# The Validity of Dimensional Regularization Method on Fractal Spacetime

Authors: Yong Tao
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Svozil developed a regularization method for quantum field theory on fractal spacetime (1987). Such a method can be applied to the low-order perturbative renormalization of quantum electrodynamics but will depend on a conjectural integral formula on non-integer-dimensional topological spaces. The main purpose of this paper is to construct a fractal measure so as to guarantee the validity of the conjectural integral formula.


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## Body

## 1. Introduction

The quantum field theory is one of the oldest fundamental and most widely used tools in physics. It is spectacularly successful that the value of theoretical calculation is precisely in agreement with experimental data, for example, the anomalous magnet moment of electron. Nevertheless, such a precise calculation is on the basis of some regularization methods, for example, dimensional regularization [1]. The dimensional regularization requires that S matrix should be calculated in a non-integer-dimensional spacetime. Following the spirit of this heuristic calculation, Svozil [2] developed the quantum field theory on fractal spacetime (QFTFS). This approach not only can be applied to the low-order perturbative renormalization of quantum electrodynamics but also preserves the gauge invariance and covariance of physical equations. Svozil’s work implied that, for a D-dimensional spacetime, there might be D=4 for macroscopic and D<4 for microscopic events [2].Interestingly, recently, the investigations for a consistent theory of quantum gravity strongly indicate that a power-counting renormalizable gravity model can be achieved in a fractional dimensional spacetime, for example, the Horava-Lifshitz (HL) gravity model [3, 4]. Unfortunately, HL gravity model is not Lorentz invariant. To maintain the Lorentz invariance, Calcagni [5, 6] extended the theoretical framework of Svozil’s QFTFS so as to include the description for gravity. Calcagni’s work showed that if the Hausdorff dimension of spacetime D~2, then the ultraviolet divergence could be removed.In fact, the notion that “the Universe is fractal” at quantum scales has become popular [5–7]. Thus, the demand for a generalized calculus theory on the non-integer-dimensional topological spaces is strongly increasing. Unfortunately, there is still not a rigorous calculus theory for analytically describing fractal so that, as Svozil has mentioned [2], some of the approaches of fractal calculus are essentially conjectural. It is worth mentioning that Svozil’s QFTFS is just on the basis of a conjectural integral formula on the non-integer-dimensional topological spaces. Because of the importance of Svozil’s approach in studying quantum gravity, we suggest to construct a fractal measure so as to guarantee the validity of Svozil’s conjectural integral formula.


