[math-ph]

# Wiener Process of Fractals and Path-Integrals I:
Emergent Lorentz Symmetry in Stochastic Process of Quantum Fields

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 first paper of a series of researches (that is followed
by varshovi E-H ; varshovi ) that aims to interpret the gravitational effects
of nature within some consistent stochastic fractal-based (intrinsically
conformal) path-integral formulation. In this paper, we initially study the
asymptotic behaviors of fractal structure of Weierstrass-like functions by
means of Hardy’s criteria for nowhere differentiability. It is proved that the
asymptotic behavior of Fourier-Laplace coefficients of such functions leads to
a non-linear differential equation which in its turn gives rise to an
exponentially increasing norm, the so called fractal norm, on the phase space.
Then, using the fractal norm the Wiener Brownian process is accomplished for
fractal functions on a flat space. By substituting non-local terms with
approximated local ones within the derived formula the d’Alembertian operator
emerges automatically in the Gaussian terms of the Wiener measure. Hence, it
is established that the Lorentz symmetry would be regarded as the first order
approximate symmetry of nature on a flat space-time manifold based on the
stochastic essence of Brownian motion of the background fractal geometry. This
leads to a well-defined framework for Feynman’s path-integral in both real and
complex formulations. It is shown that the obtained Wiener measure, the so
called Wiener fractal measure, gives rise to a consistent path-integral
formulation of scalar quantum field theories in terms of the Wiener
probability measure. Finally, we review some various significant aspects of
quantum field theories (such as renormalizability, RG flow, Wick rotation,
regularization, etc.) by means of the analytical properties of Wiener fractal
measure. We should emphasize that what makes these achievements innovative is
that the mentioned results are based solely on the basics of fractal geometry
and stochastic process, with no explicit or implicit imposed presuppositions
of quantum physics.

Keywords: Fractal Norm, Self-Similarity, Nowhere Differentiability,
Weierstrass Function, Weierstrass Power Law, Wiener Stochastic Process, Wiener
Fractal Measure, Quantum Field Theory, Renormalizability, RG Flow.

##  I Introduction

Fractals are in fact one of the most common features of nature in both macro
and micro scales. Indeed, the self-similarity of natural structures, which is
preserved in scale transformation within some specific range of spatial length
(or energy) of physical systems stems from a well-understood symmetry of
quantum field theories, known as conformal invariance sornette ; piet .
However, although this symmetry usually breaks at some scale of energies for
interacting field theories, the fundamental mathematical entities involved
within the formulation of quantum physics show some intrinsic features of
self-similarity (scale invariance) and nowhere differentiability kroger ;
feynman ; laskin . In fact, such properties are mostly figured out and
utilized via the renormalization group flow through with both Wilson’s and
Bogoliubov’s approaches wherein the high energy contributions are effectively
referred to bogo ; shirkov1 ; shirkov2 ; shirkov3 ; wilson1 ; wilson2 .

As a mathematical object a fractal structure on a $D$-dimensional Cartesian
space is basically defined upon the generalization of fractal curves. A
fractal curve is generally defined by analytic concepts such as measure,
dimension, length, or differentiability. Due to Mandelbrot mandelbrot  the
fractal curves are distinguished by a measure theoretic point of view. Through
this viewpoint a fractal curve in $\mathbb{R}^{D}$ is in fact a function
$\mathfrak{F}:I\to\mathbb{R}^{D}$, with $\text{dim}_{HB}(\mathfrak{F}(I))>1$,
wherein $\text{dim}_{HB}$ is the Hausdorff-Besicovitch dimension and
$I\in\mathbb{R}$ is an interval. However, according to the geometric-base
definition, a fractal curve is a continuous embedding of $I$ in
$\mathbb{R}^{D}$ which each of its subarc longer than a point has an infinite
length nottale1 . This point of view also leads to the definition of nowhere
rectifiable curves which are effectively known as maps from $I$ to
$\mathbb{R}^{D}$ that are non-differentiable almost everywhere on their domain
nottale1 ; falconer .111A fractal curve can be considered as a sequence of
piecewise smooth curves $\gamma_{i}:I=[0,1]\to\mathbb{R}^{D}$,
$i\in\mathbb{N}$, so that for any $t\in I$, the limit of
$\mathfrak{F}(t)=\lim_{i\to\infty}\gamma_{i}(t)$ exists and the Hausdorff-
Besicovitch dimension of $\mathfrak{F}(I)$ exceeds its topological dimension
$\text{dim}_{T}(\mathfrak{F}(I))=1$.

It is worth noting that by considering the fractal curves as the contour line
of self-similar nowhere differentiable maps
$\mathfrak{F}:\mathbb{R}^{D}\to\mathbb{R}$ the study of fractal curves is
transferred to fractal functions on Cartesian spaces. This process, from the
viewpoints of quantum mechanics, is equivalent to transfer from the fractal
trajectories in Feynman’s path-integral feynman ; feynman2  to fractal
wavefunctions (resp. fractal fields) in the Schrodinger’s wave mechanics
(resp. the theory of quantum fields).222See balankin ; calcagni ; laskin ;
kroger ; morales  and the references therein for more discussions. Then, it
would be possible to provide a firm understanding for fractal functions which
may supply a suitable framework for formulating the established fascinating
features of quantum field theories.333See for example kroger ; shirkov3 ;
wilson2 ; nottale1 . Fractal functions are indeed one of the most basic
elements of a firm extended geometric framework, usually referred to as
_fractal geometry_.

Actually, fractal geometry is extensively known as the most common natural
feature of quantum mechanics, hence of the fundamental interactions of nature.
Indeed, fractal structures which show up in most particular observables are
generated by the dynamics of quantum systems based on the uncertainty
principle. In particular, the first appearance of fractal geometry in quantum
mechanics was invoked by Feynman to demonstrate the underlying role of self-
similarity in the setting of path-integral feynman . As mentioned above, the
next but in principle, the most significant emergence of fractal geometry was
due to the renormalization group flow in quantum field theory, where the scale
(conformal) invariance and critical phenomena in condensed matter physics, and
the scale invariance properties in nuclear physics have been thoroughly
studied by addressing the problems to the fundamental structure of self-
similarity bogo ; wilson1 ; wilson2 .

Then, a vast number of discovered phenomena of quantum physics were properly
described by employing the abilities of fractal geometry, such as the critical
exponents in condensed matter, the confinement phase in QCD, the Hausdorff
dimension of quantum gravity and the Ising model, the critical clusters, and
some aspects of spin physics.444See kroger  as an interesting research about
these topics.. Since fractal geometry plays an extensive role in quantum
physics, it may be expected to have a critical contribution in fundamental
interactions of nature due to elementary particles. In his fascinating work
kroger  Kroger asks in this regard: _What is the [background fractal] geometry
of propagation for a relativistic quantum particle?_

Considering the quantum fields as fundamental objects in the path-integral
formalism of quantum field theory, their propagation in a free field theory
must be interpreted as the evolution of fractal geometric structures. In fact,
there are various elaborated mathematical tools for studying stochastic
dynamics (propagation) of geometric structures, wherein the Wiener Brownian
process is known as one of the most significant formulations due.555See li
and the references therein for other mathematical procedures. Therefore,
Kroger’s question could be perceived as: _What would happen if fractal
structures propagate stochastically via the well-known formulation of the
Wiener Brownian process?_

In the present research, we will answer the above question by employing the
celebrated Wiener stochastic process wiener ; wiener2  for fractal functions.
At the first step, with employing Hardy’s nowhere differentiability condition
for the asymptotic behavior of Weierstrass-like functions (section I) we will
work out a specific norm, the so called fractal norm, for fractal functions
(sections III and IV). Next, the Wiener measure of the fractal norm, known as
the Wiener fractal measure, is utilized to figure out the Brownian motion of
fractal structures (section V). Hereby, we will demonstrate that the
stochastic propagation of fractal functions leads to some significant physical
results including: i) The Wiener fractal measure embraces the Lorentz symmetry
at its first local approximation (section VI). ii) The Wiener fractal measure
contains the Klein-Gordon Lagrangian, hence providing a consistent framework
for path-integral formulation of scalar field theories (section VII) and for
the machinery of their renormalization group flow (section VIII).

It is worth noting that what makes this research innovative is that the
mentioned results are based solely on the basics of fractal geometry and
stochastic process, with no explicit or implicit imposed presuppositions of
quantum physics and operator algebra, or manual adjustments by mathematical
tools and techniques such as to start with a complex version of (hence non-
probabilistic) the Wiener measure or to consider the analytic continuation of
the Lagrangian terms, etc., which all have been extensively studied and
accomplished before.666See for example bodmann ; dau ; dau2 ; dau3 ; and ;
nelson ; cameron1 ; cameron2 ; chaichian ; ito1 ; ito2 ; gelfand ; kac .
Therefore, upon the above proclaim and what we have already obtained elsewhere
varshovi E-H ; varshovi , one may be suspect whether quantum field theories of
fundamental interactions of nature have only stochastic and (fractal)
geometric sources.

Whether or not such a viewpoint is correct is a matter that we will not
address here. But, however, we should emphasize that our study has not been
theoretically intended to obtain a mathematical formulation to describe a
quantum theory and its peculiar aspects (since we have not employed any
presumption from quantum mechanical mathematics) but to work out a stochastic-
geometric process that has spontaneously led to the scalar quantum field
theory. Hence, one may be optimistic that the achieved results of this perusal
may help us to perceive and to obtain more profound intuitions about the
foundations of quantum physics and fundamental interactions of nature.

##  II Fractality and Asymptotic Behaviors

As mentioned above the background fractal geometry of quantum physics is in
fact referred to _self-similarity_ and _nowhere differentiability_ of wave
functions or fields of elementary particles (such as Higgs boson). Before
constructing a general setting for fractals one should revisit the self-
similarity and nowhere differentiability conditions first. In principle, these
two properties are best understood by considering the Weierstrass function
$W(x)=\sum_{k\geq 0}a^{k}\sin(b^{k}\pi x/L):I\to\mathbb{R}$, with $I=[-L,L]$,
and $0<a<1$, wherein $b$ is an odd integer weierstrass . This function, with
no more conditions provides the simplest realization of self-similarity,
whereas it was shown by Hardy that to gain nowhere differentiability one more
condition must be considered too: $ab>1$ hardi .777In fact, Weierstrass showed
that the nowhere differentiability would appear when $b/a>1+3\pi/2$, but,
subsequently Hardy hardi  proved that this condition could be weakened to
$b\geq a^{-1}>1$. However, the case of $b=a^{-1}$ is exotic which leads to
almost everywhere smooth functions with discontinuity. Hence, we exclude this
case in our arguments, because we are mostly interested in continuous periodic
functions. See johnsen  for a more detailed discussion. Hence, to obtain both
self-similarity and nowhere differentiability the Weierstrass function $W(x)$
must obey a definite power law in its Fourier coefficients: If $n=b^{k}$, then
the $n$-th Fourier coefficient of $W(x)$ is definitely proportional to
$1/n^{s}$ wherein $s=\log_{b}(1/a)$.

a) Entire Fractals

Actually, self-similarity and nowhere differentiability put a limitation on
$s$: $0<s<1$. It is the basic criteria for recognizing the _entire fractality_
, i.e. being fractal over the entire domain. Roughly speaking, an _entire
fractal function_ is a continuous map with a power law in its Fourier
coefficients $f_{n}$s as:

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}f_{n}\propto
1/n^{s}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(0<s<1)$
(II.1)

for some definite $b\in\mathbb{N}$ and for all $n\propto b^{k}$, $k\geq 1$. We
refer to (II.1) as _Weierstrass ordinary power law_ which abbreviated as OPL.

