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

This is the first paper of a series of researches (that is followed by [42, 43]) 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 dAlembertian 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. ...


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 [34,41]. 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 [15,26,28]. 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 [4, 35-37, 47, 48].
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 [30] the fractal curves are distinguished by a measure theoretic point of view. Through this viewpoint a fractal curve in R D is in fact a function F : I → R D , with dim HB (F(I)) > 1, wherein dim HB is the Hausdorff-Besicovitch dimension and I ∈ R is an interval. However, according to the geometric-base definition, a fractal curve is a continuous embedding of I in R D which each of its subarc longer than a point has an infinite length [32]. This point of view also leads to the definition of nowhere rectifiable curves which are effectively known as maps from I to R D that are non-differentiable almost everywhere on their domain [13,32]. 1 It is worth noting that by considering the fractal curves as the contour line of self-similar nowhere differentiable maps F : R D → 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 [14,15] to fractal wavefunctions (resp. fractal fields) in the Schrodinger's wave mechanics (resp. the theory of quantum fields). 2 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. 3 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 [15]. 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 [4,47,48].
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. 4 . 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 [26] 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. 5 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 [45,46] 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. 6 Therefore, upon the above proclaim and what we have already obtained elsewhere [42,43], 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) = k≥0 a k sin(b k πx/L) : I → R, with I = [−L, L], and 0 < a < 1, wherein b is an odd integer [44]. 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 [19]. 7 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: for some definite b ∈ N and for all n ∝ b k , k ≥ 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 ∈ 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. 8 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) : , for each 0 ≤ 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 ∈ N, we easily find: a N = aa k , which leads to: a n = 1 n s a 1 , for n being some power of b ∈ N, whereas s = log b (a −1 ).
Obviously, the same relation holds for the Fourier sine coefficient b N = 1 L I f (x) sin N π L x dx. Thus, according to [19] 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 → R, which is fractal only on a subinterval I = [x 1 , x 2 ] ⊂ I = [a, b]. In fact, partial fractality causes the Fourier coefficients f n (= a n or b n ) to have an asymptotic behavior as: 9 for s = log b (a −1 ), and n a sufficiently large power of b, wherein O(1/n m , 1/n s+m ) comes from nonfractal parts, being of order 1/n m and 1/n s+m for m ≥ 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). 10 However, f (x) may contain several partial fractalities of various types (a i , b i ). In such cases we see: f n ∝ i 1/n s i + · · · , 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 [13]. In such cases, we also see that the Fourier coefficients undergo the PPL for some 0 < s < 1. 11 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 l], and W L (x) = 0 for l < |x| ≤ L, wherein 0 < a < 1, b a positive odd integer and 1 < ab, while L−l L ≈ 0. Hence, we obtain the n-th Fourier sine coefficient as; The proportionality factor could be both positive or negative numbers. 10 One 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] → R, leads to a similar power law ∝ 1/n, for each n ∈ N. But, in this case the least integer b is the unity and s = 1. Therefore, we have no PPL here, hence no fractality. 11 Actually, 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 Sup k≥1 k √ a1 · · · a k , which by assumption fulfills the condition of: 1/b < a < 1.
wherein F K is an analytic function with F K (0) = 0 and n = b K , for some large K ∈ N. Actually; 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; (II.5) Since the sine function is periodic and we are only interested in asymptotic behaviors of Fourier coefficients, then sin (nπζ/L) must be replaced by some positive constant 0 < α < π times ζ/L, which indeed refers to integers n i , i ∈ N, with n i πζ/L = 2N π + α i ζ/L, for some N ∈ N and 0 < α i ≤ α. Therefore, we see that the PPL included in F K is of the following form; wherein we define the fractal mass as; µ 2 = λ/s. In fact, for widespread fractality condition we are still being encountered with a PPL in the asymptotic behavior; 12 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 µ is proportional to ζ = L − 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 ζ → 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 inequality 13 for each function f (x) with entire fractality (i.e. OPL), wherein κ > 0 is an arbitrary constant.
