{i}$ to be block diagonal for the
spectrum of the Laplacian operator $\Delta^{g(t_{i-1})}$ as:
$F_{i}=\oplus_{\lambda}(F_{i})_{\lambda}$, with;

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(F_{i})_{\lambda,~{}mn}=\delta_{mn}+\frac{\tau_{i}\zeta}{2}\int_{M}R(t_{i})\psi_{\lambda,m}(\tau_{i})\psi_{\lambda,n}(\tau_{i})d\Omega_{g(t)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1\leq
i\leq N)$ (III.8)

for $m$ and $n$ running through the degenerate eigenfunctions of
$\Delta^{g(t_{i-1})}$ with common eigenvalue $\lambda$. Indeed, one can simply
consider (III.8) as: $F_{i}=1+r_{i}$, $i=1,\cdots,N$, where we easily gain;

$\begin{gathered}J_{i}=\det
F_{i}=\exp\left(tr\Big{\\{}\ln(1+r_{i})\Big{\\}}\right)\approx\exp\left(tr\\{r_{i}\\}\right)\\\
=\exp\left(\sum_{{\lambda\in\text{spec}\Delta^{g(t_{i-1})}}}\sum_{m\in
D_{\lambda}}\frac{\tau_{i}\zeta}{2}\int_{M}R(t_{i})\psi^{2}_{\lambda,m}(\tau_{i})d\Omega_{g(t_{i})}\right).\end{gathered}$
(III.9)

Actually, incorporating the full expression of the Jacobian determinant
(III.9) in the Wiener fractal measure (III.1) leads to a highly complicated
formulation for stochastic process of fractal functions on $M$. Therefore, we
have to consider our first approximation of the Wiener fractal measure as:

Approximation I: _The trace operator of
$\sum_{{\lambda\in\text{spec}\Delta^{g(t_{i-1})}}}\sum_{m\in D_{\lambda}}$ in
the exponent of $J_{i}$ is in fact approximated by the dominating leading term
$\sum_{\lambda=0}\sum_{m\in D_{0}}$. However, since $M$ is connected we have
$\dim H^{1}_{dR}(M,\mathbb{R})=1$, hence there is no degeneracy ($\dim
D_{0}=1$) and consequently the only considered eigenfunction is a constant. On
the other hand, upon the normalization condition (III.5) we obtain
$\psi_{0}=1/\sqrt{V_{i}}$, wherein $V_{i}$ is the volume of $M$ for metric
$g(t_{i})$; i.e. $V_{i}=\int_{M}d\Omega_{g(t_{i})}$. Therefore, we readily
obtain:_

$J_{i}=\exp\left(\frac{\tau_{i}\zeta}{2V_{i}}\int_{M}R(t_{i})d\Omega_{g(t_{i})}\right).$
(III.10)

The second approximation concerns about the volume of the space manifold $M$.
According to (III.4) one can simply compute the derivation of this volume with
respect to $t$ as:

$\frac{dV(t)}{dt}=-\zeta\int_{M}R(t)d\Omega_{g(t)}.$ (III.11)

Since $\zeta\approx 0$ we could assume that within a specific period of time
$V(t)$ evolve slightly, hence is almost constant. For instance, for fixed
$t_{0}\in[-T,T]$, say the present age of the universe, we compute:

$V_{i}=V-\zeta\int_{\mathcal{M}_{t_{0}}^{t_{i}}}R(t)dt\wedge
d\Omega_{g(t)}\approx V-(t-t_{0})\zeta\int_{M}R(t_{0})d\Omega_{g(t_{0})},$
(III.12)

with $V=V(t_{0})$, and $\mathcal{M}_{t_{0}}^{t_{i}}=[t_{0},t_{i}]\times M$ for
$t_{i}>t_{0}$ (resp. $\mathcal{M}_{t_{0}}^{t_{i}}=[t_{i},t_{0}]\times M$ for
$t_{i}<t_{0}$). Thus;

$\frac{1}{V_{i}}\approx\frac{1}{V}+\frac{(t-t_{0})\zeta}{V^{2}}\int_{M}R(t_{0})d\Omega_{g(t_{0})}+\mathcal{O}(\zeta^{2}/V^{3})\approx\frac{1}{V}.$
(III.13)

Hence, the next approximation is reasonable:151515Indeed, as we see in the
following the space volume $V$ is effectively equivalent to the Newton’s
gravitation constant $G$. In principle, the dynamics of the space volume is
formulated with a double integral on $M\times M$, hence is due to a non-local
contribution of the scalar curvature of $M$ within the Einstein-Hilbert
action. However, the slight variation of $G$ has already been formulated by
means of a local interacting scalar field theory (i.e. dilaton) via the Brans-
Dicke theory brans-dicke . See also weinberg  for more discussions.

Approximation II: _We may also assume that the variation of $V_{i}$ along the
Ricci flow is negligible during the period of time in which we concern the
Brownian stochastic evolution of quantum fields. Hence, we would simply
consider $V_{i}=V$, $1\leq i\leq N$, and write the Jacobian determinant
(III.10) as:_

$J_{i}=\exp\left(\frac{\tau_{i}\zeta}{2V}\int_{M}R(t_{i})d\Omega_{g(t_{i})}\right).$
(III.14)

In particular, upon approximations I and II one can easily extract the
contribution of the Jacobian determinants $J_{i}$s, $1\leq i\leq N$, within
the Wiener fractal measure (III.1) as:

$\prod_{i=1}^{N}J_{i}=\exp\left(\frac{1}{2\xi}\sum\tau_{i}\int_{M}R(t_{i})d\Omega_{g(t_{i})}\right),$
(III.15)

for $\xi=V/\zeta$, which in the limit of $N\to\infty$ (i.e. $\tau_{i}=2T/N\to
0$) leads to:

