Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 308691 9 pageshttpdxdoiorg1011552013308691

Research ArticleThe Validity of Dimensional Regularization Method onFractal Spacetime

Yong Tao

School of Economics and Business Administration Chongqing University Chongqing 400044 China

Correspondence should be addressed to Yong Tao taoyingyongyahoocom

Received 17 October 2013 Accepted 8 December 2013

Academic Editor Fernando Simoes

Copyright copy 2013 Yong Tao This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

The quantum field theory is one of the oldest fundamentaland most widely used tools in physics It is spectacularly suc-cessful that the value of theoretical calculation is precisely inagreement with experimental data for example the anoma-lousmagnetmoment of electron Nevertheless such a precisecalculation is on the basis of some regularization methodsfor example dimensional regularization [1]The dimensionalregularization requires that S matrix should be calculated ina non-integer-dimensional spacetime Following the spirit ofthis heuristic calculation Svozil [2] developed the quantumfield theory on fractal spacetime (QFTFS) This approachnot only can be applied to the low-order perturbative renor-malization of quantum electrodynamics but also preservesthe gauge invariance and covariance of physical equationsSvozilrsquos work implied that for a 119863-dimensional spacetimethere might be 119863 = 4 for macroscopic and 119863 lt 4 formicroscopic events [2]

Interestingly recently the investigations for a consistenttheory of quantum gravity strongly indicate that a power-counting renormalizable gravity model can be achieved in afractional dimensional spacetime for example the Horava-Lifshitz (HL) gravity model [3 4] Unfortunately HL gravitymodel is not Lorentz invariant To maintain the Lorentzinvariance Calcagni [5 6] extended the theoretical frame-work of Svozilrsquos QFTFS so as to include the description for

gravity Calcagnirsquos work showed that if the Hausdorff dimen-sion of spacetime119863 sim 2 then the ultraviolet divergence couldbe removed

In fact the notion that ldquothe Universe is fractalrdquo at quan-tum scales has become popular [5ndash7]Thus the demand for ageneralized calculus theory on the non-integer-dimensionaltopological spaces is strongly increasing Unfortunately thereis still not a rigorous calculus theory for analytically describ-ing fractal so that as Svozil has mentioned [2] some ofthe approaches of fractal calculus are essentially conjecturalIt is worth mentioning that Svozilrsquos QFTFS is just on thebasis of a conjectural integral formula on the non-integer-dimensional topological spaces Because of the importanceof Svozilrsquos approach in studying quantum gravity we suggestto construct a fractal measure so as to guarantee the validityof Svozilrsquos conjectural integral formula

2 Hausdorff Measure

The mathematical basis of QFTFS is the Hausdorff measure[2] The introduction for Hausdorff measure can be found inAppendix A If a119863-dimensional fractalΩ is embedded in119877

119899then it can be tessellated into (regular) polyhedra In partic-ular it is always possible to divide 119877

119899 into parallelepipeds of

2 Journal of Applied Mathematics

the form [2]

1198641198941119894119899

= (1199091 119909

119899) isin Ω 119909

119895= (119894119895

minus 1) Δ119909119895

+ 120572119895

0 le 120572119895

le Δ119909119895 119895 = 1 119899

(1)

Based on such a set of parallelepipeds Svozil [2] conjec-tured that the local Hausdorff measure 119889120583

119867(Ω) of Ω should

yield the following form

119889120583119867

(Ω) = lim119889(1198641198941 119894119899)rarr0

119899

prod

119895=1

(Δ119909119895)119863119899

=

119899

prod

119895=1

119889119863119899

119909119895 (2)

where119889(1198641198941119894119899

)denotes the diameter of parallelepiped1198641198941119894119899

and 119889119863119899 denotes the differential operator of order 119863119899

With these preparations above Svozil [2] proved that ifformula (2) holds then the integral of a spherically symmetricfunction 119891(119903) on Ω can be written as

intΩ

119891 (119903) 119889120583119867

=21205871198632

Γ (1198632)int

119877

0

119891 (119903) 119903119863minus1

119889119903 (3)