Note that here f n could be assumed as the Fourier complex coefficient obtained for the Fourier mode e inπx/L , hence generally a complex number. To consider any kind of self-similarity with assuming the extreme case of non-differentiability, i.e. s → 1, we conclude the inequality which must be hold asymptotically for any fixed ℓ, κ > 0. 14 That is, for each positive ℓ and κ there exists a large enough N ∈ N, so that for any n ≥ 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 is imposed to the Fourier coefficients f n s for some fixed κ and ℓ, the resulting function will not admit the entire fractality. Let κ be fixed. Then, to obtain a limit for avoiding the fractality, we should first solve The solution of (III.3) gives |f n | as a function of ℓ and n, i.e. |f n | = |f n |(n, ℓ). When we fix |f n |, then according to the implicit function theorem ℓ is given as a function of n and |f n |. This function is given by the following integral formula: 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 ≥ 0, unless replace f n by −f n . Now, according to (III.5) we obtain: Hence, the differential equation of f n , i.e. (III.4), could be simply derived with considering ∂f n /∂n = −(∂ℓ/∂n)/(∂ℓ/∂f n ), which in its own right is the immediate consequence of df n (n, ℓ(n))/dn = 0. We refer to ℓ κ (f n ) in (III.5) as the fractal κ-norm of f n , while generally we may drop the prefix κ, 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 → 0 at least with the rate induced from (III.5) for some fixed κ and the fractal norm ℓ 0 . All in all, we have the following theorem: is not an entire fractal (i.e. dismisses OPL) if and only if for some fixed κ > 0 and ℓ 0 > 0 we have: ℓ κ (f n ) ≤ ℓ 0 , for all n. In other words, f is not an entire fractal if and only if there exists some κ > 0 and ℓ 0 > 0 so that the inequality holds for any Fourier real coefficient f n , n ∈ N.

b) Widespread Fractals
For a widespread fractal with fixed a, b and l we can easily see from (II.9) that; Consequently, for each widespread fractal function fulfilling the inequality 0 < s < 1, we find the following asymptotic equation: where ℓ > 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: The solution of (III.9) works out |f n | as a function of ℓ and n, i.e. |f n | = |f n |(n, ℓ). Upon a similar reasoning to the discussion we had for the entire fractals ℓ is given as a function of f n and n: where f n is allowed to admit negative and complex values. The fractal norm ℓ (κ,µ) is conventionally called the fractal (κ, µ)-norm of f n , while we may drop the prefix (κ, µ) 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 ⊂ R is a compact interval and f (x) : I → 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 κ > 0, µ > 0, and ℓ 0 > 0 we have: ℓ (κ,µ) (f n ) ≤ ℓ 0 , for all n. In other words, f has no widespread fractality if and only if there exists some κ > 0, µ > 0, and ℓ 0 > 0 so that the inequality holds for any Fourier real coefficient f n , n ∈ 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 ≤ i ≤ D. For instance, in D = 2 dimensions, we easily construct a self-similar nowhere differentiable function as: In such cases we also see a OPL as: a n,n ′ ∝ n −s n ′−s ′ , for a n,n ′ the Fourier cosine coefficient for 2-dimensional Fourier mode (n, n ′ ), while n (resp. n ′ ) is some power of 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 ∈ N an odd integer, and 0 < a i < 1, the i-th scaling factor, while a i b i > 1 for each 1 ≤ i ≤ D. Substantially, if for D-plets u = (u 1 , · · · , u D ) and v = (v 1 , · · · , v D ) one simply sets: x D ) admits the OPL in its Fourier coefficients: f n ∝ 1/n s , for n = (n 1 , · · · , n D ) be a D-plet of integers, and s the D-plet power: s = (s 1 , · · · , s D ), with s i = log b i (a −1 i ). Also, we can go farther and define PPL accordingly for D dimensions. Thus, if f (x 1 , · · · , x D ) partly admits such self-similarities, together with non-differentiability, then f n ∝ 1/n s + O(1/n s+m , 1/n m ), m ≥ 1, for general Fourier coefficient f n and for n i some power of b i , 1 ≤ i ≤ D. It is the PPL for functions with D-dimensional domains. 