## 2. Hausdorff Measure

The mathematical basis of QFTFS is the Hausdorff measure [2]. The introduction for Hausdorff measure can be found in Appendix A. If a D-dimensional fractal Ω is embedded in Rn, then it can be tessellated into (regular) polyhedra. In particular, it is always possible to divide Rn into parallelepipeds of the form [2]:(1)Ei1,…,in={(x1,…,xn)∈Ω:xj=(ij-1)Δxj+αj,ddddddd0≤αj≤Δxj,j=1,…,n}.Based on such a set of parallelepipeds, Svozil [2] conjectured that the local Hausdorff measure dμH(Ω) of Ω should yield the following form:
(2)dμH(Ω)=limd(Ei1,…,in)→0∏j=1n(Δxj)D/n=∏j=1ndD/nxj,
where d(Ei1,…,in) denotes the diameter of parallelepiped Ei1,…,in and dD/n denotes the differential operator of order D/n.With these preparations above, Svozil [2] proved that if formula (2) holds, then the integral of a spherically symmetric function f(r) on Ω can be written as
(3)∫Ωf(r)dμH=2πD/2Γ(D/2)∫0Rf(r)rD-1dr.The integral formula (3) is the starting point of QFTFS; therefore, we must pay much attention to the validity of formula (2). Nevertheless, formula (2) does not always hold whenever D≠n; for example, using the fractional derivative [8], it is easy to check that dD/nxj/(dxj)D/n≠1 if D≠n. This means that the integral formula (3) does not always hold in the framework of Hausdorff measure.In fact, Hausdorff measure is not an ideal mathematical framework for describing fractal. Next, we will see that Hausdorff measure indeed determines the dimension of a fractal curve but does not describe its analytic properties, for example, the self-similarity between local and global shapes of a fractal curve. To realize this fact, we attempt to check the case of the Cantor set; see Figure1 [8, 9].Figure 1
The Cantor ternary set is defined by repeatedly removing the middle thirds of line segments [8, 9]. (a) One starts by removing the middle third from the interval [x2(1),x1(1)], leaving [x4(2),x3(2)] and [x2(2),x1(2)]. (b) Next, the “middle third” of all remaining intervals is removed. (c) This process is continued ad infinitum. Finally, the Cantor ternary set consists of all points in the interval [x2(1),x1(1)] that are not removed at any step in this infinite process.As shown by Figure1, the Cantor set is a fractal. Using the Hausdorff measure (A.8) (see Appendix A) we can compute the dimension of the Cantor set as [9]
(4)D=ln2ln3=0.6309⋯Nevertheless, for the Cantor set, we do not realize any correlation between its local and global segments (i.e., self-similarity) via the Hausdorff measure. For instance, the Hausdorff distance between pointsx2(3) and x1(3) is denoted by
(5)HD(x2(3),x1(3))=|x2(3)-x1(3)|D.Obviously, Hausdorff distance (5) is independent of the values of points xi(3), where i runs from 3 to 8. Nevertheless, because of the self-similarity between parts of the Cantor set, any displacement of point xi(3)(i=3,4,…,8) should influence the distance between x2(3) and x1(3). This is undoubtedly a nonlocal property. Unfortunately, Hausdorff distance (5) fails to show this property. In the next section, we will construct a new fractal measure so as to exhibit such a nonlocal property.


## 3. Fractal Measure

In Section2, we have noted that the key point of guaranteeing the validity of the integral formula (3) is that the Hausdorff measure is compelled to equal some differences of fractional order; that is, formula (2) holds. Such a fact reminds us that the differences of fractional order itself may be a type of measure. An interesting thought is that whether or not the differences of fractional order can describe the nonlocal property of a fractal curve. To this end, we attempt to check a m-dimensional volume:
(6)x(l)=ω(m)lm,