Actually, this regular behavior of Fourier coefficients of the Weierstrass
function is in principle the immediate consequence of self-similarity. In
fact, if a process of $b$-time ($1<b\in\mathbb{N}$) magnification on the
$x$-axis results in $a$-time ($0<a<1$) growth along the $y$-axis on the graph
of function $f(x)$, then the Fourier coefficients of $f(x)$ obey a similar
condition as (II.1). In such cases, we say $f(x)$ is an _$(a,b)$ -self-similar
function_.888One should note that $(a,b)$-self-similarity will also cause
$(a^{n},b^{n})$-self-similarity. But, however, in this paper we are mostly
interested in the fundamental case which thereby $f(x)$ is $(a,b)$-self-
similar if and only if it is not $(a^{1/m},b^{1/m})$-self-similar except for
$m=1$. The pair $(a,b)$ is referred to as the _type of the self-similarity_ ,
and $a$ is called the _scaling factor_. Let us see this property with more
details: Assume that $f(x):I=[0,2L]\to\mathbb{R}$ is an $(a,b)$-self-similar
function and $a_{N}=\frac{1}{{L}}\int_{I}f(x)\cos(\frac{{N\pi}}{{L}}x)dx$ be
the $N$-th Fourier cosine coefficient. Due to the $(a,b)$-self-similarity the
restriction of $f(x)$ to $[k\frac{{2L}}{{b}},(k+1)\frac{{2L}}{{b}}]$, for each
$0\leq k<b$, provides the total graph of $f(x)$ multiplied by $0<a<1$ in the
$y$-axis and by $1/b$ in the $x$-axis. Hence, if $N=kb$, $k\in\mathbb{N}$, we
easily find: $a_{N}=aa_{k}$, which leads to: $a_{n}=\frac{{1}}{{n^{s}}}a_{1}$,
for $n$ being some power of $b\in\mathbb{N}$, whereas $s=\log_{b}(a^{-1})$.

Obviously, the same relation holds for the Fourier sine coefficient
$b_{N}=\frac{{1}}{{L}}\int_{I}f(x)\sin\left(\frac{{N\pi}}{{L}}x\right)dx$.
Thus, according to hardi  this self-similarity is turned to fractality as one
considers the additional condition of nowhere differentiability: $0<s<1$. In
the following we may refer to $s$ as the _fractality scale_.

b) Widespread Fractals

One can easily check that the same condition holds for a partial fractal
function, i.e. a function $f:I\to\mathbb{R}$, which is fractal only on a
subinterval $\mathfrak{I}=[x_{1},x_{2}]\subset I=[a,b]$. In fact, partial
fractality causes the Fourier coefficients $f_{n}$ ($=a_{n}$ or $b_{n}$) to
have an asymptotic behavior as:999The proportionality factor could be both
positive or negative numbers.

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}f_{n}\propto
1/n^{s}+\mathcal{O}(1/n^{m},1/n^{s+m}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(0<s<1)$
(II.2)

for $s=\log_{b}(a^{-1})$, and $n$ a sufficiently large power of $b$, wherein
$\mathcal{O}(1/n^{m},1/n^{s+m})$ comes from _non-fractal_ parts, being of
order $1/n^{m}$ and $1/n^{s+m}$ for $m\geq 1$.

The asymptotic behavior of (II.2) together with condition $0<s<1$, which is
assumed to exclude the differentiability on the self-similarity domain, is
accordingly referred to as the _Weierstrass peculiar power law_
(PPL).101010One should note that, in the PPL, $b>1$ is the least integer
fulfilling (II.2). For example the Fourier expansion of
$f(x)=x:[-1,1]\to\mathbb{R}$, leads to a similar power law $\propto 1/n$, for
each $n\in\mathbb{N}$. But, in this case the least integer $b$ is the unity
and $s=1$. Therefore, we have no PPL here, hence no fractality. However,
$f(x)$ may contain several partial fractalities of various types
$(a_{i},b_{i})$. In such cases we see:
$f_{n}\propto\sum_{i}1/n^{s_{i}}+\cdots$, with
$0<s_{i}=\log_{b_{i}}(a^{-1}_{i})<1$. Here, one finds a summation of some PPLs
due to (II.2) for each $s_{i}$, hence $f(x)$ fulfills the PPL again. Moreover,
for each fixed $b$, there may be various scaling factors $a_{k}$s chosen
randomly at each Fourier mode, i.e. $n=b^{k}$ falconer . In such cases, we
also see that the Fourier coefficients undergo the PPL for some
$0<s<1$.111111Actually, here we have assumed $0<a_{k}<1$ and $a_{k}b>1$ for
each $k$. Thus, the PPL (II.2) is due to $s=\log_{b}(a^{-1})$ with scaling
factor $a=\lim\text{Sup}_{k\geq 1}\sqrt[k]{a_{1}\cdots a_{k}}$, which by
assumption fulfills the condition of: $1/b<a<1$.

Actually, the partial fractality and the PPL of (II.2) could be studied more
accurately for _widespread fractal functions_ , which by definition are
functions which are self-similar except over a very small subdomain. Indeed,
employing the Taylor’s expansion we can see that a more precise version of PPL
holds for the Fourier coefficients. To see this clearly let us assume the
_localized Weierstrass function_ $W_{L}(x)=\sum_{k\geq
0}a^{k}\sin(\frac{b^{k}}{\mathfrak{l}}\pi x)$ for
$x\in[-\mathfrak{l},\mathfrak{l}]$, and $W_{L}(x)=0$ for $\mathfrak{l}<|x|\leq
L$, wherein $0<a<1$, $b$ a positive odd integer and $1<ab$, while
$\frac{L-\mathfrak{l}}{L}\approx 0$. Hence, we obtain the $n$-th Fourier sine
coefficient as;

$\begin{gathered}b_{n}=\mathfrak{l}\sum_{k\geq
0}a^{k}\frac{\sin\left(\pi(b^{k}-(\mathfrak{l}/L)n)\right)}{\pi(b^{k}-(\mathfrak{l}/L)n)}-\mathfrak{l}\sum_{k\geq
0}a^{k}\frac{\sin\left({\pi(b^{k}+(\mathfrak{l}/L)n)}\right)}{\pi(b^{k}+(\mathfrak{l}/L)n)}\\\
\approx\mathfrak{l}\sum_{k\geq
0}a^{k}\frac{\sin\left({\pi(b^{k}-(\mathfrak{l}/L)n)}\right)}{\pi(b^{k}-(\mathfrak{l}/L)n)}\approx{\mathfrak{l}}a^{K}+F_{K}(L-{\mathfrak{l}}),~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{gathered}$
(II.3)

wherein $F_{K}$ is an analytic function with $F_{K}(0)=0$ and $n=b^{K}$, for
some large $K\in\mathbb{N}$. Actually;

$F_{K}(\zeta)=\frac{\mathfrak{l}\sin\left(n\pi\zeta/L\right)}{\pi}\sum_{0\leq
k\neq
K}\frac{a^{k}}{b^{k}-(\mathfrak{l}/L)n}\approx\frac{\mathfrak{l}\sin\left(n\pi\zeta/L\right)}{\pi}\sum_{0\leq
k\neq K}\frac{a^{k}}{b^{k}-n}.$ (II.4)

Actually, $F_{K}$ splits into two geometric series; an infinite series for
$k>K$ and a finite one for $k<K$. Hence, for large enough $n$ (and $K$) we
obtain;

$\begin{gathered}F_{K}(\zeta)\approx\frac{\mathfrak{l}\sin\left(n\pi\zeta/L\right)}{\pi}\left(\frac{1}{n}\left(\frac{1-a^{K}}{1-a}\right)+\frac{a^{K}}{n}\left(\frac{b}{b-a}\right)\right)~{}~{}~{}~{}~{}~{}~{}~{}\\\
=\frac{\mathfrak{l}\sin\left(n\pi\zeta/L\right)}{\pi}\left(\frac{1/n-1/n^{1+s}}{1-a}+\frac{1}{n^{s+1}}\frac{b}{b-a}\right).\end{gathered}$
(II.5)

Since the sine function is periodic and we are only interested in asymptotic
behaviors of Fourier coefficients, then $\sin\left(n\pi\zeta/L\right)$ must be
replaced by some positive constant $0<\alpha<\pi$ times $\zeta/L$, which
indeed refers to integers $n_{i}$, $i\in\mathbb{N}$, with
$n_{i}\pi\zeta/L=2N\pi+\alpha_{i}\zeta/L$, for some $N\in\mathbb{N}$ and
$0<\alpha_{i}\leq\alpha$. Therefore, we see that the PPL included in $F_{K}$
is of the following form;

$F_{K}^{PPL}(\zeta)\approx\frac{\alpha\zeta}{L}\frac{a(1-b)}{(1-a)(b-a)}\frac{1}{n^{1+s}},$
(II.6)

which leads to;

$\lim_{K(n)\to\infty}a^{-K}nF^{PPL}_{K}(\zeta)=-\lambda,$ (II.7)

for
$\lambda=\frac{\alpha\zeta}{L}\frac{a(b-1)}{(1-a)(b-a)}\in\mathbb{R}_{>0}$.
Hence, one readily deduce;

$\begin{gathered}b_{n}\approx\mathfrak{l}a^{K}\left(1-\frac{s\mu^{2}}{n}\right)+\mathcal{O}(1/n^{m})=\mathfrak{l}\left(\frac{1}{n^{s}}-s\frac{\mu^{2}}{n^{s+1}}\right)+\mathcal{O}(1/n^{m})\\\
\approx\frac{\mathfrak{l}}{\left(n^{2}+\mu^{2}\right)^{s/2}}+\mathcal{O}(1/n^{m}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{gathered}$
(II.8)

wherein we define the _fractal mass_ as; $\mu^{2}=\lambda/s$. In fact, for
widespread fractality condition we are still being encountered with a PPL in
the asymptotic behavior;121212The proportionality factor could be both
positive and negative numbers. Even for Fourier complex coefficients $f_{n}$s
could be complex numbers.

$\begin{split}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}f_{n}\propto
1/(n^{2}+\mu^{2})^{s/2}+\mathcal{O}(1/n^{m}).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(0<s<1)\end{split}$
(II.9)

We may refer to (II.9) as the _Weierstrass massive power law_ (MPL). In fact,
discovering some MPL behavior among the Fourier coefficients is a fairly
complicated process. However, although the PPL will produce the MPL, but as we
will see in the next section the MPL condition leads to a more accurate and
distinct criterion for fractal functions. Moreover, since the fractal mass
$\mu$ is proportional to $\zeta=L-\mathfrak{l}$, the asymptotic behavior of
widespread fractals in (II.9) naturally leads to that of entire fractals that
we worked out in (II.1) as $\zeta\to 0$.

##  III Fractal Norm and Fractal Functions

However, one can work out a criterion to discover and figure out the OPL and
MPL behaviors for entire and widespread fractal functions. This could be done
by putting an appropriate measure on the phase space via the Fourier
coefficients.

a) Entire Fractals

Let us first concentrate on the Weierstrass function. A direct calculation
shows that by assuming $n$ as a continuous parameter we obtain from (II.9) the
asymptotic inequality131313We allow $n$ to vary within a tiny neighborhood of
$n$, so that it never touches $b^{k}$s, $k\in\mathbb{N}$.

$\frac{{d|f_{n}|}}{{dn}}+\frac{{|f_{n}|}}{{n}}>|f_{n}|~{}e^{-\kappa
n^{2}|f_{n}|^{2}/2}$ (III.1)

for each function $f(x)$ with entire fractality (i.e. OPL), wherein $\kappa>0$
is an arbitrary constant. Note that here $f_{n}$ could be assumed as the
Fourier complex coefficient obtained for the Fourier mode $e^{in\pi x/L}$,
hence generally a complex number. To consider any kind of self-similarity with
assuming the extreme case of non-differentiability, i.e. $s\to 1$, we conclude
the inequality

$\frac{{d|f_{n}|}}{{dn}}+\frac{{|f_{n}|}}{{n}}|>\frac{{\ell}}{{n}}~{}e^{-\kappa
n^{2}|f_{n}|^{2}/2},$ (III.2)

which must be hold asymptotically for any fixed $\ell,\kappa>0$.141414This
inequality is easily inferred by comparing the heat kernel function and the
powers of $1/n$. That is, for each positive $\ell$ and $\kappa$ there exists a
large enough $N\in\mathbb{N}$, so that for any $n\geq N$ the inequality
(III.2) holds for each function with entire fractality. In principle, (III.2)
could be assumed as the main criteria for excluding the entire fractality for
each given function $f(x)$. In fact, if the equation

$\frac{{d|f_{n}|}}{{dn}}+\frac{{|f_{n}|}}{{n}}|\leq\frac{{\ell}}{{n}}~{}e^{-\kappa
n^{2}|f_{n}|^{2}/2}$ (III.3)

is imposed to the Fourier coefficients $f_{n}$s for some fixed $\kappa$ and
$\ell$, the resulting function will not admit the entire fractality. Let
$\kappa$ be fixed. Then, to obtain a limit for avoiding the fractality, we
should first solve

$\frac{{d|f_{n}|}}{{dn}}+\frac{{|f_{n}|}}{{n}}=\frac{{\ell}}{{n}}~{}e^{-\kappa
n^{2}|f_{n}|^{2}/2}.$ (III.4)

The solution of (III.3) gives $|f_{n}|$ as a function of $\ell$ and $n$, i.e.
$|f_{n}|=|f_{n}|(n,\ell)$. When we fix $|f_{n}|$, then according to the
implicit function theorem $\ell$ is given as a function of $n$ and $|f_{n}|$.
This function is given by the following integral formula:

$\ell=\ell_{\kappa}(f_{n})=\int_{0}^{|f_{n}|}~{}e^{\kappa n^{2}x^{2}/2}~{}dx,$
(III.5)

where $f_{n}$ is also allowed to admit complex values.