The integral formula (3) is the starting point of QFTFStherefore we must pay much attention to the validity offormula (2) Nevertheless formula (2) does not always holdwhenever 119863 = 119899 for example using the fractional derivative[8] it is easy to check that 119889

119863119899

119909119895(119889119909119895)119863119899

= 1 if 119863 = 119899 Thismeans that the integral formula (3) does not always hold inthe framework of Hausdorff measure

In fact Hausdorff measure is not an ideal mathematicalframework for describing fractal Next we will see thatHausdorff measure indeed determines the dimension of afractal curve but does not describe its analytic properties forexample the self-similarity between local and global shapesof a fractal curve To realize this fact we attempt to check thecase of the Cantor set see Figure 1 [8 9]

As shown by Figure 1 the Cantor set is a fractal Using theHausdorff measure (A8) (see Appendix A) we can computethe dimension of the Cantor set as [9]

119863 =ln 2

ln 3= 06309 sdot sdot sdot (4)

Nevertheless for the Cantor set we do not realize anycorrelation between its local and global segments (ie self-similarity) via the Hausdorff measure For instance theHausdorff distance between points 119909

(3)

2and 119909

(3)

1is denoted

by

119867119863

(119909(3)

2 119909(3)

1) =

10038161003816100381610038161003816119909(3)

2minus 119909(3)

1

10038161003816100381610038161003816

119863

(5)

Obviously Hausdorff distance (5) is independent of thevalues of points 119909

(3)

119894 where 119894 runs from 3 to 8 Nevertheless

because of the self-similarity between parts of the Cantor setany displacement of point 119909

(3)

119894(119894 = 3 4 8) should influ-

ence the distance between 119909(3)

2and 119909

(3)

1This is undoubtedly a

nonlocal property Unfortunately Hausdorff distance (5) failsto show this property In the next section we will construct anew fractal measure so as to exhibit such a nonlocal property

x(1)1

x(1)2

x(2)4 x(2)

3x(2)

2 x(2)1

x(3)1

x(3)2x(3)

3x(3)4

x(3)5x(3)

6x(3)7x(3)

8

x(n)2119899 x(n)

1middot middot middot

∙∙

Figure 1 The Cantor ternary set is defined by repeatedly removingthe middle thirds of line segments [8 9] (a) One starts by removingthe middle third from the interval [119909

(1)

2 119909(1)

1] leaving [119909

(2)

4 119909(2)

3] and

[119909(2)

2 119909(2)

1] (b) Next the ldquomiddle thirdrdquo of all remaining intervals

is removed (c) This process is continued ad infinitum Finally theCantor ternary set consists of all points in the interval [119909(1)

2 119909(1)

1] that

are not removed at any step in this infinite process

3 Fractal Measure

In Section 2 we have noted that the key point of guaranteeingthe validity of the integral formula (3) is that the Hausdorffmeasure is compelled to equal some differences of fractionalorder that is formula (2) holds Such a fact reminds usthat the differences of fractional order itself may be a typeof measure An interesting thought is that whether or notthe differences of fractional order can describe the nonlocalproperty of a fractal curve To this end we attempt to check a119898-dimensional volume

119909 (119897) = 120596 (119898) 119897119898

(6)

where 120596(119898) is a constant factor that depends only on thedimension 119898 and 119898 may be a fraction

The fractional derivatives of order 119898 of 119909(119897) give [8]

119889119898

119909 (119897) = Γ (119898 + 1) (119889119897)119898

sim (119889119897)119898

(7)

Obviously (119889119897)119898 as a 119898-dimensional volume is a

119898-dimensional Hausdorff measure therefore formula (7)implies that the differences of order 119898 119889

119898

119909(119897) can bealso thought of as a measure for describing the length ofa 119898-dimensional fractal curve In this case the order ofdifferences 119889

119898

119909(119897) represents the Hausdorff dimension 119898Using the differences of order 119898 we define a new

distancemdashcall it the ldquononlocal distancerdquomdashin the form (see(A19) in Appendix A)

1003816100381610038161003816Δ119898 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119909 (119897 minus 119895Δ119897)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(8)

where |Δ119898

[119909(119897) 119909(119897 minus Δ119897)]| denotes the nonlocal distancebetween points 119909(119897) and 119909(119897 minus Δ119897)