where ω(m) is a constant factor that depends only on the dimension m and m may be a fraction.The fractional derivatives of orderm of x(l) give [8]
(7)dmx(l)=Γ(m+1)(dl)m~(dl)m.Obviously,(dl)m, as a m-dimensional volume, is a m-dimensional Hausdorff measure; therefore, formula (7) implies that the differences of order m, dmx(l), can be also thought of as a measure for describing the length of a m-dimensional fractal curve. In this case, the order of differences dmx(l) represents the Hausdorff dimension m.Using the differences of orderm, we define a new distance—call it the “nonlocal distance”—in the form (see (A.19) in Appendix A):
(8)|Δm[x(l),x(l-Δl)]|=|∑j=0∞m(m-1)⋯(m-j+1)(-1)jj!x(l-jΔl)|,
where |Δm[x(l),x(l-Δl)]| denotes the nonlocal distance between points x(l) and x(l-Δl).It is carefully noted that everyx(l-jΔl) represents a point on a m-dimensional fractal curve, where, =0,1,2,….Clearly, according to (8), the distance between points x(l) and x(l-Δl) would depend on all the points x(l-jΔl), where j=0,1,2,….Moreover, it’s easy to check that|Δm=1[x(l),x(l-Δl)]|=|x(l)-x(l-Δl)|. This means that the Euclidean distance is a special case of nonlocal distance whenever the dimension of the fractal curve, m, equals 1.If we use the nonlocal distance (8) to measure the distance between points x2(3) and x1(3) (see Figure 1), then we will surprisingly find that the nonlocal distance
(9)|Δm=D[x2(3),x1(3)]|=|∑j=18D(D-1)⋯(D-j+1)(-1)jj!xj(3)|,
which remarkably differs from the Hausdorff distance (5), would depend on the values of points xi(3)(i=3,4,…,8). This means that any displacement of point xi(3)(i=3,4,…,8) would change the nonlocal distance between points x2(3) and x1(3). Consequently, nonlocal distance (8) is indeed an intrinsic way of describing self-similar fractal, since it not only determines the dimension of a fractal curve (e.g., Cantor ternary set) but also reflects the correlation between its parts. Interestingly, the nonlocal distance seems to have some connection with quantum behavior; for details see Appendix C.Using the nonlocal distance we have given a definition for fractal measure in AppendixA (see (A.23)).To study the analytic properties of a fractal curve, we define the fractal derivative (see(A.26) in Appendix A) in the form:
(10)Dlωf(x)Dlωx=limΔl→0Δω[f-(l),f-(l-Δl)]Δω[x(l),x(l-Δl)],
where f-(l)=f[x(l)] is a differentiable function with respect to coordinate l, l is a parameter (e.g., the single parameter of Peano’s curve [10]; for details see Appendix A) which completely determines the generation of a ω-dimensional fractal curve, and x(l) denotes the length of the corresponding fractal curve. (We introduce a simple way of understanding the fractal derivative (10). For the case of the Newton-Leibniz derivative of y=f(x), x is a 1-dimensional coordinate axis and hence can be measured by a Euclidean scale (ruler). Thus, the differential element of x is a 1-dimensional Euclidean length dx, which gives rise to the Newton-Leibniz derivative df(x)/dx. Nevertheless, if x is a ω-dimensional fractal curve, then it can not be measured by the Euclidean scale (ruler). In this case, the differential element of x should be a ω-dimensional volume Dlωx, which gives rise to the fractal derivative Dlωf(x)/Dlωx. For details see Appendix A and Figure 3.)In particular, the fractal derivative (10) will return to the well-known Newton-Leibniz derivative whenever ω=1.Using the formula of fractional derivative [8], the fractal derivative can be rewritten as (see (A.27)–(A.29) in Appendix A)
(11)Dlωf(x)Dlωx=dωf-(l)/dlωdωx(l)/dlω.By formula (11) we can easily compute the fractal derivative of any differentiable function using the fractional derivative; for concrete examples see Appendix B.