15 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 ∝ 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 µ 1 = · · · = µ D = µ and the fractality scale s 1 = · · · = s D = s. Hence, at the leading term we find: 1 (n 2 i +µ 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 ≤ i ≤ D and all n j s, 1 ≤ j = i ≤ 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; Therefore, by considering (IV.1) and employing the argumentation of the last section we have the following asymptotic behavior of MPL; wherein κ > 0 is fixed. Hence, we should consider the following system of PDEs to find a criteria to avoid the symmetric fractality in D-dimensions: Consequently, (III.10) is also obtained in D-dimensions for the massive fractal norm ℓ (κ,µ) as; Obviously for the entire fractals we should set µ = 0 and replace ℓ (κ,µ) with ℓ κ = ℓ (κ,0) . In principle, we simply read; 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 , · · · , x D ) is a bounded D-dimensional set, say U ⊂ 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 ⊂ R D be a bounded D-dimensional set and let f (x 1 , · · · , x D ) : U → R to be a continuous function. Then, f is said to be a fractal function if there exists a cube V ⊂ 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 ℓ 0 > 0, κ > 0 and µ ≥ 0 there exists some n = (n 1 , · · · , n D ) so that; ℓ (κ,µ) (f n ) > ℓ 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 ∈ R D at t 0 , then the probability of the particle being found in U i ⊂ R D at t i , 1 ≤ i ≤ N , with the time ordering of t 0 < t 1 < · · · < t N is given as [45,46]; 16 i , · · · , x D i ) and d D x i respectively for the coordinate system and the Lebesgue measure of 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 + · · · + (x D ) 2 for R D . Actually, the Wiener measure for time slicing t 0 < t 1 < · · · < t N is; One may be interested to consider the symmetric form of the Wiener measure as: 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: R D · · · R D dW (t N , · · · , 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 ≤ i ≤ N , to be bounded from above within the partition of time −T < t 1 < · · · < t N < T , then the mean value of the influence of V i s to the particle being located in U i ⊂ R D at t i , 1 ≤ i ≤ N , is given as: 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 R D . Hence, if 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 , · · · , U N ; t 1 , · · · , t N ) is; for any a > 0 and x 0 ∈ 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 ), · · · , x 1 (t 1 )) = e V N (x N (t N )) · · · 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 N ∈ N, set S = [−L, L] D ⊂ R D , and define C N to be the space of continuous functions f : S → R with the Fourier coefficients f n which vanish except for Fourier modes n = (n 1 , · · · , n D ) satisfying −N ≤ n i ≤ N for all 1 ≤ i ≤ 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 , · · · , 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 iπn.x/L } n∈Z D , the Fourier coefficients f n and f −n provide two independent real variables as fn+f −n 2 and fn−f −n

2i
, which may also be referred to as f n and f −n in our formalism. All in all, C N is a vector space of D N = (2N + 1) D real dimensions {f n } each of which is labeled with an allowed Fourier mode n. We refer to elements of C N as N -bounded functions. Let us define the Lebesgue fractal measure on C N by means of the fractal norm on each coordinate as: In principle, this election for C N naturally leads to the Wiener fractal measure as: which essentially describes the Brownian motion of N -bounded functions within a fractal space. Thus, the 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 ), · · · , x n (t 1 )) the following integral exists and is well-defined for any N ; wherein F N (f n (t N ), · · · , f n (t 1 )) is the restriction of F (f n (t N ), · · · , f n (t 1 )) to C N and U N i = U i C N . Actually, we are mostly interested in the behavior of the sequence {I N } ∞ N =1 as N → ∞. It is worth to note that through this viewpoint the Wiener convergence condition must be replaced by I N being a convergent sequence as N → ∞. A Wiener-integrable function F (f n (t N ), · · · , f n (t 1 )) which provides a convergent sequence I N → 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 ), · · · , f n (t 1 )).