$\prod_{i=1}^{N}J_{i}=\exp\left(\frac{1}{2\xi
c}\int_{\mathcal{M}}R~{}d\Omega\right),$ (III.16)

wherein $\mathcal{M}=[-T,T]\times M$ is the space-time continuum introduced
above, and $d\Omega=cdt\wedge d\Omega_{g(t)}$ is the Riemannian volume form
for the Lorentzian metric $\mathbf{g}=\mathbf{g}_{\mu\nu}dx^{\mu}\otimes
dx^{\nu}=c^{2}dt\otimes dt\oplus\left(-g_{ij}(t)dx^{i}\otimes dx^{j}\right)$
on $\mathcal{M}$. We should note that according to the rescaled Ricci flow
(III.3) the scalar curvature $R$ of $g$ is in essential equal to the scalar
curvature $\mathcal{R}$ of $\mathbf{g}$ up to an overall minus sign and some
negligible terms of order $\mathcal{O}(\zeta)$. In fact, upon to (III.3) we
see:

$\mathcal{R}=-R+\frac{\zeta}{c^{2}}\Big{\\{}g^{ij}\frac{\partial
Ric_{ij}}{\partial t}+\frac{\partial R}{\partial
t}\Big{\\}}-\frac{\zeta^{2}}{c^{2}}Ric_{ij}Ric^{ij}+\frac{\zeta^{2}}{c^{2}}R^{2}=-R+\mathcal{O}(\zeta).$
(III.17)

Here, we may set our third approximation law:

Approximation III: _We would simply replace $R$ by $-\mathcal{R}$ in the
Jacobian determinant $J_{i}$ and ignore the remaining terms of order
$\mathcal{O}(\zeta)$, i.e.:_

$\prod_{i=1}^{N}J_{i}=\exp\left(-\frac{1}{2\xi
c}\int_{\mathcal{M}}\mathcal{R}~{}d\Omega\right).$ (III.18)

In principle, the above three approximations would provide a sufficient set of
assumptions to work out the original formulation of general relativity via the
Einstein-Hilbert action. Actually, in a local coordinate system on the space-
time continuum $\mathcal{M}$, say $(t,x^{i})$, we would obtain the following
formula for the Jacobian term (III.18):

$\prod_{i=1}^{N}J_{i}=\exp\left(-\frac{1}{2\xi
c}\int_{\mathcal{M}}\sqrt{|\det\mathbf{g}|}~{}\mathcal{R}~{}dt\wedge
dx^{1}\wedge\cdots\wedge dx^{D}\right).$ (III.19)

Hence, the Wiener stochastic process of fractals on Riemannian manifold $M$
along with the Ricci flow automatically involves the Einstein-Hilbert action
of general relativity within the approximations I, II, and III. In fact, the
emergence of general relativity is merely due to the evolution of the cosmos
geometry, hence having nothing to do with possible background interactions of
the included matter fields. However, one should note that the analytic
structure of the spatial metric $g_{ij}$ fulfills Einstein’s field equation
via the Euler-Lagrange equations. Actually, $\mathbf{g}_{\mu\nu}$ must obey
the Ricci flow as an intrinsic property of the space-time geometry.

Following the machinery of the previous section, it can be easily seen that
the rest parts of the Wiener fractal measure on the dynamical Riemannian
manifold $(M,g)$ would be:

$\exp\left(-\frac{1}{\hbar}\int_{\mathcal{M}}\frac{1}{2}\Big{\\{}\mathbf{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{m^{2}c^{2}}{\hbar^{2}}\phi^{2}\Big{\\}}d\Omega\right)\mathfrak{D}\phi.$
(III.20)

wherein the strategy A is considered due. One should note that there is an
essential difference between (II.14) and (III.20); In the former, the space-
time metric $\mathbf{g}$ is a static tensor, but for the latter it evolves via
the Ricci flow. Moreover, in the latter the Feynman measure $\mathfrak{D}\phi$
is given in terms of the Fourier expansion of the Laplacian operator
$\Delta^{g(t_{i-1})}$ at the $i$-th time section, while in the former the
Feynman measure is static. All in all, the Wiener fractal measure along with
the Ricci flow and after the mentioned three approximations becomes:

$\begin{gathered}dW(t_{N},\cdots,t_{1})\approx\exp\left(-\frac{1}{\hbar}\int_{\mathcal{M}}\Big{\\{}\frac{1}{2\kappa}\sqrt{|\det\mathbf{g}|}~{}\mathcal{R}+\mathcal{L}_{K-G}(\phi)\Big{\\}}dx^{D+1}\right)\mathfrak{D}\phi,\end{gathered}$
(III.21)

with
$\mathcal{L}_{K-G}(\phi)=\frac{\sqrt{|\det\mathbf{g}|}}{2}\big{\\{}\mathbf{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{m^{2}c^{2}}{\hbar^{2}}\phi^{2}\big{\\}}$,
the Klein-Gordon Lagrangian density, and $\kappa=\xi c/\hbar$. Hence, we
readily obtain the Newton’s gravitational constant in terms of fractal,
geometric and physical parameters in $D=3$ dimensions: $G=\xi
c^{4}/8\pi\hbar$. Therefore, if we use the standard values of $G$, $\hbar$ and
$c$ we obtain: $\xi=V/\zeta=2.1899700\times 10^{-77}~{}(m.s)$, which shows
that the _cosmos entropic volume_ (i.e. the cosmos volume for the metric
involved in the Ricci flow) is extremely small.

One should note that this is not a problem at all, since the small amount of
the cosmos entropic volume $V$ is, in fact, equivalent to the large value of
the Einstein-Hilbert coefficient (i.e. $1/2\kappa=c^{4}/16\pi G$) in the SI
units. Actually, the small amount of $V$ stems from the smallness of the space
metric entries $g_{ij}$s, and hence the huge amplitude of the scalar curvature
$\mathcal{R}$. Hence, if we rescale the entropic metric $g_{ij}$ with some
large constant coefficient $\varsigma$ to gain an enormously large universe we
readily obtain the standard form of Einstein-Hilbert action with a large
coefficient as we expected before. In fact, in this case we have:
$g_{ij}\to\varsigma g_{ij}$, $\mathcal{R}\to\varsigma^{-1}\mathcal{R}$, and
$d\Omega\to\varsigma^{D/2}d\Omega$, and thus the Einstein-Hilbert action will
be multiplied by $\sqrt{\varsigma}$ in $D=3$ dimensions. In particular, the
extremely small amount of the cosmos entropic volume is an immediate
consequence of the Ricci flow and could be regarded as the main reason of the
hierarchy problem.161616See hier  and the references therein for interesting
discussions about the relation of hierarchy problem and the Bayesian
statistics, which could be related to our achievements in this paper.