In principle, (III.5) could be easily confirmed by employing the implicit
function theorem. To see this let $f_{n}$ be real and $f_{n}\geq 0$, unless
replace $f_{n}$ by $-f_{n}$. Now, according to (III.5) we obtain:

$\frac{\partial\ell}{\partial n}+\frac{\ell}{n}=\frac{f_{n}}{n}e^{\kappa
n^{2}f^{2}_{n}/2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{and;}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{\partial\ell}{\partial
f_{n}}=e^{\kappa n^{2}f^{2}_{n}/2}.$ (III.6)

Hence, the differential equation of $f_{n}$, i.e. (III.4), could be simply
derived with considering $\partial f_{n}/\partial n=-(\partial\ell/\partial
n)/(\partial\ell/\partial f_{n})$, which in its own right is the immediate
consequence of $df_{n}(n,\ell(n))/dn=0$. We refer to $\ell_{\kappa}(f_{n})$ in
(III.5) as the _fractal $\kappa$-norm of $f_{n}$_, while generally we may drop
the prefix $\kappa$, and use simply the _fractal norm_ for convenience.

As mentioned above the fractal norm of $f_{n}$ has critical importance in
disclosing the existence of the entire fractality. In fact, to exclude the
entire fractality we should have $f_{n}\to 0$ at least with the rate induced
from (III.5) for some fixed $\kappa$ and the fractal norm $\ell_{0}$. All in
all, we have the following theorem:

Theorem 1; _Assume that $I\subset\mathbb{R}$ is a compact interval and
$f(x):I\to\mathbb{R}$ is a continuous function with Fourier coefficients
$f_{n}$. Then, $f(x)$ is not an entire fractal (i.e. dismisses OPL) if and
only if for some fixed $\kappa>0$ and $\ell_{0}>0$ we have:
$\ell_{\kappa}(f_{n})\leq\ell_{0}$, for all $n$. In other words, $f$ is not an
entire fractal if and only if there exists some $\kappa>0$ and $\ell_{0}>0$ so
that the inequality_

$-\ell_{0}\leq\int_{0}^{f_{n}}~{}e^{\kappa n^{2}x^{2}/2}~{}dx\leq\ell_{0}.$

_holds for any Fourier real coefficient $f_{n}$, $n\in\mathbb{N}$._

b) Widespread Fractals

For a widespread fractal with fixed $a$, $b$ and $\mathfrak{l}$ we can easily
see from (II.9) that;

$\frac{{d|f_{n}|}}{{dn}}+\frac{{n}}{{n^{2}+\mu^{2}}}|f_{n}|\geq|f_{n}|~{}e^{-\kappa(n^{2}+\mu^{2})|f_{n}|^{2}/2},$
(III.7)

Consequently, for each widespread fractal function fulfilling the inequality
$0<s<1$, we find the following asymptotic equation:

$\frac{{d|f_{n}|}}{{dn}}+\frac{{n}}{{n^{2}+\mu^{2}}}|f_{n}|\geq\frac{{n\ell}}{{n^{2}+\mu^{2}}}~{}e^{-\kappa(n^{2}+\mu^{2})|f_{n}|^{2}/2},$
(III.8)

where $\ell>0$ is a constant. Therefore, with similar discussion as we had for
entire fractals, to find a measure on the phase space which excludes
widespread fractal functions we have to solve the following differential
equation:

$\frac{{d|f_{n}|}}{{dn}}+\frac{{n}}{{n^{2}+\mu^{2}}}|f_{n}|=\frac{{n\ell}}{{n^{2}+\mu^{2}}}~{}e^{-\kappa(n^{2}+\mu^{2})|f_{n}|^{2}/2}.$
(III.9)

The solution of (III.9) works out $|f_{n}|$ as a function of $\ell$ and $n$,
i.e. $|f_{n}|=|f_{n}|(n,\ell)$. Upon a similar reasoning to the discussion we
had for the entire fractals $\ell$ is given as a function of $f_{n}$ and $n$:

$\ell=\ell_{(\kappa,\mu)}(f_{n})=\int_{0}^{|f_{n}|}~{}e^{\kappa(n^{2}+\mu^{2})x^{2}/2}~{}dx,$
(III.10)

where $f_{n}$ is allowed to admit negative and complex values. The fractal
norm $\ell_{(\kappa,\mu)}$ is conventionally called the _fractal
$(\kappa,\mu)$-norm of $f_{n}$_, while we may drop the prefix $(\kappa,\mu)$
for simplicity and use the _massive fractal norm_ as the convenient
terminology. In particular, this generalized norm also provides a criterion
for recognizing non-fractal widespread function. Actually, the next theorem
could be considered as the immediate generalization of Theorem 1 for
widespread fractals.

Theorem 2; _Assume that $I\subset\mathbb{R}$ is a compact interval and
$f(x):I\to\mathbb{R}$ is a continuous widespread function with Fourier
coefficients $f_{n}$. Then, $f(x)$ has no widespread fractality if and only if
for some fixed $\kappa>0$, $\mu>0$, and $\ell_{0}>0$ we have:
$\ell_{(\kappa,\mu)}(f_{n})\leq\ell_{0}$, for all $n$. In other words, $f$ has
no widespread fractality if and only if there exists some $\kappa>0$, $\mu>0$,
and $\ell_{0}>0$ so that the inequality_

$-\ell_{0}\leq\int_{0}^{f_{n}}~{}e^{\kappa(n^{2}+\mu^{2})x^{2}/2}~{}dx\leq\ell_{0}.$

_holds for any Fourier real coefficient $f_{n}$, $n\in\mathbb{N}$._

##  IV Fractal Norm in Higher Dimensions

The same ideas could be similarly assumed for functions with $D$-dimensional
domains ($D>1$) by multiplying $D$ copies of Weierstrass function for each
variable $x^{i}$, $1\leq i\leq D$. For instance, in $D=2$ dimensions, we
easily construct a self-similar nowhere differentiable function as:
$W^{(2)}(x,y)=\sum_{m,n}a^{m}a^{\prime n}\cos(b^{m}\pi x)\cos(b^{\prime n}\pi
y)$, with $0<a,a^{\prime}<1$, and odd integers $1<b,b^{\prime}\in\mathbb{N}$,
fulfilling: $ab,a^{\prime}b^{\prime}>1$. In such cases we also see a OPL as:
$a_{n,n^{\prime}}\propto n^{-s}n^{\prime-s^{\prime}}$, for $a_{n,n^{\prime}}$
the Fourier cosine coefficient for $2$-dimensional Fourier mode
$(n,n^{\prime})$, while $n$ (resp. $n^{\prime}$) is some power of $b$ (resp.
$b^{\prime}$) and $0<s=\log_{b}(a^{-1})<1$ (resp.
$0<s^{\prime}=\log_{b^{\prime}}(a^{\prime-1})<1$). Generally, one may simply
define $W^{(D)}(x^{1},\cdots,x^{D})$ to be produced by multiplying $D$ copies
of the Weierstrass function, each of which being an $(a_{i},b_{i})$-self-
similar function, with $b_{i}\in\mathbb{N}$ an odd integer, and $0<a_{i}<1$,
the $i$-th scaling factor, while $a_{i}b_{i}>1$ for each $1\leq i\leq D$.
Substantially, if for $D$-plets $u=(u_{1},\cdots,u_{D})$ and
$v=(v_{1},\cdots,v_{D})$ one simply sets: $u^{v}=u_{1}^{v_{1}}\cdots
u_{D}^{v_{D}}$, then $W^{(D)}(x^{1},\cdots,x^{D})$ admits the OPL in its
Fourier coefficients: $f_{n}\propto 1/n^{s}$, for $n=(n_{1},\cdots,n_{D})$ be
a $D$-plet of integers, and $s$ the $D$-plet power: $s=(s_{1},\cdots,s_{D})$,
with $s_{i}=\log_{b_{i}}(a_{i}^{-1})$.

Also, we can go farther and define PPL accordingly for $D$ dimensions. Thus,
if $f(x^{1},\cdots,x^{D})$ partly admits such self-similarities, together with
non-differentiability, then $f_{n}\propto
1/n^{s}+\mathcal{O}(1/n^{s+m},1/n^{m})$, $m\geq 1$, for general Fourier
coefficient $f_{n}$ and for $n_{i}$ some power of $b_{i}$, $1\leq i\leq D$. It
is the PPL for functions with $D$-dimensional domains.151515Here, we also
assume that both self-similarity and nowhere differentiability hold for each
of the $D$ directions. Therefore, if $f$ possess the widespread fractality in
some of its $D$ variables, then it fulfills MPL in its Fourier coefficients
asymptotically; i.e.
$f_{n}\propto\prod_{i=1}^{D}1/(n_{i}^{2}+\mu_{i}^{2})^{s_{i}/2}+\mathcal{O}(1/n^{m+s_{i}},1/n^{m})$,
$m\geq 1$. Specifically, for _symmetric fractals_ which admit the same type of
self-similarity in each of the variables one discovers a beautiful formula for
the leading term in asymptotic behaviors of the Fourier coefficients.

In order to see this let $f$ to be a widespread (or entire) symmetric fractal
with the fractal mass $\mu_{1}=\cdots=\mu_{D}=\mu$ and the fractality scale
$s_{1}=\cdots=s_{D}=s$. Hence, at the leading term we find:
$f_{n}\propto\prod_{i=1}^{D}\frac{1}{(n_{i}^{2}+\mu^{2})^{s/2}}$. This
proportionality gives rise to a critical formula which helps us to work out
the corresponding differential equations for Fourier coefficients $f_{n}$. Set
$1\leq i\leq D$ and all $n_{j}$s, $1\leq j\neq i\leq D$, to be fixed.
Essentially, here we concern about the asymptotic behavior in the $i$-th
direction. Since $0<s<1$ then we asymptotically have;

$\frac{n_{i}^{2}+\mu^{2}}{n^{2}+\mu^{2}}>s,$ (IV.1)

wherein $n^{2}=\sum_{i=1}^{D}n_{i}^{2}$. Therefore, by considering (IV.1) and
employing the argumentation of the last section we have the following
asymptotic behavior of MPL;

$\frac{{\partial|f_{n}|}}{{\partial
n_{i}}}+\frac{{n_{i}}}{{n^{2}+\mu^{2}}}|f_{n}|\geq\frac{{n_{i}\ell}}{{n^{2}+\mu^{2}}}~{}e^{-\kappa(n^{2}+\mu^{2})|f_{n}|^{2}/2},$
(IV.2)

wherein $\kappa>0$ is fixed. Hence, we should consider the following system of
PDEs to find a criteria to avoid the symmetric fractality in $D$-dimensions:

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{{\partial|f_{n}|}}{{\partial
n_{i}}}+\frac{{n_{i}}}{{n^{2}+\mu^{2}}}|f_{n}|=\frac{{n_{i}\ell}}{{n^{2}+\mu^{2}}}~{}e^{-\kappa(n^{2}+\mu^{2})|f_{n}|^{2}/2}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1\leq
i\leq D)$ (IV.3)

Consequently, (III.10) is also obtained in $D$-dimensions for the massive
fractal norm $\ell_{(\kappa,\mu)}$ as;

$\ell=\ell_{(\kappa,\mu)}(f_{n})=\int_{0}^{|f_{n}|}~{}e^{\kappa(n^{2}+\mu^{2})x^{2}/2}~{}dx=\int_{0}^{|f_{n}|}~{}e^{\kappa(n_{1}^{2}+\cdots+n_{D}^{2}+\mu^{2})x^{2}/2}~{}dx.$
(IV.4)

Obviously for the entire fractals we should set $\mu=0$ and replace
$\ell_{(\kappa,\mu)}$ with $\ell_{\kappa}=\ell_{(\kappa,0)}$. In principle, we
simply read;

Corollary 1; _Assume that_ $V\subset\mathbb{R}^{D}$ _is a compact
$D$-dimensional cube and_ $f(x^{1},\cdots,x^{D}):V\to\mathbb{R}$ _is a
continuous function with Fourier coefficients_ $f_{n}$_. Then,_
$f(x^{1},\cdots,x^{D})$ _has no widespread (resp. entire) fractality if and
only if there exists some_ $\ell_{0}>0$, $\kappa>0$ _and_ $\mu>0$_, so that_
$\ell_{(\kappa,\mu)}(f_{n})\leq\ell_{0}$ _(resp._
$\ell_{\kappa}(f_{n})\leq\ell_{0}$_) for each_ $f_{n}$.