Journal of Applied Mathematics 3

It is carefully noted that every 119909(119897minus119895Δ119897) represents a pointon a 119898-dimensional fractal curve where 119895 = 0 1 2

Clearly according to (8) the distance between points 119909(119897)

and 119909(119897minusΔ119897) would depend on all the points 119909(119897minus119895Δ119897) where119895 = 0 1 2

Moreover itrsquos easy to check that |Δ119898=1

[119909(119897) 119909(119897 minus Δ119897)]| =

|119909(119897) minus 119909(119897 minus Δ119897)| This means that the Euclidean distance is aspecial case of nonlocal distance whenever the dimension ofthe fractal curve 119898 equals 1

If we use the nonlocal distance (8) tomeasure the distancebetween points 119909

(3)

2and 119909

(3)

1(see Figure 1) then we will

surprisingly find that the nonlocal distance

10038161003816100381610038161003816Δ119898=119863

[119909(3)

2 119909(3)

1]10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

8

sum

119895=1

119863 (119863 minus 1) sdot sdot sdot (119863 minus 119895 + 1) (minus1)119895

119895119909(3)

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(9)

which remarkably differs from the Hausdorff distance (5)would depend on the values of points 119909

(3)

119894(119894 = 3 4 8)

This means that any displacement of point 119909(3)

119894(119894 = 3 4

8) would change the nonlocal distance between points119909(3)

2and 119909

(3)

1 Consequently nonlocal distance (8) is indeed

an intrinsic way of describing self-similar fractal since itnot only determines the dimension of a fractal curve (egCantor ternary set) but also reflects the correlation betweenits parts Interestingly the nonlocal distance seems to havesome connection with quantum behavior for details seeAppendix C

Using the nonlocal distance we have given a definition forfractal measure in Appendix A (see (A23))

To study the analytic properties of a fractal curve wedefine the fractal derivative (see (A26) in Appendix A) in theform

119897119863120596119891 (119909)

119897119863120596119909

= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (10)

where 119891(119897) = 119891[119909(119897)] is a differentiable function with respectto coordinate 119897 119897 is a parameter (eg the single parameter ofPeanorsquos curve [10] for details see Appendix A) which com-pletely determines the generation of a 120596-dimensional fractalcurve and 119909(119897) denotes the length of the correspondingfractal curve (We introduce a simple way of understandingthe fractal derivative (10) For the case of the Newton-Leibnizderivative of 119910 = 119891(119909) 119909 is a 1-dimensional coordinate axisand hence can bemeasured by a Euclidean scale (ruler)Thusthe differential element of 119909 is a 1-dimensional Euclideanlength 119889119909 which gives rise to the Newton-Leibniz derivative119889119891(119909)119889119909 Nevertheless if 119909 is a 120596-dimensional fractal curvethen it can not be measured by the Euclidean scale (ruler)In this case the differential element of 119909 should be a 120596-dimensional volume

119897119863120596119909 which gives rise to the fractal

derivative119897119863120596119891(119909)

119897119863120596119909 For details see Appendix A and

Figure 3)In particular the fractal derivative (10) will return to the

well-known Newton-Leibniz derivative whenever 120596 = 1

Using the formula of fractional derivative [8] the frac-tal derivative can be rewritten as (see (A27)ndash(A29) inAppendix A)

119897119863120596119891 (119909)

119897119863120596119909

=119889120596

119891 (119897) 119889119897120596

119889120596119909 (119897) 119889119897120596 (11)

By formula (11) we can easily compute the fractal deriva-tive of any differentiable function using the fractional deriva-tive for concrete examples see Appendix B

4 Fractal Integral

In Section 3 we have proposed a definition for fractal deriva-tive Correspondingly we can now present a convenientdefinition for fractal integral as follow

Definition 1 If119897119863119898

119891(119909)119897119863119898

119909 = 119892(119909) then the fractalintegral of 119892(119909) on a 119898-dimensional fractal curve 120573

119898(119897) is

defined in the form

int119897119863119898

119891 (119909) = int120573119898(119897)