To work out a well-defined theory from Wiener fractal measure (III.21) we must
employ the strategy B once again. We readily obtain:

$\begin{gathered}dW(t_{N},\cdots,t_{1})\approx\exp\left(\frac{i}{\hbar}\int_{\mathcal{M}}\Big{\\{}\frac{1}{2\kappa}\sqrt{|\det\mathbf{g}|}~{}\mathcal{R}+\mathcal{L}_{K-G}(\phi)+\sqrt{|\det\mathbf{g}|}~{}\frac{i\varepsilon}{2}\phi^{2}\Big{\\}}dx^{D+1}\right)\mathfrak{D}\phi,\end{gathered}$
(III.22)

for $\varepsilon\to 0$. Therefore, one can consider the Wiener fractal process
(III.1) for the Lebesgue fractal measure (II.2) through with the Ricci flow as
the more fundamental formulation of quantum field theory in dynamical curved
spacetime, i.e. in presence of semi-classical gravity. Hence, we would claim
that the basic source of the Einstein’s theory of gravity stems from the
dynamical Wiener fractal measure via the Ricci flow. We refer to this
stochastic derivation of gravitational effect as the _Wiener fractal gravity_.
As we discussed in last section, although including non-local terms the Wiener
fractal gravity leads to finite predictable expectation values

$\lim_{\mathcal{N},N\to\infty}\int_{C_{\mathcal{N}}}F^{(\mathcal{N})}(t_{N},\cdots,t_{1})\big{\\{}\phi(t_{i_{1}})\cdots\phi(t_{i_{n}})\big{\\}}dW(t_{N},\cdots,t_{1})$
(III.23)

for
$F^{(\mathcal{N})}(t_{N},\cdots,t_{1})=\prod_{k=1}^{N}\exp\left(\tau_{k}\int_{M}V^{(\mathcal{N})}(\phi(t_{k-1}))d\Omega_{g(t_{k-1})}\right)$
with negative (bounded from above) and renormlizable interaction terms
$V^{(\mathcal{N})}(\phi(t_{i}))$. This ensures the implication of the Wiener
fractal gravity as a more fundamental formulation of quantum field theory in
presence of semi-classical gravity. Through the next two sections, we try to
study the theoretical and empirical properties of the Wiener fractal gravity.

##  IV Ricci Flow, Cosmological Constant and $\Lambda$CDM Model

Upon the Hamilton’s celebrated theorem we may consider the metric of the
universe to be almost Einstein with $Ric_{ij}(t)=Kg_{ij}(t)+\varepsilon_{ij}$,
for infinitesimal $\varepsilon_{ij}\ll Kg_{ij}$, and $t>t_{0}$, where $t_{0}$
is considered as the current age of the universe.171717In fact, according to
experimental achievements $K$ has a very tiny amplitude; i.e. $K\approx 0$.
Indeed, the experimental data from different laboratories (for example due to
WMAP, BOOMERanG, and Planck) confirm that the universe manifold $M$ is flat
with about $0.4$ percent margin of deviation yoo . See also dem ; levin . With
this assumption, the Ricci flow is;

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{\partial
g_{ij}(t)}{\partial t}=-2\zeta
Kg_{ij}(t)-2\zeta\varepsilon_{ij},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}t>t_{0}.$
(IV.1)

which has a unique solution as:181818We may temporary assume that
$\varepsilon_{ij}$ is constant. Indeed, we suppose that $\varepsilon_{ij}$
varies slightly through the cosmological time $t$.

$g_{ij}(t)=e^{-2\zeta
K(t-t_{0})}g_{ij}(t_{0})-\frac{\varepsilon_{ij}}{K}~{}~{}~{}~{}~{}\text{and;}~{}~{}~{}~{}~{}Ric_{ij}(t)=Ke^{-2\zeta
k(t-t_{0})}g_{ij}(t_{0}).$ (IV.2)

Hence, the higher order terms of (III.17) would become;

$\begin{gathered}\frac{\zeta}{c^{2}}\Big{\\{}g^{ij}\frac{\partial
Ric_{ij}}{\partial t}+\frac{\partial R}{\partial
t}\Big{\\}}-\frac{\zeta^{2}}{c^{2}}Ric_{ij}Ric^{ij}+\frac{\zeta^{2}}{c^{2}}R^{2}\\\
=(D-3)Dc^{-2}\zeta^{2}K^{2}+2(D-3)\zeta^{2}c^{-2}Ke^{-2\zeta
k(t-t_{0})}tr\\{\varepsilon g_{0}\\}+2\Lambda,\end{gathered}$ (IV.3)

for $g_{0}=g(t_{0})$, and the cosmological constant $\Lambda$;191919Actually,
$\Lambda$ is not a constant in general, but upon to Hamilton’s theorem
hamilton1  for some appropriate interval of time it evolve slightly on $M$,
hence could be considered as a constant at the first order of approximation.

$\Lambda=\frac{1}{2}\zeta^{2}c^{-2}e^{-4\zeta
K(t-t_{0})}\Big{\\{}\left(tr\\{\varepsilon
g_{0}\\}\right)^{2}-tr\\{\left(\varepsilon g_{0}\right)^{2}\\}\Big{\\}}.$
(IV.4)

Actually, upon Hamilton’s theorem hamilton1 , the deviation matrix
$\varepsilon$ is positive definite for $D=3$ dimensions, hence the
cosmological constant $\Lambda$ is a positive number on closed 3-manifolds. On
the other hand, we see surprisingly that in $D=3$ dimensions the constant
component and the $\varepsilon$-linear term of (IV.3) vanish and we are left
with only the cosmological constant $\Lambda\propto e^{-4\zeta
Kt}\zeta^{2}O(\varepsilon^{2})$ which is in essential an extremely small
amplitude. Thus, on three-dimensional space $M$ we have:
$R=\mathcal{R}-2\Lambda$, and consequently the cosmological term $-2\Lambda$
is simply inserted beside the scalar curvature $\mathcal{R}$ within the
Einstein-Hilbert Lagrangian density:

$\prod_{i=1}^{N}J_{i}=\exp\left(-\frac{1}{2\xi
c}\int_{\mathcal{M}}\sqrt{|\det\mathbf{g}|}~{}\left(\mathcal{R}-2\Lambda\right)~{}dt\wedge
dx^{1}\wedge\cdots\wedge dx^{D}\right).$ (IV.5)

Therefore, according to the above calculations the cosmological constant
emerges only upon the matric evolution of the space manifold, hence
independent from the matter fields and the zero-point energy fluctuations of
quantum field theory. In principle, the cosmological constant $\Lambda$ has an
entropic source due to the Ricci flow of the universe three-dimensional
geometry, but not an inevitable consequence of matter fields. Therefore, it
seems that the Wiener fractal gravity would give a reliable consistent answer
to the cosmological constant problem hobson ; weinberg-cos  and the nature of
dark energy.202020See cai  for another entropic viewpoint to the cosmological
constant via the holographic approach in quantum gravity.

Moreover, for almost the canonical asymptotic (limit) Einstein metric $g_{c}$
of the Ricci flow, i.e. $Ric_{ij}\approx K{g_{c}}_{ij}$, we readily obtain an
inflating universe $M$. To see this one has to employ the Jacobi field
equation for Einstein metric $g_{c}$. Let $\gamma_{s}(t)$ be a continuous
family of geodesics on $\mathcal{M}$ labeled by $s\in\mathbb{R}$, each of
which with initial condition $\dot{\gamma}_{s}(0)=\partial/\partial t$. Then,
according to the Jacobi field equation we read for small enough $t$:

$\frac{D^{2}}{dt^{2}}J(t)=\mathcal{R}\left(\dot{\gamma}_{s}(t),J(t)\right)\dot{\gamma}_{s}(t)\approx{\mathcal{R}_{tit}}^{j}J^{i}(t)\frac{\partial}{\partial
x^{j}},$ (IV.6)

wherein $J(t)=\frac{\partial\gamma_{s}(t)}{\partial s}$ and $D/dt$ is the
covariant derivative along $t$, while ${\mathcal{R}_{\mu\nu\sigma}}^{\lambda}$
is the space-time Riemannian curvature computed for the metric
$\mathbf{g}=c^{2}dt\otimes dt\oplus(-g_{c})$.212121Here, we assume for the
coordinate index $\mu$ to vary among $\\{t,1,\cdots,D\\}$. Let us compute the
relative acceleration of inertial celestial bodies due to the space-time
metric $\mathbf{g}$. We simply set $J(0)=r\frac{\partial}{\partial x^{i}}$,
assuming the observed galaxies are located along the $i$-th direction of the
local coordinate system $(x^{\mu})=(x^{0}=t,x^{j}=x^{1},\cdots,x^{D})$. We
compute the relative acceleration as:222222Upon the Ricci flow (III.3) we
have: $\Gamma^{0}_{ij}=\zeta Ric_{ij}$,
$\Gamma^{i}_{0j}=\Gamma^{i}_{j0}=-\zeta{g_{c}}^{im}Ric_{mj}$, and
$\Gamma^{0}_{0i}=\Gamma^{0}_{i0}=\Gamma^{i}_{00}=\Gamma^{0}_{00}=0$.

$\begin{gathered}a_{r}=a_{r}^{\mu}\frac{\partial}{\partial
x^{\mu}}={\mathcal{R}_{0i0}}^{\mu}\frac{\partial}{\partial
x^{\mu}}=r\Bigg{\\{}\frac{\partial}{\partial
t}\Gamma^{\mu}_{ti}-\frac{\partial}{\partial
x^{i}}\Gamma^{\mu}_{tt}+\Gamma^{\mu}_{t\nu}\Gamma^{\nu}_{it}-\Gamma^{\mu}_{i\nu}\Gamma^{\nu}_{tt}\Bigg{\\}}\frac{\partial}{\partial
x^{\mu}}=\zeta^{2}K^{2}r\frac{\partial}{\partial x^{i}},\end{gathered}$ (IV.7)

which is in fact equal to:232323Here we assume $t^{\prime}$ to be close enough
to the ultimate time for the Ricci flow to reach to the Einstein’s metric
$g_{c}$ up to the extent of the experimental and observational accuracy.

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\ddot{r}(t)=H_{\infty}^{2}r(t),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}t>t^{\prime},$
(IV.8)

for radial distance $r(t)$ and _ultimate Hubble’s constant_
$H_{\infty}=\zeta|K|$.242424Although, according to the Hamilton’s theorem
hamilton1  $K$ is considered to be positive, here we prefer to employ its
absolute value to involve Einstein manifolds with negative curvature as is
considered in the FLRW metric. In principal, (IV.8) has a unique dominating
solution for $t\to\infty$:

$r(t)=r_{0}e^{H_{\infty}(t-t^{\prime})}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\to~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\dot{r}(t)=H_{\infty}r(t).$
(IV.9)

These conclusions could be slightly modified via the assumption of (IV.1) for
the present age of the universe $t_{0}$ and the upcoming future of the
cosmological time $t>t_{0}$. In this case we obtain the accurate form of
(IV.7) as:252525As we see once again the $\varepsilon$-linear term cancels out
automatically.

$a_{r}=a_{r}^{\mu}\frac{\partial}{\partial
x^{\mu}}=\zeta^{2}K^{2}r\frac{\partial}{\partial
x^{i}}+2\zeta^{2}\varepsilon_{jm}\varepsilon_{mi}r\frac{\partial}{\partial
x^{j}}+\zeta^{2}e^{4K\zeta(t-t_{0})}{A^{j}_{0}}_{k}{A^{k}_{0}}_{i}r\frac{\partial}{\partial
x^{j}}+\mathcal{A}^{j}_{i}(\dot{\varepsilon})r\frac{\partial}{\partial
x^{j}}+\mathcal{O}(\varepsilon^{3}),$ (IV.10)

with ${A^{j}_{0}}_{i}=g_{0}^{jm}\varepsilon_{mi}$ and

$\mathcal{A}^{j}_{i}(\dot{\varepsilon})=-\zeta
e^{2K\zeta(t-t_{0})}g_{0}^{jm}\dot{\varepsilon}_{mi}+\frac{1}{K}\left(\dot{\varepsilon}_{jm}\varepsilon_{mi}+{\varepsilon}_{jm}\dot{\varepsilon}_{mi}\right).$
(IV.11)