To enjoy the simplicity of the Fourier analysis on cubes, we are mostly
interested to functions with cubic domains. Even, if the domain of the
function $f(x^{1},\cdots,x^{D})$ is a bounded $D$-dimensional set, say
$U\subset\mathbb{R}^{D}$, we prefer to restrict it to some large enough cube
$V$ which is included in $U$ as a proper subset. Now we are ready to define a
fractal function:

Definition 1: _Assume that $U\subset\mathbb{R}^{D}$ be a bounded
$D$-dimensional set and let $f(x^{1},\cdots,x^{D}):U\to\mathbb{R}$ to be a
continuous function. Then, $f$ is said to be a fractal function if there
exists a cube $V\subset U$, on which $f$ is self-similar and nowhere
differentiable except on a subset with very small measure. Equivalently, $f$
is referred to as a fractal function if for each $\ell_{0}>0$, $\kappa>0$ and
$\mu\geq 0$ there exists some $n=(n_{1},\cdots,n_{D})$ so that;
$\ell_{(\kappa,\mu)}(f_{n})>\ell_{0}$, wherein $f_{n}$ is the $n$-th Fourier
coefficient of $f$ restricted to $V$._

##  V Wiener Fractal Measure and Path-Integrals

It is well-known that the Brownian motion of a particle in $D$-dimensional
space is given by means of the celebrated Wiener probability measure. More
precisely, if the particle is at $x_{0}\in\mathbb{R}^{D}$ at $t_{0}$, then the
probability of the particle being found in $U_{i}\subset\mathbb{R}^{D}$ at
$t_{i}$, $1\leq i\leq N$, with the time ordering of $t_{0}<t_{1}<\cdots<t_{N}$
is given as wiener ; wiener2 ;161616See also folland ; durret  as two nice
presentations of the issue.

$\begin{gathered}P(U_{1},\cdots,U_{N};t_{1},\cdots,t_{N})=\left(\frac{1}{2\pi\tau_{N}}\right)^{D/2}\cdots\left(\frac{1}{2\pi\tau_{1}}\right)^{D/2}\\\
\times\int_{U_{N}}\cdots\int_{U_{1}}e^{-|x_{N}-x_{N-1}|^{2}/2\tau_{N}}\cdots
e^{-|x_{2}-x_{1}|^{2}/2\tau_{2}}e^{-|x_{1}-x_{0}|^{2}/2\tau_{1}}~{}d^{D}x_{N}\cdots
d^{D}x_{1},\end{gathered}$ (V.1)

wherein $\tau_{i}=t_{i}-t_{i-1}$, $1\leq i\leq N$. Also, we used
$x_{i}=(x_{i}^{1},\cdots,x_{i}^{D})$ and $d^{D}x_{i}$ respectively for the
coordinate system and the Lebesgue measure of $\mathbb{R}^{D}$ at $t_{i}$
(i.e. $x_{i}=x_{i}(t_{i})=x(t_{i})$), and employed the Euclidean metric
$|x|^{2}=(x^{1})^{2}+\cdots+(x^{D})^{2}$ for $\mathbb{R}^{D}$. Actually, the
Wiener measure for time slicing $t_{0}<t_{1}<\cdots<t_{N}$ is;

$dW(t_{N},\cdots,t_{1})=\left(\frac{1}{2\pi\tau_{N}}\right)^{D/2}~{}e^{-|x_{N}-x_{N-1}|^{2}/2\tau_{N}}~{}d^{D}x_{N}\times\cdots\times\left(\frac{1}{2\pi\tau_{1}}\right)^{D/2}~{}e^{-|x_{1}-x_{0}|^{2}/2\tau_{1}}d^{D}x_{1}.$
(V.2)

One may be interested to consider the symmetric form of the Wiener measure as:

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

which evaluates the corresponding amount of probability of a particle
initiating its Brownian motion from $x_{I}$ at $t_{0}=-T$, and terminating its
random journey at $x_{f}$ at $t_{N+1}=T$. Obviously, we easily see that:
$\int_{\mathbb{R}^{D}}\cdots\int_{\mathbb{R}^{D}}dW(t_{N},\cdots,t_{1})=1$. In
the following we only consider the symmetrized form of the Wiener measure
(V.3) for working out the physical interpretations due. Hence, if the particle
is subject to some external force given in terms of some measurable functions
$e^{V_{i}(x)}$ for potentials $V_{i}(x)$, $1\leq i\leq N$, to be bounded from
above within the partition of time $-T<t_{1}<\cdots<t_{N}<T$, then the mean
value of the influence of $V_{i}$s to the particle being located in
$U_{i}\subset\mathbb{R}^{D}$ at $t_{i}$, $1\leq i\leq N$, is given as:

$P(U_{1},\cdots,U_{N};t_{1},\cdots,t_{N})=\int_{U_{N}}\cdots\int_{U_{1}}e^{-V_{N}(x_{N})}\cdots
e^{-V_{1}(x_{1})}dW(t_{N},\cdots,t_{1}).$ (V.4)

However, the probabilistic interpretation of the above Wiener integral
entirely depends on the theory. Also, the term of particle which we used above
addresses essentially to an element of the target space $\mathbb{R}^{D}$.
Hence, if $\mathbb{R}^{D}$ is the space of some special Fourier modes
satisfying some specific condition, the above Brownian interpretation of the
Wiener integral must be translated to the space of the corresponding functions
included in the integral.

It is worth to note that the sufficient condition for potentials $V_{i}$ in
(V.4) to lead to some finite value for
$P(U_{1},\cdots,U_{N};t_{1},\cdots,t_{N})$ is;

$\int_{\mathbb{R}^{D}}e^{-a|x-x_{0}|^{2}+V_{i}(x)}d^{D}x<\infty$ (V.5)

for any $a>0$ and $x_{0}\in\mathbb{R}$, which is certainly fulfilled by
potentials that are bounded from above. We may refer to (V.5) as the _Wiener
convergence condition_. Moreover, the measurable function
$F(x_{N}(t_{N}),\cdots,x_{1}(t_{1}))=e^{V_{N}(x_{N}(t_{N}))}\cdots
e^{V_{1}(x_{1}(t_{1}))}$ or each of the potentials $V_{i}(x)$ which satisfies
(V.5) is said to be a _Wiener convergent_ Function.

Now, let $\mathcal{N}\in\mathbb{N}$, set $S=[-L,L]^{D}\subset\mathbb{R}^{D}$,
and define $C_{\mathcal{N}}$ to be the space of continuous functions
$f:S\to\mathbb{R}$ with the Fourier coefficients $f_{n}$ which vanish except
for Fourier modes $n=(n_{1},\cdots,n_{D})$ satisfying $-\mathcal{N}\leq
n_{i}\leq\mathcal{N}$ for all $1\leq i\leq D$. Here, for simplicity and
without loss of generality one can assume that each $f_{n}$ is a real number.
For instance, one may suppose that a negative (resp. non-negative) component
of $n=(n_{1},\cdots,n_{D})$ corresponds to the same Fourier sine (resp.
cosine) coefficient in that dimension. Similarly, for the Fourier complex
coefficients calculated for the Fourier basis $\\{e^{i\pi
n.x/L}\\}_{n\in\mathbb{Z}^{D}}$, the Fourier coefficients $f_{n}$ and $f_{-n}$
provide two independent real variables as $\frac{f_{n}+f_{-n}}{2}$ and
$\frac{f_{n}-f_{-n}}{2i}$, which may also be referred to as $f_{n}$ and
$f_{-n}$ in our formalism. All in all, $C_{\mathcal{N}}$ is a vector space of
$D_{\mathcal{N}}=(2\mathcal{N}+1)^{D}$ real dimensions $\\{f_{n}\\}$ each of
which is labeled with an allowed Fourier mode $n$. We refer to elements of
$C_{\mathcal{N}}$ as _$\mathcal{N}$ -bounded functions_. Let us define the
_Lebesgue fractal measure_ on $C_{\mathcal{N}}$ by means of the fractal norm
on each coordinate as:

$d\ell_{(\kappa,\mu)}(f_{n})=e^{\kappa(n^{2}+\mu^{2})f_{n}^{2}/2}df_{n}.$
(V.6)

In principle, this election for $C_{\mathcal{N}}$ naturally leads to the
_Wiener fractal measure_ as:

$\begin{gathered}dW_{\mathcal{N}}(t_{N},\cdots,t_{1})=\left(4\pi
T\right)^{D_{\mathcal{N}}/2}~{}\text{exp}\left(\frac{1}{4T}\sum_{n}\left[\int_{f_{n}(-T)}^{f_{n}(T)}e^{\kappa(n^{2}+\mu^{2})x^{2}/2}dx\right]^{2}\right)\\\
\times\left(\frac{1}{2\pi\tau_{N+1}}\right)^{D_{\mathcal{N}}/2}~{}\text{exp}\left(\sum_{n}\Bigg{\\{}-\frac{1}{2\tau_{N+1}}\left[\int_{f_{n}(t_{N})}^{f_{n}(T)}e^{\kappa(n^{2}+\mu^{2})x^{2}/2}dx\right]^{2}\Bigg{\\}}\right)\times\prod_{i=1}^{N}\left(\frac{1}{2\pi\tau_{i}}\right)^{D_{\mathcal{N}}/2}\\\
\times~{}\text{exp}\left(\sum_{n}\Bigg{\\{}-\frac{1}{2\tau_{i}}\left[\int_{f_{n}(t_{i-1})}^{f_{n}(t_{i})}e^{\kappa(n^{2}+\mu^{2})x^{2}/2}dx\right]^{2}+\frac{1}{2}\kappa(n^{2}+\mu^{2})f_{n}^{2}(t_{i})\Bigg{\\}}\right)~{}d^{D_{\mathcal{N}}}f_{n}(t_{i}),\end{gathered}$
(V.7)

which essentially describes the Brownian motion of $\mathcal{N}$-bounded
functions within a fractal space. Thus, the $\mathcal{N}$-bounded functions
are subject to a Wiener stochastic process of fractals while they evolve in
time. On the other hand, for any Wiener convergent measurable function
$F(x_{n}(t_{N}),\cdots,x_{n}(t_{1}))$ the following integral exists and is
well-defined for any $\mathcal{N}$;

$I_{\mathcal{N}}:=\int_{U^{\mathcal{N}}_{N}}\cdots\int_{U^{\mathcal{N}}_{1}}F_{\mathcal{N}}(f_{n}(t_{N}),\cdots,f_{n}(t_{1}))dW_{\mathcal{N}}(t_{N},\cdots,t_{1})$
(V.8)

wherein $F_{\mathcal{N}}(f_{n}(t_{N}),\cdots,f_{n}(t_{1}))$ is the restriction
of $F(f_{n}(t_{N}),\cdots,f_{n}(t_{1}))$ to $C_{\mathcal{N}}$ and
$U^{\mathcal{N}}_{i}=U_{i}\bigcap C_{\mathcal{N}}$. Actually, we are mostly
interested in the behavior of the sequence
$\\{I_{\mathcal{N}}\\}_{\mathcal{N}=1}^{\infty}$ as $\mathcal{N}\to\infty$. It
is worth to note that through this viewpoint the Wiener convergence condition
must be replaced by $I_{\mathcal{N}}$ being a convergent sequence as
$\mathcal{N}\to\infty$. A Wiener-integrable function
$F(f_{n}(t_{N}),\cdots,f_{n}(t_{1}))$ which provides a convergent sequence
$I_{\mathcal{N}}\to I$ is called a _Wiener renormalizable_ function. The well-
defined value $I$ is also referred to as the _path-integral of
$F(f_{n}(t_{N}),\cdots,f_{n}(t_{1}))$_.

