The Validity of Dimensional Regularization Method on Fractal Spacetime

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.Themain purpose of this paper is to construct a fractal measure so as to guarantee the validity of the conjectural integral formula.


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 -dimensional spacetime, there might be = 4 for macroscopic and < 4 for microscopic events [2].
Interestingly, recently, the investigations for a consistent theory of quantum gravity strongly indicate that a powercounting 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 ∼ 2, then the ultraviolet divergence could be removed.
In fact, the notion that "the Universe is fractal" at quantum scales has become popular [5][6][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-integerdimensional 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.

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 -dimensional fractal Ω is embedded in , then it can be tessellated into (regular) polyhedra. In particular, it is always possible to divide into parallelepipeds of (1) Based on such a set of parallelepipeds, Svozil [2] conjectured that the local Hausdorff measure (Ω) of Ω should yield the following form: where ( 1 ,..., ) denotes the diameter of parallelepiped 1 ,..., and / denotes the differential operator of order / . With these preparations above, Svozil [2] proved that if formula (2) holds, then the integral of a spherically symmetric function ( ) on Ω can be written as 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 ̸ = ; for example, using the fractional derivative [8], it is easy to check that / /( ) / ̸ = 1 if ̸ = . 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 Figure 1 [8,9].
As shown by Figure 1, 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] = ln 2 ln 3 = 0.6309 ⋅ ⋅ ⋅ .
Nevertheless, for the Cantor set, we do not realize any correlation between its local and global segments (i.e., selfsimilarity) via the Hausdorff measure. For instance, the Hausdorff distance between points (3) 2 and (3) 1 is denoted by Obviously, Hausdorff distance (5) is independent of the values of points (3) , where runs from 3 to 8. Nevertheless, because of the self-similarity between parts of the Cantor set, any displacement of point (3) (3) 1 . 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. 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

Fractal Measure
In Section 2, 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 -dimensional volume: where ( ) is a constant factor that depends only on the dimension and may be a fraction. The fractional derivatives of order of ( ) give [8] ( ) = Γ ( + 1) ( ) ∼ ( ) .
Obviously, ( ) , as a -dimensional volume, is a -dimensional Hausdorff measure; therefore, formula (7) implies that the differences of order , ( ), can be also thought of as a measure for describing the length of a -dimensional fractal curve. In this case, the order of differences ( ) represents the Hausdorff dimension . Using the differences of order , we define a new distance-call it the "nonlocal distance"-in the form (see (A.19) in Appendix A): where |Δ [ ( ), ( − Δ )]| denotes the nonlocal distance between points ( ) and ( − Δ ).
If we use the nonlocal distance (8) to measure the distance between points (3) 2 and (3) 1 (see Figure 1), then we will surprisingly find that the nonlocal distance which remarkably differs from the Hausdorff distance (5) (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 Appendix A (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: where ( ) = [ ( )] is a differentiable function with respect to coordinate , 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 -dimensional fractal curve, and ( ) 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 = ( ), is a 1-dimensional coordinate axis and hence can be measured by a Euclidean scale (ruler). Thus, the differential element of is a 1-dimensional Euclidean length , which gives rise to the Newton-Leibniz derivative ( )/ . Nevertheless, if is a -dimensional fractal curve, then it can not be measured by the Euclidean scale (ruler). In this case, the differential element of should be adimensional volume , which gives rise to the fractal derivative ( )/ . For details see Appendix A and Figure 3.) In particular, the fractal derivative (10) will return to the well-known Newton-Leibniz derivative whenever = 1.
Using the formula of fractional derivative [8], the fractal derivative can be rewritten as (see (A.27)-(A.29) in Appendix A) By formula (11) we can easily compute the fractal derivative of any differentiable function using the fractional derivative; for concrete examples see Appendix B.

Fractal Integral
In Section 3, we have proposed a definition for fractal derivative. Correspondingly, we can now present a convenient definition for fractal integral as follow.
where denotes the definitional domain of the characteristic parameter ; also, the parameter completely determines the generation of the -dimensional fractal curve ( ).
Using such a definition of fractal integral we can prove the following proposition.
Before proceeding to arrive at the main result of this paper, let us consider three measurable sets with the dimension , where = 1, 2, 3. According to Fubini's theorem, the Cartesian product of the sets can produce a set with the dimension The integration over a function ( 1 , 2 , 3 ) on can be written in the form: Then we have the following lemma.
With the preparations above, we can introduce the main result of this paper as follow.
The proof is complete.

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.

A. Mathematical Preparations
In Euclidean geometry, the dimension of a geometric graph is determined by the number of independent variables (i.e., the number of degrees of freedom). For example, every point on a plane can be represented by 2-tuples real number ( 1 , 2 ); then the dimension of the plane is denoted by Journal of Applied Mathematics 5 2. Nevertheless, the existence of Peano's curve powerfully refutes this viewpoint. Peano's curve, which is determined by an independent characteristic parameter (i.e., fill parameter), would fill up the entire plane [10]. Therefore, mathematicians have to reconsider the definition of dimension. The most famous one of all definitions of dimension is the Hausdorff dimension, which is defined through the Hausdorff measure [8].