Indeed, the dominating diagonal term of (IV.11) (for
$\varepsilon_{ij}\approx\varepsilon\delta_{ij}$ and
$g_{0}^{ij}\approx\delta^{ij}$) is:

$a_{r}=a_{r}^{\mu}\frac{\partial}{\partial
x^{\mu}}=r\Bigg{\\{}\zeta^{2}K^{2}+\zeta^{2}{\varepsilon^{2}}\left(2+De^{4K\zeta(t-t_{0})}\right)-\zeta\dot{\varepsilon}\left(e^{2K\zeta(t-t_{0})}+\frac{2\varepsilon}{K}\right)\Bigg{\\}}\frac{\partial}{\partial
x^{i}},$ (IV.12)

which leads to a time-dependent Hubble’s parameter;

$H(t)=H_{\infty}\sqrt{1+\frac{\varepsilon^{2}}{K^{2}}\left(2+De^{4K\zeta(t-t_{0})}\right)+\frac{|\dot{\varepsilon}|}{\zeta
K^{2}}\left(e^{2K\zeta(t-t_{0})}+\frac{2\varepsilon}{K}\right)},$ (IV.13)

since upon the Hamilton’s theorem $\dot{\varepsilon}<0$ hamilton1 . Set

$\Omega_{1}(t)=\frac{\varepsilon^{2}}{K^{2}}\left(2+De^{4K\zeta(t-t_{0})}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Omega_{2}(t)=\frac{|\dot{\varepsilon}|}{\zeta
K^{2}}\left(e^{2K\zeta(t-t_{0})}+\frac{2\varepsilon}{K}\right).$ (IV.14)

Thus, we gain approximately:262626We are working in a period of time
$[t_{0},T]$ in which $\Omega_{i}(t_{0})\Delta\Omega_{j}(t)\ll\Omega_{k}(t)$,
with $t_{0}\leq t\leq T$, for $i,j,k=1,2$.

$H(t)=H_{0}\sqrt{1+\Delta\Omega_{1}(t)+\Delta\Omega_{2}(t)},$ (IV.15)

for $\Delta\Omega_{i}(t)=\Omega_{i}(t)-\Omega_{i}(t_{0})$, $i=1,2$, and the
Hubble’s constant $H_{0}$, given as;

$H_{0}=H_{\infty}\sqrt{1+\Omega_{1}(t_{0})+\Omega_{2}(t_{0})}.$ (IV.16)

All in all, we consequently find out some major results:

a) _The universe is expanding radially, hence radial velocities must be
observed among extra-galactic nebulae and the amplitudes of these velocities
must be proportional to distances from the fiducial observer, just as Hubble
pointed out in his seminal 1929 paperhubble ._

b) _The acceleration is positive, hence galaxies flee from each other and the
universe is inflating with an increasing rate of;_ 272727One should note that
the inflation of the universe does not mean that the volume $V(t)$ of $M$ is
increasing along the Ricci flow. Indeed, according to (III.4) the volume of
$M$ for Einstein metric $g_{c}$ changes with the rate of $dV(t)/dt=-\zeta
KDV(t)$. Hence, for positive $K$ the volume is decreasing through the Ricci
flow.

$\frac{\dot{a}}{a}=H_{0}\sqrt{1+\Delta\Omega_{1}(t)+\Delta\Omega_{2}(t)},$
(IV.17)

_where $a(t)=\left(1+z\right)^{-1}$ is the scale factor for observed redshift
$z$._282828See mccann  as an intersting research which shows that the root-
mean-square distance of two density volume terms $\omega_{1}(t)\geq 0$ and
$\omega_{2}(t)\geq 0$ evolving with the heat (diffusion) equation
$\frac{\partial}{\partial t}\omega_{i}=\Delta^{g(t)}\omega_{i}$, $i=1,2$, is
non-increasing along the backward Ricci flow $\frac{\partial}{\partial
t}g_{ij}=2Ric_{ij}$. Indeed, this would lead to an expansion theorem along the
Ricci flow for density fields via transformations $t\to-t$. We regard this
conclusion as an entropic force, which obviously include the ordinary pressure
in thermodynamics. This result, in principle, establishes the entropic root of
the Ricci flow in its own turn.

c) _According to the arguments of the next section for similarities between
the second law of thermodynamics and Wiener fractal gravity, hereby we
conclude that the dark energy (and the negative pressure due) stems from an
entropic source due to the Ricci flow._ 292929See cai ; chang  for more
discussions about the entropic source of inflation.

Indeed, the fundamental assumptions of Wiener fractal gravity resembles the
well-understood Friedmann-Lamitre-Robertson-Walker solution (besides the
assumed properties of homogeneity and isotropy), for which the Einstein’s
field equations are only mandatory to work out the scale factor of the
universe metric $a(t)$ as a function of cosmological time $t$. This provides
more common results between Wiener fractal gravity and the $\Lambda$CDM model.
In particular, for the space metric of the FLRW solution we have:303030For
example one can read the equality for the FLRW spatial metric in reduced-
circumference polar coordinates
$g_{ij}=a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\right)$. $Ric_{ij}=-2kg_{ij}$; hence $g_{ij}$ is an Einstein
metric which is the promised limit metric $g_{cij}$ of the Ricci flow (for
$K=-2k$):

$Ric_{ij}=Kg_{ij}$ (IV.18)

Thus, upon the above arguments the FLRW metric would describe the universe
with conclusions in an admissable agreement with the Wiener fractal gravity.