It is easy to see that if
$F(f_{n}(t_{N}),\cdots,f_{n}(t_{1}))=e^{V_{N}(f_{n}(t_{N}))}\cdots
e^{V_{1}(f_{n}(t_{1}))}$ is bounded to the unity from above, which happens for
negative potentials $V_{i}(f_{n}(t_{i}))$, then $\\{I_{\mathcal{N}}\\}$ would
be a decreasing sequence, hence convergent to some definite
$I\in\mathbb{R}$.171717Obviously, if $V_{i}(f_{n}(t_{i}))\leq 0$, then
integration on more Fourier modes with the Wiener fractal measure will lead to
a less amount, hence $I_{\mathcal{N}}$ is decreasing and convergent. This
condition is definitely equivalent to the set of energy eigenvalues being
bounded from below, which is mandatory to gain a well-defined quantum field
theory. However, as we will see in the following, all local potentials
$V_{i}(f_{n})$ that are capable to be written via an integral over the domain
space $S$ as the interaction term of some renormalizable scalar quantum field
theory lead to integrable functions $F(f_{n}(t_{N}),\cdots,f_{n}(t_{1}))$
which are of this class. Therefore, all the local renormalizable interaction
terms of scalar quantum field theories have intimate correlations to Wiener
renormalizable functions.181818As we will see in the following there is a
slight difference between Wiener renormalizability and the renormalizability
of a quantum field theory, while actually, the former contains the later as a
proper subclass.

In the following, we see that the above formulations definitely provide the
same definition of the path-integral approach in quantum field theory. This is
done by transferring to the complex space via the analytic continuation
mechanism and imposing an infinitesimal heat kernel into the fractal mass to
control the possible divergences which may emerge in approximating the Wiener
fractal measure. Actually, (V.7) and (V.8) provide one of the most consistent
formulations of path-integral framework for scalar quantum field theories. The
approximated version of (V.7) will lead to two interesting achievements for
relativistic quantum field theories which are the main subjects of the next
two sections.

##  VI Emergence of Lorentz Symmetry

Actually, according to the included highly non-local terms calculating the
Wiener path-integral (V.8) is a highly complicated process unless we consider
some appropriate approximations of the Wiener fractal measure. However, we
should emphasize that the only correct and exact measure which must be
employed to extract the Green’s functions of the quantum field theory is the
original formula of (V.7), hence any approximation of the Wiener fractal
measure will lead to some approximated amplitudes for expectation values of
the quantum field theory. In principle, we are interested in a local
approxmated version of $dW_{\mathcal{N}}(t_{N},\cdots,t_{1})$ which is
actually derived for two simplifying assumptions as following:

i) Ignore the contributions at high amounts of $f_{n}$s, which upon Theorem 1
and Theorem 2 it means to exclude quantum fields with highly fractality. In
principle, it is equivalent to suppose each
$U^{\mathcal{N}}_{i}\subset\mathbb{R}^{D_{\mathcal{N}}}$ in (V.8) is a compact
set.

ii) Set $N\to\infty$, which leads to: $\tau_{i}=\theta\to 0$, $1\leq i\leq N$.

With these assumptions we have;

$\Bigg{\\{}\int_{f_{n}(t_{i-1})}^{f_{n}(t_{i})}e^{\kappa(n^{2}+\mu^{2})x^{2}/2}dx\Bigg{\\}}/\tau_{i}~{}~{}~{}\mapsto~{}~{}~{}\partial_{t}f_{n}(t_{i})e^{\kappa(n^{2}+\mu^{2})f_{n}^{2}(t_{i})/2}=\partial_{t}f_{n}(t_{i})+\cdots,$
(VI.1)

thus (V.7) goes to

$\begin{gathered}dW_{\mathcal{N}}(t_{N},\cdots,t_{1})\\\
\approx\text{exp}\left(\sum_{n}\Bigg{\\{}-\frac{1}{2}\left(\partial_{t}f_{n}(t_{i})\right)^{2}\tau_{i}+\frac{1}{2}\kappa(n^{2}+\mu^{2})f_{n}^{2}(t_{i})\Bigg{\\}}\right)\times
C_{I}^{F}\times\prod_{i=1}^{N}\left(\left(\frac{1}{\sqrt{2\pi\theta}}\right)^{D_{\mathcal{N}}}d^{D_{\mathcal{N}}}f_{n}(t_{i})\right)\\\
\approx\text{exp}\left(\frac{c}{\mathfrak{h}}\int_{X}\Bigg{\\{}-\frac{1}{2c^{2}}\left(\partial_{t}f(t,x)\right)^{2}+\frac{1}{2}\sum_{i=1}^{D}\left(\partial_{i}f(t,x)\right)^{2}+\frac{1}{2}\frac{\mathfrak{m}^{2}c^{2}}{{\mathfrak{h}}^{2}}f^{2}(t,x)\Bigg{\\}}d^{D}xdt\right)~{}\mathfrak{D}f~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
=\text{exp}\left(\frac{1}{\mathfrak{h}}\int_{X}\Bigg{\\{}-\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}f(t,x)\partial_{\nu}f(t,x)+\frac{1}{2}\frac{\mathfrak{m}^{2}c^{2}}{{\mathfrak{h}}^{2}}f^{2}(t,x)\Bigg{\\}}d^{D}xdx^{0}\right)\mathfrak{D}f,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{gathered}$
(VI.2)

wherein we have set:

$\mathfrak{m}=\frac{\pi^{3}\mu\theta
L^{D-3}}{\kappa},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}c=\frac{L\sqrt{\kappa}}{\pi\sqrt{\theta}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathfrak{h}=\frac{\pi\sqrt{\theta}L^{D-1}}{\sqrt{\kappa}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X=[-T,T]\times
S,$ (VI.3)

while
$\mathfrak{D}f:=C_{I}^{F}\left(\frac{1}{2\pi\theta}\right)^{ND_{\mathcal{N}}/2}\prod_{i=1}^{N}d^{D_{\mathcal{N}}}f_{n}(t_{i})$
for the overall (initial-to-final) normalization factor $C_{I}^{F}$ given as:
$C_{I}^{F}=\left(4\pi
T\right)^{D_{\mathcal{N}}/2}~{}\text{exp}(\sum_{n}[\int_{f_{n}(-T)}^{f_{n}(T)}e^{\kappa(n^{2}+\mu^{2})x^{2}/2}dx]^{2}/4T)$.
Moreover, in the last line of (VI.2) use has been made of $dx^{0}=cdt=c\theta$
and $\eta^{\mu\nu}=\text{diag}(1,-1,\cdots,-1)$, the Lorentz metric in $D+1$
dimensions. Actually, to transfer to physical scales the dimension of $f$,
i.e. $T^{\frac{1}{2}}$, must be changed to
$M^{\frac{1}{2}}T^{-\frac{1}{2}}L^{\frac{1}{2}(3-D)}$. Let $\gamma$ be such an
scaling factor of dimension $M^{\frac{1}{2}}T^{-1}L^{\frac{1}{2}(3-D)}$ and
define $\phi=\gamma f$. Thus, with such redefinitions the Wiener fractal
measure turns to:

$\begin{gathered}dW_{\mathcal{N}}(t_{N},\cdots,t_{1})\\\
=\text{exp}\left(\frac{1}{\hbar}\int_{X}\Bigg{\\{}-\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi(t,x)\partial_{\nu}\phi(t,x)+\frac{1}{2}\frac{m^{2}c^{2}}{{\hbar}^{2}}\phi^{2}(t,x)\Bigg{\\}}d^{D+1}x\right)\mathfrak{D}\phi,\end{gathered}$
(VI.4)

for the physical mass $m=\mathfrak{m}\gamma^{2}$ and the Dirac constant
$\hbar=\mathfrak{h}\gamma^{2}$, while we have simply considered
$\mathfrak{D}\phi=C_{I}^{F}\frac{1}{\left(2\pi\theta\gamma^{2}\right)^{ND_{\mathcal{N}}/2}}\prod_{i=1}^{N}d^{D_{\mathcal{N}}}\phi_{n}(t_{i})$
as the _Feynman measure_. In fact, if $D=3$ and $L,T\to\infty$, then the
Wiener fractal measure becomes:

$dW_{\mathcal{N}}(t_{N},\cdots,t_{1})=\text{exp}\left(-\frac{1}{\hbar}\int_{\mathbb{R}^{4}}\mathcal{L}_{K-G}~{}d^{4}x\right)\mathfrak{D}\phi$
(VI.5)

for $\mathcal{L}_{K-G}$ the Klein-Gordon Lagrangian density. Therefore, the
Wiener fractal measure admits a natural Lorentz symmetry due to the
substantial fractality of the quantum scalar fields. More precisely, the
different sign of the time and the spatial dimensions within the Lorentz
metric is the immediate consequence of the fractality of $\phi$s. However, the
partial derivations in (VI.4) are absolutely well-defined and justifiable
since as far as we concentrate on ${\mathcal{N}}$-bounded functions the
included quantum fields in the Wiener fractal measure are thoroughly smooth.
Therefore, although the process of $\mathcal{N}\to\infty$ covers the whole
space of fractal fields, according to the definition of the path-integral via
(V.8), the smoothness of quantum fields is involved within the entire
framework of the path-integration.

Obviously, any scalar quantum field theory could be similarly formulated by
path-integration of some Wiener fractal convergent interacting function
$F(x_{n}(t_{N}),\cdots,x_{n}(t_{1}))=e^{V(x_{n}(t_{N}))}\cdots
e^{V(x_{n}(t_{1}))}$, by means of the approximated Wiener fractal measure
(VI.4). For example, for $\phi^{4}$-theory in $\mathbb{R}^{4}$ we have;
$V(x_{n})\propto-\sum_{n,m}\left(x_{m}x_{n-m}\right)^{2}\leq 0$. Consequently,
as we discussed above this interaction term leads to a Wiener fractal
convergent function with path-integral

$\int\text{exp}\left(-\frac{1}{\hbar}\int_{\mathbb{R}^{4}}\mathcal{L}_{\phi^{4}}~{}d^{4}x\right)\mathfrak{D}\phi:=\int_{U_{N}^{\mathcal{N}}:=\mathbb{R}^{D_{\mathcal{N}}}}\cdots\int_{U_{1}^{\mathcal{N}}=\mathbb{R}^{D_{\mathcal{N}}}}\text{exp}\left(-\frac{1}{\hbar}\int_{\mathbb{R}^{4}}\mathcal{L}_{\phi^{4}}~{}d^{4}x\right)\mathfrak{D}\phi$
(VI.6)

wherein
$\mathcal{L}_{\phi^{4}}=\mathcal{L}_{K-G}-\frac{\lambda\hbar^{-1}}{4!}\phi^{4}$
is the Lagrangian density of $\phi^{4}$-theory for some positive constant
$\lambda$.191919As we stated above $\lambda$ has to be a positive number for
the Wiener path-integral (VI.6) being a well-defined amplitude as
$\mathcal{N}\to\infty$. This is equivalent to the stability of the
corresponding $\phi^{4}$-theory. See zee  for more discussions about unstable
vacuum state of $\phi^{4}$-theory for negative $\lambda$. Actually, despite
assumption (i), here we considered the intermediate integral domains
$U^{\mathcal{N}}_{i}$s to be the whole space of $\mathcal{N}$-bounded
functions $C_{\mathcal{N}}$. Roughly speaking we have considered that the
contributions of highly fractal quantum fields could be ignored within a
concrete manner.