It is easy to see that if F (f n (t N ), · · · , f n (t 1 )) = e V N (fn(t N )) · · · e V 1 (fn(t 1 )) is bounded to the unity from above, which happens for negative potentials V i (f n (t i )), then {I N } would be a decreasing sequence, hence convergent to some definite I ∈ R. 17 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 ), · · · , 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. 18 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 N (t N , · · · , 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 With these assumptions we have; for the overall (initial-to-final) normalization factor C F I given as: fn(−T ) e κ(n 2 +µ 2 )x 2 /2 dx] 2 /4T ). Moreover, in the last line of (VI.2) use has been made of dx 0 = cdt = cθ and η µν = diag(1, −1, · · · , −1), the Lorentz metric in D + 1 dimensions. Actually, to transfer to physical scales the dimension of f , i.e. T for the physical mass m = mγ 2 and the Dirac constant = hγ 2 , while we have simply considered for 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 φs. However, the partial derivations in (VI.4) are absolutely welldefined and justifiable since as far as we concentrate on N -bounded functions the included quantum fields in the Wiener fractal measure are thoroughly smooth. Therefore, although the process of N → ∞ 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 ), · · · , x n (t 1 )) = e V (xn(t N )) · · · e V (xn(t 1 )) , by means of the approximated Wiener fractal measure (VI.4). For example, for φ 4 -theory in R 4 we have; V (x n ) ∝ − n,m (x m x n−m ) 2 ≤ 0. Consequently, as we discussed above this interaction term leads to a Wiener fractal convergent function with path-integral is the Lagrangian density of φ 4 -theory for some positive constant λ. 19 Actually, despite assumption (i), here we considered the intermediate integral domains U N i s to be the whole space of N -bounded functions C 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 φ I , given by Fourier coefficients f n (−T )s, to the final state φ 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 ), · · · , x n (t 1 )) is multiplied by a given functional O(x n (t N )) × · · · × O(x n (t 1 )), then the corresponding Wiener path-integral would be wherein O(φ) VEV is the vacuum expectation value of the local operator O(φ). On the other hand, according to the LSZ reduction formula for Fourier modes n 1 , · · · , n l , m 1 , · · · , m k ∈ Z D , and with Fourier coefficients φ n (t) = γf n (t), effectively describes the scattering matrix element out n 1 , · · · , n k |m 1 , · · · , m l 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 N -bounded functions we see that the theory admits the parity and the charge conjugate symmetries due. 20 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. κ and µ, or on the structure of space-time, i.e. the spatial size L and the time slicing width θ, which the latter is substantially imposed by the potential exponential function F . Actually, κ is of dimension T −1 , hence plays the role of frequency and includes some fundamental information about the particle. In particular, κ would provide an important criterion for the energy levels of the particle. Moreover, since κ couples to the Laplacian term n 2 and the fractal mass µ, 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 κ could be compared to 1/θ, 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 µ. As we saw above µ 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: 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. 21 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 ] × S, which appeared in (VI.2) and (VI.3). We will retain this convention in upcoming perusals [42,43].

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 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 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; or more generally: for any analytic function f (x) : R → R. Therefore, in order to have a well-defined approximated Wiener fractal measure we should employ the following replacement; However, since the causality of quantum field theories is an immediate consequence of the heat kernel of iε-term [33], we see that the replacement (VII.4) successfully revives the natural causality of the Wiener path-integral. An augmented beneficial iε-term also appears in demonstrating the Feynman propagators or expectation values of some local operators via the limit of T → (1−iε)∞ or equivalently for setting p 0 → p 0 + iε with p 0 the energy of virtual particles [33]. This mechanism is actually an alternative procedure for the replacement of m → m − iε within the Lagrangian density. More precisely, the iε-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 iε-term plays the role of the removed nonlocal fractal norm from the Lagrangian density in the exponent of the Gaussian function of the Wiener measure. 22 On the other hand, according to the significance of the iε-term in figuring out the structure of the vacuum state of a quantum field theory [50], 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 n independent quantum fields φ (i) , 1 ≤ i ≤ n. In this case the well-defined complexified approximated Wiener fractal measure would be; Specially, this well-defined measure naturally leads to the Maxwell theory for n = 3 and m = 0 in the Coulomb gauge; ∂ 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. 23 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 [20]: 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 2 n (t)(∂ t f n (t)) 2 for compact U 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 [42] 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. 24 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 [33]. 25

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 (VIII.1) for some y n ∈ R and with ∇f the gradient of f . Therefore, I N is roughly written as; I N ∝ J 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(N ), for g(N ) : N → N a monotone increasing function, and fix t g(N ) = T and t −g(N ) = −T for all 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 N (f n (t g(N ) ), · · · , f n (t −g(N ) )}, which has two properties; N ) ), · · · , f n (t −g(N ) )) is a Wiener fractal convergent function in C N .