However, if one readily imposes the Ricci flow on the FLRW metric we simply
obtain a first order differential equation (for $k=1$) for the scale factor
$a(t)$ as; $\dot{a}(t)=2\zeta a(t)=H_{\infty}a(t)$, with the immediate
solution $a(t)=e^{H_{\infty}(t-t_{0})}$, which is in agreement with (IV.9).
Nevertheless, upon the Hamilton’s theorem for 3-manifolds hamilton1 , the
Einstein metric is an ultimate (limit) solution of the Ricci flow, hence the
FLRW metric is indeed an approximate solution for the Wiener fractal gravity.
Anyway, when the Ricci flow is employed this approximation leads to simple
equalities of $\frac{\dot{a}}{a}=H_{\infty}$ and
$\frac{\ddot{a}}{a}=H_{\infty}^{2}$, hence the Friedmann equations turn to:

$H_{\infty}^{2}+\frac{kc^{2}}{a^{2}}-\frac{\Lambda c^{2}}{3}=\frac{8\pi
G}{3}\rho,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{and;}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}H_{\infty}^{2}+\frac{kc^{2}}{3a^{2}}-\frac{\Lambda
c^{2}}{3}=-\frac{8\pi G}{3c^{2}}p,$ (IV.19)

which result in an immediate approximate equality;

$\rho+\frac{p}{c^{2}}=\frac{kc^{2}}{4\pi G}e^{-2H_{\infty}(t-t_{0})}.$ (IV.20)

Combining (IV.20) with313131This equation is derived from the Ricci flow and
Friedmann’s equation: $\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}\left(\rho+\frac{3p}{c^{2}}\right)+\frac{\Lambda c^{2}}{3}$.

$\rho+\frac{3p}{c^{2}}=\frac{\Lambda c^{2}}{4\pi
G}-\frac{3H_{\infty}^{2}}{4\pi G}$ (IV.21)

leads to explicit albeit approximate solutions for the density $\rho(t)$ and
the pressure $p(t)$ as:

$\begin{gathered}\rho(t)=\frac{1}{8\pi
G}\Big{\\{}3H_{\infty}^{2}+c^{2}\left(3ke^{-2H_{\infty}(t-t_{0})}-\Lambda\right)\Big{\\}},\\\
p(t)=\frac{c^{2}}{8\pi G}\Big{\\{}c^{2}\left(\Lambda-
ke^{-2H_{\infty}(t-t_{0})}\right)-3H_{\infty}^{2}\Big{\\}}.\end{gathered}$
(IV.22)

Before closing this section we prefer to return to (IV.15) to have a brief
look at the Hubble’s parameter $H(t)$. It is well-known that the Friedmann
equation is rewritten in terms of the density parameters as

$H(t)=H_{0}\sqrt{\Omega_{mat}a^{-3}+\Omega_{rad}a^{-4}+\Omega_{k}a^{-2}+\Omega_{\Lambda}a^{-3(1+\omega)}},$
(IV.23)

wherein $\Omega_{mat}$ is the matter density including baryons and the cold
dark matter, $\Omega_{rad}$ is for radiation including
neutrinos,323232Actually, the radiation has a negligible amplitude fri . See
also peebles  as a nice presentation of the issue. $\Omega_{k}$ is due to the
curvature and $\Omega_{\Lambda}$ is the contribution of dark energy with the
equation of state parameter $\omega$. To impose the Ricci flow on Friedmann
equations for positive curvature $k$ (i.e. $K<0$ in (IV.18)), one simply reads
from (IV.14); $\Delta\Omega_{1}\propto\varepsilon^{2}a^{-4}$ and
$\Delta\Omega_{2}\propto\dot{\varepsilon}a^{-2}$. We should note that the
matter (including the cold dark matter) and the radiation contributions are
encoded in $\varepsilon$ due to Einstein’s field equation. Therefore, the
above argument leads to an agreement between the $\Lambda$CDM model formula
(IV.23) and the equation (IV.15) which we obtained for the Wiener fractal
gravity.

##  V Discussion: Ricci Flow, Thermodynamics and Gravity

According to the Perelman’s seminal paper perelman  the Ricci flow is an
entropic based differential equation. In fact, employing an appropriate
diffeomorphism $\varphi:M\to M$, it can be seen that the Ricci flow is the
gradient of Perelman’s $\mathcal{F}$-entropy (i.e.
$\mathcal{F}(g_{ij}(t),f(t))=\int_{M}\left(R+|\nabla
f|^{2}\right)e^{-f}d\Omega_{g(t)}$) via a volume preserving variation of the
metric $g_{ij}(t)$.333333For more discussion about the appropriate
diffeomorphism see deturck  and hamilton4 . The entropy functional
$\mathcal{F}$ has a natural interpretation in terms of Bochner-Lichnerovicz
formulas which we will use to formulate the Yang-Mills gauge theories within
the Wiener fractal measure in an independent paper varshovi2 . However, from
the theoretical physics insights, $\mathcal{F}$ and its first variation
describes the low energy effective action in the string theory, wherein $f$ is
called dilaton field.343434See for example ts , where the author shows that
the generalization of the Perelman’s $\mathcal{F}$-entropy to all loop orders
equals to minus the central charge at the fixed points, in agreement with the
general claim of the Zamolodchikov’s c-theorem.

On the other hand, Perelman’s generalized entropy functional, the so called
Perelman’s $\mathcal{W}$-entropy (i.e.
$\mathcal{W}(g_{ij}(t),f(t),\tau)=\frac{1}{\left(4\pi\tau\right)^{n/2}}\int_{M}\left(\tau\left(R+|\nabla
f|^{2}\right)+f-D\right)e^{-f}d\Omega_{g(t)}$) has a specific interpretation
as the entropy of a canonical ensemble in statistical mechanics perelman
.353535Perelman’s $\mathcal{W}$-entropy also describes the RG flow in quantum
field theories. See also li-entropy ; li-entropy2  for more discussions due.
Thus, it seems that considering the Ricci flow for the evolution of the space
geometry stem from an entropic source, which give rise to the reminiscence of
thermodynamic interpretation of gravity. In principle, now a day there is no
doubt that classical (and semi-classical) general relativity is intimately
correlated to thermodynamics; there are already enough theoretical evidences
which lead to a thermodynamical interpretation of the Einstein’s theory of
gravity pad ; visser ; jacobson .363636The correlation between thermodynamics
and general relativity was first put forward by Bekenstein and Hawking via
studying the black hole entropy bekenstein ; hawking . Moreover, as we
explained above, the Wiener fractal measure is based on the Brownian motion
which is in its turn subject to an entropic force neu .