Upon the above assumption the Wiener path-integral (VI.6) is in principle, a
criterion for evaluating the probabilistic process for transferring the
initial state $\phi_{I}$, given by Fourier coefficients $f_{n}(-T)$s, to the
final state $\phi_{F}$, expanded by $f_{n}(T)$ in the phase space. However, if
the initial and the final states are set to be the vacuum field as
$f_{n}(T)=f_{n}(-T)=0$, for all $n$, and the potential exponential function
$F(x_{n}(t_{N}),\cdots,x_{n}(t_{1}))$ is multiplied by a given functional
$O(x_{n}(t_{N}))\times\cdots\times O(x_{n}(t_{1}))$, then the corresponding
Wiener path-integral would be

$\int\text{exp}\left(-\frac{1}{\hbar}\int_{\mathbb{R}^{4}}\mathcal{L}_{\phi^{4}}~{}d^{4}x\right)O(\phi)\mathfrak{D}\phi=\langle
O(\phi)\rangle_{\text{VEV}}$ (VI.7)

wherein $\langle O(\phi)\rangle_{\text{VEV}}$ is the vacuum expectation value
of the local operator $O(\phi)$. On the other hand, according to the LSZ
reduction formula

$\begin{gathered}\int\text{exp}\left(-\frac{1}{\hbar}\int_{\mathbb{R}^{4}}\mathcal{L}_{\phi^{4}}~{}d^{4}x\right)\Big{\\{}\phi_{n_{1}}(t_{N})\cdots\phi_{n_{k}}(t_{N})\phi_{m_{1}}(t_{1})\cdots\phi_{m_{l}}(t_{1})\Big{\\}}~{}\mathfrak{D}\phi\end{gathered}$
(VI.8)

for Fourier modes $n_{1},\cdots,n_{l},m_{1},\cdots,m_{k}\in\mathbb{Z}^{D}$,
and with Fourier coefficients $\phi_{n}(t)=\gamma f_{n}(t)$, effectively
describes the scattering matrix element ${{}_{\text{out}}}\langle
n_{1},\cdots,n_{k}|m_{1},\cdots,m_{l}\rangle_{\text{in}}$ modulo some overall
factors on the Fourier modes. The precise formula and the unitarity of this
matrix are better understood via the complexification of the Wiener fractal
measure in the next section. We emphasize that here we are working with
quantum fields in Heisenberg’s picture. According to the analytic structure of
the symmetric form of the Wiener measure, the emergent quantum field theory is
essentially time reversal, while upon the construction of the Wiener fractal
measure over the space of $\mathcal{N}$-bounded functions we see that the
theory admits the parity and the charge conjugate symmetries
due.202020However, in varshovi E-H  we will see that the gravity is an
entropic-geometric force that intrinsically respects the time’s arrow of the
second law of thermodynamics.

Before closing this section we may wish to have a new look at the physical
constants emerged via approximating the Wiener fractal measure in (VI.3). As
we see all these constants depend on either the parameters of the massive
fractal norm, i.e. $\kappa$ and $\mu$, or on the structure of space-time, i.e.
the spatial size $L$ and the time slicing width $\theta$, which the latter is
substantially imposed by the potential exponential function $F$. Actually,
$\kappa$ is of dimension $T^{-1}$, hence plays the role of frequency and
includes some fundamental information about the particle. In particular,
$\hbar\kappa$ would provide an important criterion for the energy levels of
the particle. Moreover, since $\kappa$ couples to the Laplacian term $n^{2}$
and the fractal mass $\mu$, it would tie the information of fractality within
the spatial and the time directions, which leads to space-time being as a unit
continuum via the approximation (VI.1).

However, because of its dimension $\kappa$ could be compared to $1/\theta$,
which in its turn is included in the interaction term due. Thus, the only
constant that explicitly inherits the fractal structure of quantum fields
through the spatial dimensions is the fractal mass $\mu$. As we saw above
$\mu$ dependends partly on the spatial extension of the fractal field and
partly on the type of the self-similarity due. For instance for a localized
Weierstrass function of $(a,b)$-self-similarity we have:

$\mu\propto\sqrt{\frac{a(b-a)}{s(1-a)(b-a)}}\sqrt{\frac{L-\mathfrak{l}}{L}}=\text{Fractality}\times\text{Spacial
Property}.$ (VI.9)

This confirms our physical intuition since it shows that the fractality of a
wave function would be an equivalent parameter for the physical quantity of
energy distributed through the spatial dimensions. Thus, the energy has a
background amount corresponding to the type of self-similarity $(a,b)$,
whereas the rest amount is included in the Laplacian term $n^{2}$ coming from
the general fractal structure due to the fractal norm.212121This resembles the
energy formula of a quantum harmonic oscillator
$E=\hbar\omega(\frac{D}{2}+\sum_{i=1}^{D}n_{i})$, which has a ground state
with $E=\frac{D}{2}\hbar\omega$ comparable to the mass term $\kappa\mu$.

However, it is worth noting that based on distinct natures of spatial and time
dimensions in the Wiener Brownian process, we deliberately use the terminology
of _space-time_ , but not _spacetime_ , for addressing the product manifold
$X=[-T,T]\times S$, which appeared in (VI.2) and (VI.3). We will retain this
convention in upcoming perusals varshovi E-H ; varshovi .

##  VII Wiener Complex Fractal Measure and Quantum Field Theory

Actually, the Wiener fractal measure loses its consistency after the
approximation mentioned within the last section. In fact, the negative term in
the exponential function of the original Wiener fractal measure (V.7) always
exceeds the corresponding positive parts, hence the Wiener fractal measure is
substantially given by means of some numbers of Gaussian functions. Therefore,
the formulation in (V.7) is well-defined and bounded for any $f_{n}$. However,
when one transfers from (V.7) to (VI.2) via approximation (VI.1) this
consistency is lost. In principle, the replacement of

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\int_{f_{n}(t_{i-1})}^{f_{n}(t_{i})}e^{\kappa(n^{2}+\mu^{2})x^{2}/2}dx~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mapsto~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}f_{n}(t_{i})-f_{n}(t_{i-1})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1\leq
i\leq N)$ (VII.1)

in the exponent of the Gaussian function via the approximation approach causes
the Wiener fractal measure to lose its powerful damping mechanism and get
infinite values, hence being inconsistent. To overcome this problem, we have
to compensate the lost dominated heat kernel which was replaced by a linear
term through the approximation process (VII.1). In fact, incorporating any
heat kernel beside the approximated exponent term would not guarantee the
convergence of the approximated measure, unless the exponent term is a pure
imaginary function which in this case any small quadratic component will
restore the lost consistency.

In principle, although the integration of $e^{-ax^{2}}$ on $\mathbb{R}$ is not
always well-defined for general $a$, its complexified version $e^{iax^{2}}$
would have a finite integral but at the cost of adding some augmented (but
infinitesimal) quadratic component into the exponent. Actually, the main idea
for treating the pathology of the approximated Wiener fractal measure stems
from the analytic continuation of the measure and employing an infinitesimal
heat kernel term according to the fallowing simple equation;

$\Big{|}\int_{\mathbb{R}}e^{(ia-\varepsilon)x^{2}}dx\Big{|}=\sqrt{\frac{\pi}{\sqrt{a^{2}+\varepsilon^{2}}}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\underrightarrow{\varepsilon\to
0^{+}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sqrt{\frac{\pi}{|a|}}=\int_{\mathbb{R}}e^{-|a|x^{2}}dx,$
(VII.2)

or more generally:

$\lim_{\varepsilon\to
0^{+}}\Big{|}\int_{\mathbb{R}}e^{(ia-\varepsilon)x^{2}}f(x)dx\Big{|}=\Big{|}\int_{\mathbb{R}}e^{-|a|x^{2}}f(x)dx\Big{|},$
(VII.3)

for any analytic function $f(x):\mathbb{R}\to\mathbb{R}$. Therefore, in order
to have a well-defined approximated Wiener fractal measure we should employ
the following replacement;

$\begin{gathered}\text{exp}\left(-\frac{1}{\hbar}\int_{\mathbb{R}^{D+1}}\Bigg{\\{}\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi(t,x)\partial_{\nu}\phi(t,x)-\frac{1}{2}\frac{m^{2}c^{2}}{{\hbar}^{2}}\phi^{2}(t,x)\Bigg{\\}}d^{D+1}x\right)\\\
\mapsto\\\
\text{exp}\left(\frac{i}{\hbar}\int_{\mathbb{R}^{D+1}}\Bigg{\\{}\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi(t,x)\partial_{\nu}\phi(t,x)-\frac{1}{2}\frac{(m^{2}-i\varepsilon)c^{2}}{{\hbar}^{2}}\phi^{2}(t,x)\Bigg{\\}}d^{D+1}x\right).\end{gathered}$
(VII.4)

However, since the causality of quantum field theories is an immediate
consequence of the heat kernel of $i\varepsilon$-term peskin , we see that the
replacement (VII.4) successfully revives the natural causality of the Wiener
path-integral. An augmented beneficial $i\varepsilon$-term also appears in
demonstrating the Feynman propagators or expectation values of some local
operators via the limit of $T\to(1-i\varepsilon)\infty$ or equivalently for
setting $p_{0}\mapsto p_{0}+i\varepsilon$ with $p_{0}$ the energy of virtual
particles peskin . This mechanism is actually an alternative procedure for the
replacement of $m\mapsto m-i\varepsilon$ within the Lagrangian density.

More precisely, the $i\varepsilon$-term is an appropriate alternative for the
fractal norm in the exponent of the Wiener fractal measure. More generally,
upon to the above argument it can be inferred that in all path-integral
formulations of quantum physics the $i\varepsilon$-term plays the role of the
removed non-local fractal norm from the Lagrangian density in the exponent of
the Gaussian function of the Wiener measure.222222See witten  for the role of
$i\varepsilon$-term in string theory. On the other hand, according to the
significance of the $i\varepsilon$-term in figuring out the structure of the
vacuum state of a quantum field theory weinberg , it seems that the vacuum
state has an intimate correlation to the fractal structure of quantum fields.

Actually, the above machinery for working out the path-integral formulation of
scalar field theory could be simply generalized for a quantum field theory of
$\mathfrak{n}$ independent quantum fields $\phi_{(i)}$, $1\leq
i\leq\mathfrak{n}$. In this case the well-defined complexified approximated
Wiener fractal measure would be;

$\text{exp}\left(\frac{i}{\hbar}\int_{\mathbb{R}^{D+1}}\sum_{j=1}^{\mathfrak{n}}\Bigg{\\{}\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi_{(j)}(t,x)\partial_{\nu}\phi_{(j)}(t,x)-\frac{1}{2}\frac{(m^{2}-i\varepsilon)c^{2}}{{\hbar}^{2}}\phi_{(j)}^{2}(t,x)\Bigg{\\}}d^{D+1}x\right)\mathfrak{D}\phi.$
(VII.5)

Specially, this well-defined measure naturally leads to the Maxwell theory for
$\mathfrak{n}=3$ and $m=0$ in the Coulomb gauge; $\partial_{i}A_{i}=A_{0}=0$.
Therefore, it seems that the Wiener fractal process could be similarly
considered for gauge field theories via some appropriate modifications within
the fractal norm and consequently the Lebesgue fractal measure.232323This is
the main subject of varshovi . Moreover, Feynman’s path-integral formulation
of quantum mechanics can be worked out via the machinery we used above to
extract the scalar quantum field theories in a natural way within the real
analysis framework based on purely geometric and stochastic foundations. This
is the subject of Appendix A.

In particular, the main conclusion of the last two sections is that the
Lorentz symmetry in the background of relativistic quantum field theories
stems from a natural feature of fractal wave functions, which emerges as the
fractal norm within the phase space of Fourier modes. As we saw above, the
fundamental assumption for extracting the symmetry is the classical separation
of space and time as two distinct entities. This idea has two major
similarities to the Horava-Lifschitz theory of gravity horava :

i) _Spacial dimensions are non-equivalent (anisotropic) at high energy
levels._

ii) _Space is actually defined via the time foliations of the space-time
4-manifold._

Although the Horava’s theory is a candidate formulation for quantum gravity it
has fundamental similarities to our derived formulas for scalar quantum field
theory. Actually, the latter Horava’s assumption is located at the center of
our formalism which defines the causality via the foliation of space-time.
But, the former has an intimate correlation to the original Wiener fractal
measure (V.7). In fact, the approximation process (VI.1) of the Wiener fractal
measure ceases for high values of Laplacian term $n^{2}$ and must contain next
to the leading terms within the action. In fact, one has to consider at least
$n^{2}f_{n}^{2}(t)(\partial_{t}f_{n}(t))^{2}$ for compact
$U^{\mathcal{N}}_{i}$s within the Fourier transform of the Lagrangian density,
which is in fact a non-local term. However, Horava’s prescription is to employ
local terms with higher spatial derivatives at the Lifshitz point $z=3$ and to
impose the symmetry of foliation preserving diffeomorphism invariance on the
theory. Actually, in varshovi E-H  where we have worked out the Einstein-
Hilbert theory from the Wiener fractal measure, more similarities have been
discussed and established between the Horava-Lifshits gravity and the Wiener
Brownian process of fractals.