119892 (119909)119897119863119898

119909

= int119882

119892 (119909)119889119898

119909

119889119897119898(119889119897)119898

(12)

where119882denotes the definitional domain of the characteristicparameter 119897 also the parameter 119897 completely determines thegeneration of the 119898-dimensional fractal curve 120573

119898(119897)

Using such a definition of fractal integral we can prove thefollowing proposition

Proposition 2 If 119909(119903) = 120596(119898)119903119898 not only describes the

length of a 119898-dimensional fractal curve 120573119898

(119903) but also denotesthe volume of a 119898-dimensional sphere (for example the 119898-dimensional fractal curve 120573

119898(119903) fills up the entire sphere Ω) Ω

and if119891(119909) = 119891(119903) is a spherically symmetric function then thefractal integration of 119891(119909) on the 119898-dimensional fractal curve120573119898

(119903) equals

int120573119898(119903)

119891 (119909)119903119863119898

119909 =21205871198982

Γ (1198982)int

119877

0

119891 (119903) 119903119898minus1

119889119903 (13)

where 120596(119898) = 1205871198982

Γ((1198982) + 1)

Proof The Riemann-Liouville fractional integrals of order 119898

of 119891(119897) are defined in the form [8]

int 119891 (119897) (119889119897)119898

=1

Γ (119898)int

119877

119910

(119897 minus 119910)119898minus1

119891 (119897) 119889119897 (14)

Using (A29) and (B3) we have

119903119863119898

119909 =119889119898

119909

119889119903119898(119889119903)119898

= 120596 (119898) Γ (1 + 119898) (119889119903)119898

(15)

Using (12) and (15) we arrive at

int120573119898(119903)

119891 (119909)119903119863119898

119909 = intΩ

119891 (119903) 120596 (119898) Γ (1 + 119898) (119889119903)119898

(16)

4 Journal of Applied Mathematics

Inserting (14) into (16) leads to

int120573119898(119903)

119891 (119909)119903119863119898

119909 = 120596 (119898)Γ (1 + 119898)

Γ (119898)

times int

119877

119910

(119903 minus 119910)119898minus1

119891 (119903) 119889119903

(17)

Using 120596(119898) = 1205871198982

Γ((1198982) + 1) and the formula (B7)(17) can be rewritten as

int120573119898(119903)

119891 (119909)119903119863119898

119909 =1205871198982

119898

Γ ((1198982) + 1)int

119877

119910

(119903 minus 119910)119898minus1

119891 (119903) 119889119903

(18)

For 119910 = 0 (18) yields

int120573119898(119903)

119891 (119909)119903119863119898

119909 =1205871198982

119898

Γ ((1198982) + 1)int

119877

0

119903119898minus1

119891 (119903) 119889119903 (19)

Using the formula (B7) we have119898

2Γ (

119898

2) = Γ (

119898

2+ 1) (20)

Substituting (20) into (19) leads to

int120573119898(119903)

119891 (119909)119903119863119898

119909 =21205871198982

Γ (1198982)int

119877

0

119891 (119903) 119903119898minus1

119889119903 (21)

The proof is complete

Clearly for 119891(119909) = 1 formula (13) gives the volume of a119898-dimensional sphere 120587

1198982

Γ((1198982) + 1)119877119898

Before proceeding to arrive at the main result of thispaper let us consider three measurable sets 119882

119894with the

dimension 119898119894 where 119894 = 1 2 3 According to Fubinirsquos the-

orem the Cartesian product of the sets 119882119894can produce a set

119882 with the dimension 119898 = 1198981

+ 1198982

+ 1198983 where 119882 =

1198821

otimes 1198822

otimes 1198823 The integration over a function 119891(119903

1 1199032 1199033)

on 119882 can be written in the form

int119882

119891 (1199031 1199032 1199033) 119889120583 (119882)

= int1198821

int1198822

int1198823

119891 (1199031 1199032 1199033) 119889120583 (119882

1) sdot 119889120583 (119882

2) sdot 119889120583 (119882

3)

(22)