Indeed, a renormalizable quantum field theory is a Wiener renormalizable sequence that fulfills three more conditions; 26 iii) The potential exponential functions F N (f n (t g(N ) ), · · · , f n (t −g(N ) )) could be written by means of a local potential V N (f n ) within the time slicing t −g(N ) < · · · < t g(N ) as; is a polynomial of f n s of order ≥ 2 which contains a fixed number of constants. The coefficient of f 2 n in V 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. F N (f n (t g(N ) ), · · · , f n (t −g(N ) )) −→ F N −k (f n (t g(N −k) ), · · · , f n (t −g(N −k) )), (k ∈ N) (VIII. 4) as N → ∞, T → ∞, and L → ∞. Through this process b = lim N →∞ g(N −1) g(N ) effectively describes the RG flow scaling parameter, and the survived polynomials in the limit of b → 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 [4] and the Wilson's [47,48] viewpoints to the RG flow machinery give rise to profound understandings of and guidelines towards the fractality of quantum fields. 28 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 pathintegral (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 φ 4 -theory [33]. 29 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, function asf : [−T, T ] → R via the inverse Fourier transformation as: n=−1 Hence, transferring from N to N − 1 will causef(t) to lose ∆E = (g(N ) − g(N − 1)) π T amount of energy. We use this idea for potential exponential functions FN (fn(t g(N ) ), · · · , fn(t −g(t N ) ), and set ∆E ∝ π L . This, revives the Wilsonian process of integrating out the highest amount of energy included in a cutoff shell with fixed radius and thickness ∆E in the Euclidean momentum space given by the Wick rotation. 28 See also [35,36,52] for more discussion about Bogoliubov's special insights toward the RG flow for quantum fields. 29 And the Yukawa interaction, just when the spinor fields are involved in the Wiener fractal measure.
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 [20]. 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, 30 regularizing the measure and geometric approximations which have already been done before. 31 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 iε-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 [42]) and Maxwell's and Yang-Mills theories (which is accomplished in [43]).
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 pathintegral 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 30 Which both cease the Wiener measure to be still a well-defined probability measure. 31 See for example [1,3,[6][7][8][9][10][11]31]. For another viewpoint to Feynman's path-integral formulation see [18,21,22,24].
The readers are also referred to [25] for a historical review on path-integral and its various stochastic formulation via the Wiener measure.
for X = [0, T ]×S. However, the mixed termḊ x n ∂ 0 φ n (t) is odd for variables φ n s, hence cancels out from the measure due to orthogonality of D N 0 and D ⊥ N 0 . Moreover, via a similar reasoning we explained above one can see that: The mixed term consists ofẋ iẋj , 1 ≤ i = j ≤ D, which is odd for each variable x i , hence cancels out at the first approximation via path-integrating. Therefore, we obtain; In fact, P 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 N 0 ∝ N 3 0 , hence the ε-term in dW D N 0 is also of order m. Thus; If the quantum particle is subject to some external force given as potential V (x) + V (φ ⊥ ) the Wiener path-integral would be; which leads to the celebrated Feynman's path-integral formula for the Schrodinger's wave equation [14,15]; wherein L is the classical Lagrangian.