Before closing this section we prefer to have a brief look at the Wick
rotation process. Wick rotation as a reliable method for obtaining solutions
in Minkowski space in terms of those in Euclidean space via replacing some
imaginary variables with real ones, is indeed an immediate consequence of
complexification of the approximated Wiener fractal measure (VI.2). This
process could be in fact regarded as an (almost) equivalence relation between
Lorentzian and Euclidean signatures of the spacetime
metric.242424Nevertheless, there are physical phenomena that reject this
equivalence relation within some relativistic effects of quantum field theory.
See sorkin  for a more detailed discussion about the problem. Actually, the
original formulation of the Wiener fractal measure (V.7) explains obviously
why nature does not embrace the Euclidean signature intrinsically. In
particular, there is no correct signature of the space-time continuum and all
we obtain within local Lagrangian formulations of nature are approximated
structures at low energy regimes of the Wiener Brownian process. However,
after accomplishing the approximation procedure (VI.1) and obtaining (VI.2) it
would be natural to employ the regularization method of (VII.3), which would
lead us to thermal quantum field theories and the functional methods in
condensed matter peskin .252525See landsman  for an alternative approach to
real-time formalisms in thermal field theory via Bogoliubov transformations,
the so called thermo field dynamics.

##  VIII Wiener Fractal Measure and Renormalization

Now we are ready to have a new look to the renormalizability problem of
quantum field theories. Actually, the Wiener fractal measure provides a
profound understanding of renormalizability. To see this within a better way
let us turn back to the original (real valued) Wiener fractal measure and the
equatuin of (V.8). One should note that each individual integral of (V.8) for
a local potential $V_{i}(f_{n})$ is actually of the following form

$\mathcal{J}_{\mathcal{N}}=\int_{\mathbb{R}^{D_{\mathcal{N}}}}\text{exp}\left(\Bigg{\\{}-\frac{1}{2\tau}\sum_{n}\Big{|}\int^{f_{n}}_{y_{n}}e^{\kappa(n^{2}+\mu^{2})x^{2}/2}dx\Big{|}^{2}\Bigg{\\}}+\int_{S}\Big{\\{}\frac{\kappa}{2}\left(|\nabla
f|^{2}+\mu^{2}f^{2}\right)+V_{i}(f)\Big{\\}}~{}d^{D}x\right)d^{D_{\mathcal{N}}}f_{n},$
(VIII.1)

for some $y_{n}\in\mathbb{R}$ and with $\nabla f$ the gradient of $f$.
Therefore, $I_{\mathcal{N}}$ is roughly written as;
$I_{\mathcal{N}}\propto\mathcal{J}_{\mathcal{N}}^{N}$. In fact, in physics we
readily assume that the number of the time slices $N$ could be enlarged
arbitrarily, hence we may simply replace the role of $N$ with
$g(\mathcal{N})$, for $g(\mathcal{N}):\mathbb{N}\to\mathbb{N}$ a monotone
increasing function, and fix $t_{g(\mathcal{N})}=T$ and
$t_{-g(\mathcal{N})}=-T$ for all $\mathcal{N}$. This assumption may cause a
slight redefinition of the Wiener renormalizable functions, which is now
referred to as the _Wiener renormalizable sequence_. Actually, a Wiener
renormalizable sequence is a set of functions
$\\{F_{\mathcal{N}}(f_{n}(t_{g(\mathcal{N})}),\cdots,f_{n}(t_{-g(\mathcal{N})})\\}$,
which has two properties;

i) Each
$F_{\mathcal{N}}(f_{n}(t_{g(\mathcal{N})}),\cdots,f_{n}(t_{-g(\mathcal{N})}))$
is a Wiener fractal convergent function in $C_{\mathcal{N}}$.

ii) The sequence of the Wiener path-integrals

$I_{\mathcal{N}}=\int
F_{\mathcal{N}}(f_{n}(t_{g(\mathcal{N})}),\cdots,f_{n}(t_{-g(\mathcal{N})}))dW_{\mathcal{N}}(t_{g(\mathcal{N})},\cdots,t_{-g(\mathcal{N})})$
(VIII.2)

converges to some definite value $I\in\mathbb{R}$, the so called _Wiener path-
integral of $F_{\mathcal{N}}$s_.

Indeed, a renormalizable quantum field theory is a Wiener renormalizable
sequence that fulfills three more conditions;262626However, we should
emphasize the these three properties are in fact fractal-geometric/stochastic
interpretation of RG flow. For example, the property (iv) effectively provides
an equivalent definition for effective field theories according to the RG
flow.

iii) The potential exponential functions
$F_{\mathcal{N}}(f_{n}(t_{g(\mathcal{N})}),\cdots,f_{n}(t_{-g(\mathcal{N})}))$
could be written by means of a local potential $V_{\mathcal{N}}(f_{n})$ within
the time slicing $t_{-g(\mathcal{N})}<\cdots<t_{g(\mathcal{N})}$ as;

$F_{\mathcal{N}}(f_{n}(t_{g(\mathcal{N})}),\cdots,f_{n}(t_{-g(t_{\mathcal{N}})})=\exp\left({\int_{S}V_{\mathcal{N}}(f_{n}(t_{g(\mathcal{N})})~{}d^{D}x}\right)\cdots\exp\left({\int_{S}V_{\mathcal{N}}(f_{n}(t_{-g(\mathcal{N})})~{}d^{D}x}\right).$
(VIII.3)

iv) For each $\mathcal{N}$ the potential $V_{\mathcal{N}}(f_{n})$ is a
polynomial of $f_{n}$s of order $\geq 2$ which contains a fixed number of
constants. The coefficient of $f_{n}^{2}$ in $V_{\mathcal{N}}(f_{n})$ is
effectively the _mass renormalization_ and the coefficients of the higher
order terms are the _running coupling constants._ The emergent coefficient $Z$
before the Gaussian term of the Wiener fractal measure is indeed the _field
strength renormalization._

v) If $g(\mathcal{N})-g(\mathcal{N}-1)\propto\frac{cT}{L}$, as
$\mathcal{N}\to\infty$, then integrating out the highest frequencies and the
intermediate time intervals causes the following _renormalization group_
process;272727Actually, if
$\mathfrak{L}=\\{-g(\mathcal{N}),\cdots,g(\mathcal{N})\\}$ is a lattice, then
any real function $\mathfrak{f}:L\to\mathbb{R}$ could be simply converted to a
real function as $\tilde{\mathfrak{f}}:[-T,T]\to\mathbb{R}$ via the inverse
Fourier transformation as:
$\tilde{\mathfrak{f}}(t)=\sum_{n=1}^{g(\mathcal{N})}\mathfrak{f}(n)\cos\left(\frac{n\pi
t}{T}\right)+\sum_{n=-1}^{-g(\mathcal{N})}\mathfrak{f}(n)\sin\left(\frac{|n|\pi
t}{T}\right)+\frac{1}{2}\mathfrak{f}(0).$ Hence, transferring from
$\mathcal{N}$ to $\mathcal{N}-1$ will cause $\tilde{\mathfrak{f}}(t)$ to lose
$\Delta E=\left(g(\mathcal{N})-g(\mathcal{N}-1)\right)\frac{\hbar\pi}{T}$
amount of energy. We use this idea for potential exponential functions
$F_{\mathcal{N}}(f_{n}(t_{g(\mathcal{N})}),\cdots,f_{n}(t_{-g(t_{\mathcal{N}})})$,
and set $\Delta E\propto\frac{\hbar\pi}{L}$. This, revives the Wilsonian
process of integrating out the highest amount of energy included in a cutoff
shell with fixed radius and thickness $\Delta E$ in the Euclidean momentum
space given by the Wick rotation.

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}F_{\mathcal{N}}(f_{n}(t_{g(\mathcal{N})}),\cdots,f_{n}(t_{-g(\mathcal{N})}))\stackrel{{\scriptstyle}}{{\longrightarrow}}F_{\mathcal{N}-k}(f_{n}(t_{g(\mathcal{N}-k)}),\cdots,f_{n}(t_{-g(\mathcal{N}-k)})),~{}~{}~{}~{}~{}(k\in\mathbb{N})$
(VIII.4)

as $\mathcal{N}\to\infty$, $T\to\infty$, and $L\to\infty$. Through this
process $b=\lim_{\mathcal{N}\to\infty}\frac{g(\mathcal{N}-1)}{g(\mathcal{N})}$
effectively describes the RG flow _scaling parameter_ , and the survived
polynomials in the limit of $b\to 0$ are the _relevant operators_.

Actually, the above process in performing the Wiener path-integrals can be
similarly converted to the complex version of the approximated Wiener fractal
measure, the so called _Wiener complex fractal measure_. Hence, the process of
integrating out the highest frequencies and the intermediate time sections in
(v) will naturally lead to the renormaliation group method for relativistic
quantum field theories. It is worth to emphasize that both the Bogoliubov’s
bogo  and the Wilson’s wilson1 ; wilson2  viewpoints to the RG flow machinery
give rise to profound understandings of and guidelines towards the fractality
of quantum fields.282828See also zimm ; shirkov1 ; shirkov2  for more
discussion about Bogoliubov’s special insights toward the RG flow for quantum
fields.

The original Wiener fractal measure (V.7) is a pure probability measure, hence
its integrals would end up with finite values for appropriate renormalizable
potentials terms. Therefore, the infinities appearing in loop calculations in
interacting quantum field theories stem from the approximation we employed
above to extract a local Lagrangian formulation from the Wiener fractal
measure (V.7). In other words, the regularization methods are indeed
mathematical techniques to compensate for the fatal pathology of approximation
(VI.2).

On the other hand, as we explained above the renormalization group flow of the
Wiener path-integral (VIII.2) will cause the higher non-local terms to
disappear (approximately) and then the Wiener path-integral would lead to an
effective local field theory at low energy levels which contains only stable
renormalizable local interactions, i.e. the $\phi^{4}$-theory peskin
.292929And the Yukawa interaction, just when the spinor fields are involved in
the Wiener fractal measure. Thus, one may be optimistic that the Wiener path-
integral of Brownian motion of fractal structures on space-time could help us
with better understanding the original theory for scalar quantum fields at
high energy levels as the promised original theory within the RG flow
procedure.

##  IX Summery and Conclusions

This paper is in fact the first part of a series of studies which aims to
figure out and explains the fundamental features of gravity within a
consistent framework of mathematics which stands on a footing that consists of
Horava-Lifshitz gravity (that stems from quantum field theory), Verlinde’s
entropic gravity (with backgrounds in string theory and Holographic
principle), and the theory of stochastic gravity (that is a stochastic
modification of semi-classical gravity). In this present perusal, we are
essentially concerned about two principal aspects of quantm physics: 1) _The
Heisenberg’s uncertainty principle as one of the most essential features of
nature_ , and b) _The background fractal geometry of nature and path-integral
formulation of fundamental forces of nature_. We combined these two basic
foundations of quantum physics by allowing the fractal structures to evolve
stochastically within the Wiener Brownian process.