Then we have the following lemma

Lemma 3 If 119889120583(119882119894) is denoted by

119889120583 (119882119894) =

21205871198981198942

Γ (1198981198942)

119903119898119894minus1

119894119889119903119894

(119894 = 1 2 3) (23)

and if 119891(1199031 1199032 1199033) is specified by a spherically symmetric

function 119891(119903) with 1199032

= 1199032

1+ 1199032

2+ 1199032

3 then we have

int1198821

int1198822

int1198823

119891 (1199031 1199032 1199033) 119889120583 (119882

1) sdot 119889120583 (119882

2) sdot 119889120583 (119882

3)

=21205871198982

Γ (1198982)int

119877

0

119891 (119903) 119903119898minus1

119889119903

(24)

The proof of Lemma 3 can be found in page 28 in theliterature [8]

With the preparations above we can introduce the mainresult of this paper as follow

Proposition 4 If 119909119894(119903119894) = 120596(119863

119894)119903119863119894

119894(119894 = 1 2 3) not only

describes the length of a 119863119894-dimensional fractal curve 120573

119863119894

(119903119894)

but also denotes the volume of a 119863119894-dimensional sphere Ω

119894and

if 119891(119903) = 119891(1199031 1199032 1199033) is a spherically symmetric function then

the fractal integration of 119891(119903) on the fractal graph 1205731198631

(1199031) otimes

1205731198632

(1199032) otimes 1205731198633

(1199033) equals

int1205731198631(1199031)

int1205731198632(1199032)

int1205731198633(1199033)

119891 (1199031 1199032 1199033)

3

prod

119894=1

119903119894119863119863119894

119909119894

=21205871198632

Γ (1198632)int

119877

0

119891 (119903) 119903119863minus1

119889119903

(25)

where 120596(119863119894) = 120587

1198631198942

Γ((1198631198942) + 1) 119903

2

= 1199032

1+ 1199032

2+ 1199032

3 and

119863 = 1198631

+ 1198632

+ 1198633

Proof Using (13) and (24) we easily arrive at (25)The proof is complete

Proposition 4 shows that we have arrived at the con-jectural integral formula (3) through the fractal measure(A23) In particular if 119877 rarr infin then the integral for-mula (25) will exhibit linearity translational invariance andscaling property These properties are natural and necessaryin applying dimensional regularization to quantum fieldtheory [2] It is worth mentioning that Proposition 4 canbe generalized so that 119894 might take any positive integer forexample 119894 = 1 2 infin If 119894 = 1 2 3 4 then 120573

119863119894

(119903119894) may

denote coordinate axiswith the dimension119863119894and so120573

1198631

(1199031)otimes

1205731198632

(1199032) otimes 1205731198633

(1199033) otimes 1205731198634

(1199034) may denote spacetime

5 Conclusion

Hausdorff measure is not an ideal mathematical frameworkfor describing fractal since it fails to describe the nonlocalproperty of fractal (eg self-similarity) However the fractalmeasure constructed by this paper not only shows the dimen-sion of a fractal but also describes its analytic properties (egnonlocal property) Not only so using this fractal measurewe can derive Svozilrsquos conjectural integral formula (3) whichis the starting point of quantum field theory on fractalspacetimeTherefore our fractal measure may be regarded asa possible mathematical basis of establishing quantum fieldtheory on fractal spacetime

Appendices

A Mathematical Preparations

In Euclidean geometry the dimension of a geometric graphis determined by the number of independent variables (iethe number of degrees of freedom) For example everypoint on a plane can be represented by 2-tuples real number(1199091 1199092) then the dimension of the plane is denoted by

Journal of Applied Mathematics 5

2 Nevertheless the existence of Peanorsquos curve powerfullyrefutes this viewpoint Peanorsquos curve which is determined byan independent characteristic parameter (ie fill parameter)would fill up the entire plane [10]Therefore mathematicianshave to reconsider the definition of dimension The mostfamous one of all definitions of dimension is the Hausdorffdimension which is defined through the Hausdorff measure[8]

A1 Hausdorff Measure and Hausdorff Dimension In orderto bring the definition of Hausdorff dimension we firstlyintroduce the Hausdorff measure [8]