Thus, one may suspect that the Wiener fractal gravity has a lot of properties
in common with Verlinde’s entropic gravity verlinde11 ; verlinde16 . However,
there is a profound distinction between these two approaches to gravitational
force. In particular, the entropic source of the Wiener fractal gravity is due
to the evolution of the space geometry via the Ricci flow,373737Note that we
preserve the topology of the space manifold $M$ along the Ricci flow.
regardless of the existence of matter fields involved in the space-time
continuum $\mathcal{M}$, whereas the entropic source of the Verlinde’s theory
is solely due to the (quantum) matter fields interactions. Actually, the
latter is based on a combination of Jacobson’s thermodynamical viewpoint to
gravity with ’t Hooft’s holographic principle and interpret gravity as a
consequence of the information associated with the positions of material
bodies. Hence, Verlinde’s theory claims that gravity is not in fact a
fundamental interaction, but an emergent phenomenon.

On the other hand, as we established by working out the Wiener fractal measure
for evolving metric $g_{ij}(t)$, the Einstein-Hilbert action has an entropic
origin. Nevertheless, this entropic source is independent of the interaction
of matter fields and the information of material positions. Actually, although
the space metric $g_{ij}$ (or the space-time metric $\mathbf{g}_{\mu\nu}$)
fits with the distribution of matter fields and their interactions via the
semi-classical solution of Einstein’s field equation, its time evolution has
an entropic essence due to the Ricci flow. Therefore, despite their seemingly
common interpretation based on entropic origin, the Wiener fractal gravity is
fundamentally different from Verlinde’s theory of entropic gravity.

Moreover, including squared terms of time derivation, the Wiener fractal
measure is essentially symmetric with respect to transformation $t\to-t$, but,
however, imposed for evolving the geometry of the background manifold $M$, the
Ricci flow is in fact asymmetric versus reversing the time direction, hence is
the only _time’s arrow-dependent_ term (i.e. entropic term) included in the
Wiener fractal measure. Therefore, one may readily regard the second law of
thermodynamics (hence all entropic forces) as an immediate aspect of the Ricci
flow of the cosmos geometry.

Actually, the most significant strategy to work out the Einstein-Hilbert
action in the Wiener fractal measure is due to the approximation I which let
us restrict the trace of the Ricci transformations $F_{i}$s in (III.9) to the
lowest eigenvalue of the Laplacian.383838As we show in varshovi2 , this is the
main idea to obtain the same result for Yang-Mills fields in presence of
gravity. In principle, the emergence of the Einstein-Hilbert action is a mere
consequence of evolving the geometric structure on $M$ and has nothing to do
with the quantum field theory (i.e. the interaction terms and their low and
high energy scenarios) involved in the space-time continuum $\mathcal{M}$.
Hence, the only fundamental law which we assumed within the Wiener Brownian
process is due to the Ricci flow of the universe. Therefore, as we emphasized
above, and demonstrated in varshovi1 ; varshovi2 , one may conclude that the
second law of thermodynamics is a fundamental law of nature, either more
fundamental than gravitation or coming with gravity from the same basic
source.

The stochastic base of the Wiener fractal gravity puts forward the idea of the
existence of an intimate correlation between our path-integral formulation of
general relativity and the theory of stochastic gravity hu ; hu2 . In fact,
stochastic gravity stands on the same footing as the Wiener fractal gravity.
Actually, in stochastic gravity, the semi-classical solutions are effectively
modified by including background noise from quantum fields via the Langevin
equation. For instance, the familiar moving charge quantum field system in the
framework of stochastic gravity leads to the so called Abraham-Lorentz-Dirac-
Langevine equation (a generalized version of ALD equation) which describes the
stochastic dynamics of the moving charge within a correct and pathology-free
setting hu .

Historically, stochastic (semi-classical) gravity commenced from the 1990s as
the third step of a theoretical evolution of quantum gravity which its two
first development stages was _quantum field theory in curved spacetime_
(accomplished in the 1970s) and _semi-classical gravity_ (established in the
1980s). Actually, the best introduction to stochastic gravity is found in hu
where the authors explain: _”While semi-classical gravity is based on the
semi-classical Einstein equation with the source given by the expectation
value of the stress-energy tensor of quantum fields, stochastic semi-classical
gravity includes also its fluctuations in a new stochastic semi-classical
Einstein-Langevin equation. If the centerpiece of semi-classical gravity is
the vacuum expectation value of the stress-energy tensor of a quantum field,
the centerpiece in stochastic semi-classical gravity is the symmetrized
stress-energy bi-tensor and its expectation value known as the noise kernel.
The mathematical properties of this quantity, its physical contents in
relation to the behavior of fluctuations of quantum fields in curved
spacetimes and their backreaction in the spacetime dynamics engendering
induced metric fluctuations are the main focus of this theory.”_

Indeed, stochastic gravity could be simply regarded as a rigorous mathematical
technique to work out the most accurate solution of Einstein’s field equation
extracted from the approximated Wiener fractal gravity. In fact, as we
established above, according to the original form of the Wiener fractal
measure subject to the dynamics of space geometry due to the Ricci flow, the
quantum field fluctuations are basically non-local effects, admitting no
definite Lagrangian density, whereas the gravitational features come from an
explicit action which at the fist approximation leads to the celebrated
Einstein-Hilbert theory. In other words, stochastic gravity is in principle
one of the best understood frameworks to work out the approximated expectation
values of the Wiener fractal gravity as (III.23). However, computing these
expectation values could be accomplished with some parallel methods of real
analysis for static space-time and the ordinary Lebesgue measure folland .
Therefore, another parallel effort besides stochastic gravity to work out
(III.23) is to develop the present analytical methods to dynamical space-time
and the Lebesgue fractal measure.

The substantial distinguishing features between space and time dimensions and
considering the space-time continuum as a product manifold
$\mathcal{M}=[-T,T]\times M$ (or considering the space manifolds as the
foliation leaves of $\mathcal{M}$), may also resemble the main ideas of
Horava-Lifshitz gravity horava . However, we should stress that this
similarity only includes the fundamental viewpoints about the time and spatial
dimensions and their distinction in physical equations.393939In other words,
the separation of space and time dimensions is presupposed in both Horava-
Lifshitz and Wiener fractal theories of gravity. However, it can be seen that
some modified versions of Horava-Lifshitz gravity could give rise to dynamical
mechanisms for breaking the Lorentz symmetry in the UV contribution. See for
example vag1 ; vag2 . In particular, there is a vast difference between
Horava’s theory and the Wiener fractal gravity in their basic technology
strategies and the mathematics of formulation, while, however, it is apparent
that the Wiener fractal gravity does not contain quantum fluctuations of the
metric components in its present form, hence still being a semi-classical
theory.