A.1. Hausdorff Measure and Hausdorff Dimension.
In order to bring the definition of Hausdorff dimension, we firstly introduce the Hausdorff measure [8].
Let be a nonempty subset of -dimensional Euclidean space ; the diameter of is defined as where ( , ), which is the distance between points and , is a real-valued function on ⊗ , such that the following four conditions are satisfied: Now, let us consider a countable set { } of subsets of diameter at most that covers ; that is, For a positive and each > 0, we consider covers of by countable families { } of (arbitrary) sets with diameter less than and take the infimum of the sum of [diam( )] . Then we have

A.2. Shortcoming of Hausdorff
Measure. In general, may be a fraction. In 1967, Mandelbrot realized that [11] the length of coastline can be measured using Hausdorff measure (A.8) rather than Euclidean measure (A.6), and then the dimension of coastline is a fraction. Mandelbrot called such geometric graphs the "fractal".
The fractal is self-similar between its local and global shapes. Unfortunately, Hausdorff measure can determine the dimension of fractal but not reflect the connection (e.g., selfsimilarity) among the parts of the corresponding fractal. To see this, we consider Koch's curve in Figure 2.
Clearly, the congruent triangle Δ 1 2 3 is similar to Δ 4 5 6 . If we use the Hausdorff measure (A.8) to measure the local distance of Koch's curve (e.g., the distance between points 1 and 3 ), then we have where is the dimension of theKoch curve. Equation (A.10) shows that the Hausdorff distance between points 1 and 3 depends only on the positions of points ( = 1, 2, 3) and is thereby independent of the positions of points ( = 4, 5, 6). Nevertheless, because of the self-similarity of Koch's curve, any displacements of points ( = 4, 5, 6) would influence the positions of ( = 1, 2, 3) and hence change the distance between points 1 and 3 . That is to say, the local shape (e.g., Δ 1 2 3 ) is closely related to the global shape (e.g., Δ 4 5 6 ). Unfortunately, the Hausdorff distance (A.10) undoubtedly fails to reflect this fact. Therefore, we need to find a new measure of describing the analytic properties of fractal.
A.3. Definition of Fractal Measure. Hausdorff measure (A.8) does not reflect the self-similarity of fractal, so we cannot establish the calculus theory of fractal using the Hausdorff measure. In general, people often use the fractional calculus to approximately describe the analytic properties of fractal [8,12].
The fractional calculus is a theory of integrals and derivatives of any arbitrary real order. For example, the fractional derivatives of order of the function ( ) = equal [8] ( ) = Γ ( + 1) where Γ( ) denotes the Gamma function and is an arbitrary real number. Now, let us consider a -dimensional volume where ( ) is a constant which depends only on the dimension .
Using formula (A.11), the fractional derivatives of order of (A.12) equal (A.13) x 1 x 2 x 4 x 3 x 5 x 6 Figure 2: Koch's curve, which is similar to the generation of Cantor ternary set (see Figure 1), is defined by repeatedly adding the middle thirds of line segments [11]. Obviously, (Δ ) is a -dimensional Hausdorff measure, which can describe the length of a -dimensional fractal curve. Consequently, (A.15) and (A.16) together imply that Δ ( ) can be also thought of as a -dimensional measure. In this case, the order of differences Δ ( ) represents the Hausdorff dimension . Because of this fact, we next attempt to use the differences of order to define a new measure.
Let us consider the left-shift operator with step Δ and the identity operator as follows: Using the left-shift operator Δ and the identity operator 0 , we can define the difference operator of order in the form: We call (A.19) the "nonlocal distance, " which describes the length of a -dimensional fractal curve. When = 1, the nonlocal distance (A.19) returns to the Euclidean distance; that is, For instance, in Figure 2, the nonlocal distance |Δ [ 5 , 3 ]| between points 3 and 5 would depend on the positions of points ( = 1, 2, 3, 4, 5) rather than only on points 3 and 5 . Therefore, the nonlocal distance (A.19) is indeed an intrinsic way of describing fractal, since it not only shows the dimension but also reflects the connection between local and global segments of fractal.
Using the nonlocal distance (A. 19), we can propose a definition for fractal measure.
Let be a nonempty subset of -dimensional Euclidean space . We consider a countable set { } of subsets of diameter at most that covers ; that is, where diam ( ) defined by using the nonlocal distance (A.19) denotes the diameter of ; that is, Fractal Measure. For a positive and each > 0, we consider covers of by countable families { } of (arbitrary) sets with diameter less than and take the infimum of the sum of diam ( ). Then we have A.4. Definition of Fractal Derivative. Obviously, to describe the analytic properties of fractal, we need the corresponding calculus theory. Before proceeding to introduce the definition of fractal derivative, let us consider a -dimensional fractal curve ( ) (see Figure 3), which is determined by an independent characteristic parameter (e.g., the fill parameter of Peano's curve), filling up a -dimensional region. Assume that the length of the fractal curve is specified by a -dimensional volume ( ), then the (nonlocal) length between points = ( 0 ) and = ( ) in Figure 3 should be denoted by where we have used the fractal measure (A.23).
It is carefully noted that the length between points = ( 0 ) and = ( ) can not be measured by Euclidean scale; see Figure 3.
As such, we can present a definition for fractal derivative as follows. Clearly, if = 1, then the formula (A.26) will return to the Newton-Leibniz derivative, and meanwhile =1 ( ) is restored to a 1-dimensional coordinate axis.
In general, the fractional derivative of order of any differentiable function ( ) is defined in the form [13]: Formula (A.29) indicates that we can compute the fractal derivative using the fractional derivative.

B. Computation Examples
In Appendix A, we have noted that the fractal derivative can be computed using the formula (A.29). In this appendix, we present two computing examples.