Let 119882 be a nonempty subset of 119899-dimensional Euclideanspace 119877

119899 the diameter of 119882 is defined as

diam (119882) = sup 119889 (119909 119910) 119909 119910 isin 119882 (A1)

where 119889(119909 119910) which is the distance between points 119909 and 119910is a real-valued function on 119882 otimes 119882 such that the followingfour conditions are satisfied

119889 (119909 119910) ge 0 forall119909 119910 isin 119882 (A2)

119889 (119909 119910) = 0 iff 119909 = 119910 (A3)

119889 (119909 119910) = 119889 (119910 119909) forall119909 119910 isin 119882 (A4)

119889 (119909 119911) le 119889 (119909 119910) + 119889 (119910 119911) forall119909 119910 119911 isin 119882 (A5)

For example the distance of 119899-dimensional Euclideanspace 119877

119899 can be defined as

119889119864

(119909 119910) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 = (

119899

sum

119894=1

1003816100381610038161003816119909119894 minus 119910119894

1003816100381610038161003816

2

)

12

(A6)

Then it is easy to check that (A6) satisfies conditions(A2)ndash(A5)

Now let us consider a countable set 119864119894 of subsets of

diameter at most 120576 that covers 119882 that is

119882 sub

infin

⋃

119894=1

119864119894 diam (119864

119894) le 120576 forall119894 (A7)

For a positive 119863 and each 120576 gt 0 we consider covers of 119882

by countable families 119864119894 of (arbitrary) sets 119864

119894with diameter

less than 120576 and take the infimum of the sum of [diam(119864119894)]119863

Then we have

119867119863

120576(119882)

= inf

infin

sum

119894=1

[diam (119864119894)]119863

119882 sub

infin

⋃

119894=1

119864119894 diam (119864

119894) le 120576

(A8)

If the following limit exists

119867119863

(119882) = lim120576rarr0

119867119863

120576(119882) = finite (A9)

then the value119867119863

(119882) is called the119863-dimensional Hausdorffmeasure

A2 Shortcoming of Hausdorff Measure In general 119863 maybe a fraction In 1967 Mandelbrot realized that [11] the lengthof coastline can be measured using Hausdorff measure (A8)rather than Euclideanmeasure (A6) and then the dimensionof coastline is a fraction Mandelbrot called such geometricgraphs the ldquofractalrdquo

The fractal is self-similar between its local and globalshapes Unfortunately Hausdorff measure can determine thedimension of fractal but not reflect the connection (eg self-similarity) among the parts of the corresponding fractal Tosee this we consider Kochrsquos curve in Figure 2

Clearly the congruent triangle Δ119909111990921199093is similar to

Δ119909411990951199096 If we use the Hausdorff measure (A8) to measure

the local distance of Kochrsquos curve (eg the distance betweenpoints 119909

1and 119909

3) then we have

119867119863

(1199091 1199093) =

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816

119863

+10038161003816100381610038161199092 minus 119909

3

1003816100381610038161003816

119863

(A10)

where 119863 is the dimension of theKoch curveEquation (A10) shows that the Hausdorff distance

between points 1199091and 119909

3depends only on the positions of

points 119909119894

(119894 = 1 2 3) and is thereby independent of thepositions of points 119909

119895(119895 = 4 5 6) Nevertheless because

of the self-similarity of Kochrsquos curve any displacements ofpoints 119909

119895(119895 = 4 5 6) would influence the positions of 119909

119894(119894 =

1 2 3) and hence change the distance between points 1199091and

1199093 That is to say the local shape (eg Δ119909

111990921199093) is closely

related to the global shape (eg Δ119909411990951199096) Unfortunately

the Hausdorff distance (A10) undoubtedly fails to reflect thisfact Therefore we need to find a new measure of describingthe analytic properties of fractal

A3 Definition of Fractal Measure Hausdorff measure (A8)does not reflect the self-similarity of fractal so we cannotestablish the calculus theory of fractal using the Hausdorffmeasure In general people often use the fractional calculusto approximately describe the analytic properties of fractal[8 12]