Actually, Horava’s theory respects some significant admitted properties for
quantum field theory such as the necessity for the Lagrangian density to only
consist of local terms and to maintain the renormalizability of the theory by
power-counting at the Lifshitz point $z=3$, but the Wiener fractal gravity is
a highly non-local theory at high energy levels which does not even admit a
definite form of Lagrangian density in its original expression. Although both
try to remove the problem of obtaining a consistent theory of gravity within
the path-integral formulation, and also both address the problem to breaking
the (local) Lorentz symmetry and general diffeomorphism invariance of the
theory, but the Horava’s strategy to overcome the subtlety is somehow active.
He breaks the Lorentz invariance (and diffeomorphism invariance) by hand, puts
forward the fixed Lifshitz point $z=3$ according to some evidence coming from
condensed matter, and introduces and incorporates some foliation-preserving
diffeomorphism invariant terms for bringing gravity into the quantum field
theory in some renormalizable manner.

On the other hand, in Wiener fractal gravity we behave as passive. We do not
impose any preferred geometric structure (such as some definite symmetries)
and theoretical constraint into the theory. We only assume that the quantum
fields are mostly distributed over the space $M$ by fractal structures and
their propagation along time is fundamentally a stochastic process that is
partly subject to some dominating entropic force due to the Ricci flow.
Therefore, the most geometric structures and theoretical constraints of the
theory emerge naturally from the essential properties of the Wiener Brownian
process and the entropic effect of the Ricci flow. In fact, the theory
basically respects Heisenberg’s uncertainty principle of quantum mechanics
(due to its intrinsic stochastic property) and is fundamentally entropic
(breaks the time-reversal symmetry).

Moreover, in Wiener fractal gravity we surprisingly showed that the source of
gravity is intrinsically different from that of the three fundamental forces
of nature which have been already formulated in the Standard Model, i.e. the
electromagnetic and the weak and strong nuclear forces. Thus, the viewpoint of
the Wiener fractal gravity to quantum interactions of nature may resolve and
interpret the question of: _”Why we cannot unify gravity with other
fundamental forces of nature?”_ Indeed, upon what we have done above the
answer is: _Because the quantum fundamental forces of nature stem from
Brownian propagation of fractal quantum fields along time, whereas the gravity
is basically rooted in entropic evolution of the universe_. Obviously, they
address two fiercely distinct scales of nature, as we expected before.

##  VI Summary and Conclusion

In this paper we established a more fundamental formulation for quantum field
theory in presence of gravity by employing three basic mathematical
mechanisms:

a) _The asymptotic properties of Fourier-Laplace coefficients of Weierstrass-
like fractal functions_ ,

b) _The Wiener stochastic process of fractal functions on a closed Riemannian
manifold_ ,

c) _The entropic based Ricci flow of the Riemannian metric on a closed
Riemannian 3-manifold._

We proved that the stochastic Brownian motion of fractal functions on a closed
Riemannian manifold with dynamical geometry due to the Ricci flow causes the
emergence of the Einstein-Hilbert action and the path-integral formulation of
the corresponding quantum field theory within the Wiener probability measure.
The resulting theory, the so called _Wiener fractal gravity_ , could be
regarded as a more fundamental formulation of gravity. We also established
that the emergence of the Einstein-Hilbert Lagrangian density has nothing to
do with the background quantum field theory and has a merely entropic source
due to the Ricci flow.

We discussed the different sources of gravity and the other fundamental forces
of nature based on the mentioned stochastic framework on dynamical Riemannian
manifolds. Also, by computing the coupling constant of the Einstein-Hilbert
action we interpreted the hierarchy problem in terms of the background
geometric structures and the Ricci flow. Then, we worked out the cosmological
constant according to Hamilton’s theorem for 3-manifolds and showed that in
$D=3$ dimensions this constant is positive and surprisingly extremely small.
We also argued that the cosmological constant is a pure entropic/geometric
term and is independent of the zero-point energy of the involved quantum field
theories.

The correlation of the solutions of the corresponding Ricci flow and the
achievements of the $\Lambda$CDM Model was discussed and we found some
specific similarities to the FLRW metric and its cosmological features.
Hubble’s law and the acceleration of the universe expansion were worked out
due to the Ricci flow and it was argued that the dark energy (hence the
inflation) has an entropic source. Afterward, we compared our achievements
with three important theories of gravity; 1) Verlinde’s entropic gravity, 2)
Stochastic gravity, and 3) Horava-Lifshitz gravity. We found both basic
similarities and fundamental differences between these theories and the Wiener
fractal gravity in some appropriate approximated versions.

Upon the above results, we are optimistic that the above formulation could
provide a well-defined framework of quantum gravity. Indeed, within the
mentioned formulation, the gravitational fields have no quantum (fractal)
fluctuations and their effects are regarded in terms of expectation values via
some semi-classical viewpoint. Actually, to insert these quantum effects
within the theory one should localize the non-local fluctuations of the metric
which give rise to appearance of Grassmannian superfields just similar to the
mechanism of the Faddeev-Popov quantization method. Anyway, as we have shown
in varshovi2  obtaining a probability measure is only possible by employing
Grassmannian spinor fields. This causes the emergence of fermionic Majorana
quantum fields which accompany the bosonic gravitational fields within the
interaction term of the Wiener path-integral. This could be regarded as a
possible procedure to produce a path-integral formulation of supergravity by
means of a probability measure.

##  VII Acknowledgments

The author would like to dedicate this paper to Saya Ziaee for her birthday on
January 22nd; This research would not have been possible without her emotional
support. Moreover, the author would like to thank Shahroud Azami for his
fruitful guidelines. Also, it should be noted that this research was in part
supported by a grant from IPM (No. 1400810418).

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