## 4. Fractal Integral

In Section3, we have proposed a definition for fractal derivative. Correspondingly, we can now present a convenient definition for fractal integral as follow.Definition 1.
IfDlmf(x)/Dlmx=g(x) then the fractal integral of g(x) on a m-dimensional fractal curve βm(l) is defined in the form:
(12)∫Dlmf(x)=∫βm(l)g(x)Dlmx=∫Wg(x)dmxdlm(dl)m,
where W denotes the definitional domain of the characteristic parameter l; also, the parameter l completely determines the generation of the m-dimensional fractal curve βm(l).Using such a definition of fractal integral we can prove the following proposition.Proposition 2.
Ifx(r)=ω(m)rm not only describes the length of a m-dimensional fractal curve βm(r) but also denotes the volume of a m-dimensional sphere (for example, the m-dimensional fractal curve βm(r) fills up the entire sphere Ω) Ω and if f(x)=f-(r) is a spherically symmetric function, then the fractal integration of f(x) on the m-dimensional fractal curve βm(r) equals
(13)∫βm(r)f(x)Drmx=2πm/2Γ(m/2)∫0Rf-(r)rm-1dr,
where ω(m)=πm/2/Γ((m/2)+1).Proof.
The Riemann-Liouville fractional integrals of orderm of f(l) are defined in the form [8]:
(14)∫f(l)(dl)m=1Γ(m)∫yR(l-y)m-1f(l)dl.
Using(A.29) and (B.3) we have
(15)Drmx=dmxdrm(dr)m=ω(m)Γ(1+m)(dr)m.
Using (12) and (15) we arrive at
(16)∫βm(r)f(x)Drmx=∫Ωf-(r)ω(m)Γ(1+m)(dr)m.
Inserting (14) into (16) leads to
(17)∫βm(r)f(x)Drmx=ω(m)Γ(1+m)Γ(m)×∫yR(r-y)m-1f-(r)dr.
Usingω(m)=πm/2/Γ((m/2)+1) and the formula (B.7), (17) can be rewritten as
(18)∫βm(r)f(x)Drmx=πm/2mΓ((m/2)+1)∫yR(r-y)m-1f-(r)dr.
Fory=0, (18) yields
(19)∫βm(r)f(x)Drmx=πm/2mΓ((m/2)+1)∫0Rrm-1f-(r)dr.
Using the formula(B.7) we have
(20)m2Γ(m2)=Γ(m2+1).
Substituting (20) into (19) leads to
(21)∫βm(r)f(x)Drmx=2πm/2Γ(m/2)∫0Rf-(r)rm-1dr.
The proof is complete.Clearly, forf(x)=1, formula (13) gives the volume of a m-dimensional sphere, πm/2/Γ((m/2)+1)Rm.Before proceeding to arrive at the main result of this paper, let us consider three measurable setsWi with the dimension mi, where i=1,2,3. According to Fubini’s theorem, the Cartesian product of the sets Wi can produce a set W with the dimension m=m1+m2+m3, where W=W1⊗W2⊗W3. The integration over a function f(r1,r2,r3) on W can be written in the form:
(22)∫Wf(r1,r2,r3)dμ(W)=∫W1∫W2∫W3f(r1,r2,r3)dμ(W1)·dμ(W2)·dμ(W3).Then we have the following lemma.Lemma 3.
Ifdμ(Wi) is denoted by
(23)dμ(Wi)=2πmi/2Γ(mi/2)rimi-1dri(i=1,2,3)
and if f(r1,r2,r3) is specified by a spherically symmetric function f(r) with r2=r12+r22+r32, then we have
(24)∫W1∫W2∫W3f(r1,r2,r3)dμ(W1)·dμ(W2)·dμ(W3)=2πm/2Γ(m/2)∫0Rf(r)rm-1dr.The proof of Lemma3 can be found in page 28 in the literature [8].With the preparations above, we can introduce the main result of this paper as follow.Proposition 4.
Ifxi(ri)=ω(Di)riDi(i=1,2,3) not only describes the length of a Di-dimensional fractal curve βDi(ri) but also denotes the volume of a Di-dimensional sphere Ωi and if f(r)=f(r1,r2,r3) is a spherically symmetric function, then the fractal integration of f(r) on the fractal graph βD1(r1)⊗βD2(r2)⊗βD3(r3) equals
(25)∫βD1(r1)∫βD2(r2)∫βD3(r3)f(r1,r2,r3)∏i=13DriDixi=2πD/2Γ(D/2)∫0Rf(r)rD-1dr,
where ω(Di)=πDi/2/Γ((Di/2)+1), r2=r12+r22+r32, and D=D1+D2+D3.Proof.
Using (13) and (24) we easily arrive at (25).
The proof is complete.Proposition4 shows that we have arrived at the conjectural integral formula (3) through the fractal measure (A.23). In particular, if R→∞, then the integral formula (25) will exhibit linearity, translational invariance, and scaling property. These properties are natural and necessary in applying dimensional regularization to quantum field theory [2]. It is worth mentioning that Proposition 4 can be generalized so that i might take any positive integer, for example, i=1,2,…,∞. If i=1,2,3,4, then βDi(ri) may denote coordinate axis with the dimension Di and so βD1(r1)⊗βD2(r2)⊗βD3(r3)⊗βD4(r4) may denote spacetime.


## 5. Conclusion

Hausdorff measure is not an ideal mathematical framework for describing fractal since it fails to describe the nonlocal property of fractal (e.g., self-similarity). However, the fractal measure constructed by this paper not only shows the dimension of a fractal but also describes its analytic properties (e.g., nonlocal property). Not only so, using this fractal measure we can derive Svozil’s conjectural integral formula (3) which is the starting point of quantum field theory on fractal spacetime. Therefore, our fractal measure may be regarded as a possible mathematical basis of establishing quantum field theory on fractal spacetime.



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