At the first step, we considered Hardy’s nowhere differentiability condition
for Weierstrass-like fractal functions and obtained a non-linear differential
equation for the asymptotic behavior of Fourier-Laplace coefficients due.
Finding the solution of the differential equation, we obtained an
exponentially increasing norm on the phase space, the so called fractal norm,
for both entire and widespread fractals (where the latter includes a matter
term). Then, we employed the Wiener measure of the fractal norm, known as the
Wiener fractal measure, to study the Brownian motion of fractal structures on
a flat space-time. We established that the Brownian motion of fractal
structures automatically leads to a number of interesting achievements:

a) _The Wiener measure shows up an intrinsic Lorentz symmetry at its first
local approximation_. Hence, it is established that Lorentz invariance could
be regarded as an approximate symmetry of local interactions of nature, as has
already been shown in some aspects of string theory horava . This conclusion
is obtained without any augmentation of mathematical tools that have imposed
to the Wiener measure by hand, such as the analytic continuation of the
measure or of the mass term,303030Which both cease the Wiener measure to be
still a well-defined probability measure. regularizing the measure and
geometric approximations which have already been done before.313131See for
example bodmann ; dau ; dau2 ; dau3 ; and ; nelson ; cameron1 ; cameron2 ;
chaichian . For another viewpoint to Feynman’s path-integral formulation see
ito1 ; ito2 ; gelfand ; kac . The readers are also referred to klauder  for a
historical review on path-integral and its various stochastic formulation via
the Wiener measure.

b) _The Wiener measure automatically gives rise to scalar field theory as the
admitted theory of Higgs boson in Standard Model_. This leads to a
probabilistic formalism of scalar quantum field theories within both real
(including non-local terms that retain and guarantee the finiteness of the
theory) and complex analysis (that consists of only local terms which are
accompanied with an $i\varepsilon$-mass term for consistency).

Actually, the naturalness of the extracted Wiener fractal measure guarantees
that all path-integral formulations of quantum physics could be figured out
via the Wiener Brownian stochastic process of the underlying fractal geometry
of fine structures of nature. Therefore, it seems that the original form of
the Wiener fractal measure can be regarded as a reliable framework to
formulate the fundamental interactions of nature including gravity (as we have
already established in varshovi E-H ) and Maxwell’s and Yang-Mills theories
(which is accomplished in varshovi ).

The distinguishing feature of this research with respect to previous similar
works that aimed to find a correlation between Feynman’s path integral and the
Wiener measure is that our achievements have been entirely independent of any
assumptions related to quantum physics and its mathematics (such as operator
algebra, analytic continuation, and even fundamental basics of Feynman’s path-
integral formalism), and the whole results have been obtained from a purely
geometric and stochastic cornerstone. Hence, one may be optimistic that the
achieved results of this perusal may help us to perceive and to obtain more
profound intuitions about the foundations of quantum physics and fundamental
interactions of nature.

##  X Acknowledgments

The author says his special gratitude to Saya Ziaee who was the main
motivation for appearing this article. Also, it should be noted that this
research was in part supported by a grant from IPM (No. 1400810418). Finally,
the author is proud to dedicate this article to Ferdowsi, the great Persian-
speaking literary and the poet of Shahname, for his respectful status in
Iranian culture on the occasion of his birthday on January 21st.

##  Appendix A
Quantum Mechanics Revisited via the Wiener Measure

Actually, the Feynman path-integral formulation of quantum mechanics is also
understood via the complexification (VII.3). In fact, despite some viewpoints
which attempt to work out the Feynman path-integral formulation within complex
analysis methods, the machinery of the Wiener fractal measure can extract the
whole framework of quantum mechanical path-integral through with a natural way
given via entirely real analysis mechanism.323232See for example nelson .
Through this way the Wiener fractal measure is focused on the space of
$C_{\mathcal{N}_{0}}$, for some fixed $\mathcal{N}_{0}$, so that the whole
reserved energy in $C_{\mathcal{N}_{0}}$ is of order $mc^{2}$, for $m$ the
mass of the quantum particle; i.e.:

$\sum_{n}\frac{\pi^{2}\hbar^{2}n^{2}}{L^{2}}=\left(\frac{2D\pi^{2}\hbar^{2}}{L^{2}}\right)\frac{\mathcal{N}_{0}(\mathcal{N}_{0}+1)(2\mathcal{N}_{0}+1)}{6}\propto
mc^{2}.$ (A.1)

Then, one may neglect the high energy contributions. In principle, in quantum
field theory, one is mostly concerned about the asymptotic behavior of the
Wiener path-integrals $I_{\mathcal{N}}$ introduced in (V.8) as
$\mathcal{N}\to\infty$, hence the renormalization approach is mandatory. But,
here, we will not wonder such asymptotic behaviors and therefore, the quantum
mechanics needs no renormalization and the fractality assumptions due within
our framework.

To work out the Feynman’s quantum mechanical path-integral we have to consider
a semi-classical decomposition. To see this we need some simple definitions.
Let $D^{x_{0}}(x)$, for $x_{0}\in S$, be the restriction of the Dirac delta
function $\delta\left((x-x_{0})/L\right)$ to $C_{\mathcal{N}_{0}}$ multiplied
by $\gamma\sqrt{\theta}$, which is inserted for adjusting the physical
dimensions. Set $D_{\mathcal{N}_{0}}:=\\{D^{x_{0}}(x)|x_{0}\in S\\}$, the so
called _$\mathcal{N}_{0}$ -bounded Dirac space_. Actually,
$D_{\mathcal{N}_{0}}$ is a $D$-dimensional subspace of $C_{\mathcal{N}_{0}}$
with Fourier coefficient;333333Note that here we have no summation rule on
similar indices inside the arguments of trigonometric functions..

${D^{x_{0}}}_{n}=\gamma\sqrt{\theta}\left(\prod_{n_{i}=0}\frac{1}{2}\times\prod_{n_{i}>0}\cos\left(n_{i}\pi
x^{i}_{0}/L\right)\times\prod_{n_{i}<0}\sin\left(|n_{i}|\pi
x^{i}_{0}/L\right)\right).$ (A.2)

for Fourier mode $n=(n_{1},\cdots,n_{D})$. Let us get more information about
the $\mathcal{N}_{0}$-bounded Dirac space. Set
$A_{n}:=\\{n^{\prime}=(n^{\prime}_{1},\cdots,n^{\prime}_{D})|n^{\prime}_{i}=\pm
n_{i}\\}$ with $n=(n_{1},\cdots,n_{D})$ which fulfills $0\leq
n_{i}\leq\mathcal{N}_{0}$ for all $1\leq i\leq D$. Then, we see that:
$\sum_{n^{\prime}\in
A_{n}}{D^{x_{0}}_{n}}^{2}=\frac{\gamma^{2}\theta}{4^{z_{n}}}$, where $z_{n}$
is the number of zeros in $n$. Therefore, we obtain:
$\sum_{n}{D^{x_{0}}_{n}}^{2}=\gamma^{2}\theta\left(\frac{1}{4}+\mathcal{N}_{0}\right)^{D}=R_{\mathcal{N}_{0}}^{2}$,
thus, the $\mathcal{N}_{0}$-bounded Dirac space is a $D$-dimensional subspace
of $C_{\mathcal{N}_{0}}$, included in the sphere of radius
$R_{\mathcal{N}_{0}}$. Therefore, the $\mathcal{N}_{0}$-bounded Dirac space is
a compact subset. Now, let $D_{\mathcal{N}_{0}}^{\perp}$ be the perpendicular
dimensions to $D_{\mathcal{N}_{0}}$ in $C_{\mathcal{N}_{0}}$. Hence, the
Wiener fractal measure is decomposed as:

$dW_{\mathcal{N}_{0}}=dW_{D_{\mathcal{N}_{0}}}\times dW^{\perp}.$ (A.3)

Actually, the $\mathcal{N}_{0}$-bounded Dirac space is compact with bounded
values of Fourier coefficients, hence admits no fractal norm in its
dimensions. Thus, we can easily set $\kappa=0$ and ignore the Laplacian and
the mass terms in $dW_{D_{\mathcal{N}_{0}}}$. On the other hand, since
$D_{\mathcal{N}_{0}}$ is a $D$-dimensional space we simply read;
$dW_{D_{\mathcal{N}_{0}}}\propto\mathfrak{D}x$, for
$\mathfrak{D}x:=\prod\frac{1}{\left(2\pi\theta\right)^{D/2}}d^{D}x$.
Therefore, the complexified approximated form of the Wiener fractal measure
via the decomposition (A.3) is;343434Actually, since $\mathcal{N}_{0}$ is
fixed one could easily forget the Laplacian term in $dW^{\perp}$. However, we
prefer to keep it here to insist on the background fractal structures due.

$\begin{gathered}dW_{D_{\mathcal{N}_{0}}}\times
dW^{\perp}=\exp\left(\frac{iL^{D}}{\hbar}\int_{0}^{T}\sum_{n}\Bigg{\\{}\frac{1}{2c}\dot{D^{x}_{n}}^{2}+\dot{D^{x}_{n}}\partial_{0}\phi_{n}(t)+i\varepsilon\frac{c^{2}}{\hbar^{2}}{D_{n}^{x}}^{2}\Bigg{\\}}dt\right)\mathfrak{D}x\\\
\times~{}\text{exp}\left(\frac{i}{\hbar}\int_{X}\Bigg{\\{}\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi(t,x)\partial_{\nu}\phi(t,x)-\frac{1}{2}\frac{(m^{2}-i\varepsilon)c^{2}}{{\hbar}^{2}}\phi^{2}(t,x)\Bigg{\\}}d^{D+1}x\right)\mathfrak{D}\phi^{\perp},\end{gathered}$
(A.4)

for $X=[0,T]\times S$. However, the mixed term
$\dot{D^{x}_{n}}\partial_{0}\phi_{n}(t)$ is odd for variables $\phi_{n}$s,
hence cancels out from the measure due to orthogonality of
$D_{\mathcal{N}_{0}}$ and $D_{\mathcal{N}_{0}}^{\perp}$. Moreover, via a
similar reasoning we explained above one can see that:

$L^{D}\sum_{n}\dot{D^{x}_{n}}^{2}=P_{\mathcal{N}_{0}}\sum_{i=1}^{D}\dot{x^{i}}^{2}+\text{mixed
term}.$ (A.5)

with;

$P_{\mathcal{N}_{0}}=\frac{\hbar\kappa}{c}\left(\frac{\mathcal{N}_{0}(\mathcal{N}_{0}+1)(2\mathcal{N}_{0}+1)}{6}\right)\left(\frac{5}{4}\right)^{D-1}.$
(A.6)

The mixed term consists of $\dot{x^{i}}\dot{x^{j}}$, $1\leq i\neq j\leq D$,
which is odd for each variable $x^{i}$, hence cancels out at the first
approximation via path-integrating. Therefore, we obtain;

$dW_{D_{\mathcal{N}_{0}}}\approx\exp\left(\frac{i}{\hbar}\int_{0}^{T}\Bigg{\\{}\frac{P_{\mathcal{N}_{0}}}{2c}\sum_{i=1}^{D}\dot{x^{i}}^{2}+i\varepsilon\frac{R_{\mathcal{N}_{0}}^{2}L^{D}c^{3}}{\hbar^{2}}\Bigg{\\}}dt\right)\mathfrak{D}x.$
(A.7)

In fact, $P_{\mathcal{N}_{0}}/c$ is of the mass dimension, comparable to the
particle’s mass $m$ computed in (A.1). Also, for $D=3$ spatial dimensions
$R^{2}_{\mathcal{N}_{0}}\propto\mathcal{N}_{0}^{3}$, hence the
$\varepsilon$-term in $dW_{D_{\mathcal{N}_{0}}}$ is also of order $m$. Thus;

$dW_{D_{\mathcal{N}_{0}}}\approx\text{exp}\left(\frac{i}{\hbar}\int_{0}^{T}\Bigg{\\{}\frac{m}{2}\left(\frac{dx}{dt}\right)^{2}+i\epsilon
m\Bigg{\\}}dt\right)\mathfrak{D}x.$ (A.8)

If the quantum particle is subject to some external force given as potential
$V(x)+V(\phi^{\perp})$ the Wiener path-integral would be;

$I\approx\int\text{exp}\left(\frac{i}{\hbar}\int_{0}^{T}\Bigg{\\{}\frac{m}{2}\left(\frac{dx}{dt}\right)^{2}-V(x)+i\epsilon
m\Bigg{\\}}dt\right)\mathfrak{D}x\times\exp\left(\int_{X}V(\phi)d^{4}x\right)dW^{\perp},$
(A.9)

which leads to the celebrated Feynman’s path-integral formula for the
Schrodinger’s wave equation feynman ; feynman2 ;

$I\propto\int\text{exp}\left(\frac{i}{\hbar}\int_{0}^{T}L~{}dt\right)\mathfrak{D}x,$
(A.10)

wherein $L$ is the classical Lagrangian.

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