The fractional calculus is a theory of integrals and deriva-tives of any arbitrary real order For example the fractionalderivatives of order 119898 of the function 119910(119897) = 119888119897

119899 equal [8]

119889119898

119910 (119897)

119889119897119898= 119888

Γ (119899 + 1)

Γ (119899 minus 119898 + 1)119897119899minus119898

(A11)

where Γ(119909) denotes the Gamma function and 119898 is an arbi-trary real number

Now let us consider a 119898-dimensional volume

119909 (119897) = 120596 (119898) 119897119898

(A12)

where 120596(119898) is a constant which depends only on the dimen-sion 119898

Using formula (A11) the fractional derivatives of order119898

of (A12) equal

119889119898

119909 (119897)

119889119897119898= Γ (119898 + 1) 120596 (119898) (A13)

6 Journal of Applied Mathematics

x1

x2

x4

x3

x5

x6

Figure 2 Kochrsquos curve which is similar to the generation of Cantorternary set (see Figure 1) is defined by repeatedly adding themiddlethirds of line segments [11]

Equation (A13) can be written as

Δ119898

119909 (119897) = Γ (119898 + 1) 120596 (119898) (Δ119897)119898

+ 119900 [(Δ119897)119898

] (A14)

whereΔ119898

119909(119897) denotes the differences of order119898 and 119900[(Δ119897)119898

]

denotes the infinitesimal terms of higher order compared to(Δ119897)119898Equation (A14) implies that

Δ119898

119909 (119897) sim (Δ119897)119898

(A15)

and thereby

119873

sum

119894=1

Δ119898

119909 (119897119894) sim

119873

sum

119894=1

(Δ119897)119898

(A16)

Obviously (Δ119897)119898 is a 119898-dimensional Hausdorff measure

which can describe the length of a 119898-dimensional fractalcurve Consequently (A15) and (A16) together imply thatΔ119898

119909(119897) can be also thought of as a 119898-dimensional measureIn this case the order of differences Δ

119898

119909(119897) represents theHausdorff dimension119898 Because of this fact we next attemptto use the differences of order 119898 to define a new measure

Let us consider the left-shift operator with stepΔ119897 and theidentity operator as follows

119871Δ119897

119909 (119897) = 119909 (119897 minus Δ119897)

1198710119909 (119897) = 119909 (119897)

(A17)

Using the left-shift operator 119871Δ119897and the identity operator

1198710 we can define the difference operator of order 119898 in the

form

(1198710

minus 119871Δ119897

)119898

=

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119871119895Δ119897

(A18)

Using (A18) we define a new distance between points119909(119897) and 119909(119897 minus Δ119897) in the form

1003816100381610038161003816Δ119898 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119909 (119897 minus 119895Δ119897)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(A19)

We call (A19) the ldquononlocal distancerdquo which describesthe length of a 119898-dimensional fractal curve

When 119898 = 1 the nonlocal distance (A19) returns to theEuclidean distance that is

1003816100381610038161003816Δ119898=1 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816 = |119909 (119897) minus 119909 (119897 minus Δ119897)| (A20)

In general the nonlocal distance (A19) does not satisfythe general properties of distance (A4)-(A5) but reflects theconnection between local and global segments of fractal (Forexample the nonlocal distance between points 119909(119897) and 119909(119897 +

Δ119897) that is |Δ119898

[119909(119897) 119909(119897+Δ119897)]| needs to be defined using theright-shift operator119877

Δ119897which leads to119877

Δ119897119909(119897) = 119909(119897+Δ119897)The

corresponding difference operator reads (119877Δ119897

minus 1198770)119898 Then

the nonlocal distance would not satisfy condition (A4))To understand the latter we need to realize that the outputvalue of (A19) would depend on the values of all points119909(119897 minus 119895Δ119897) (119895 = 0 1 2 ) rather than only on points 119909(119897) and119909(119897 minus Δ119897)

For instance in Figure 2 the nonlocal distance |Δ119863

[1199095

1199093]| between points 119909

3and 119909

5would depend on the positions

of points 119909119894

(119894 = 1 2 3 4 5) rather than only on points 1199093

and 1199095 Therefore the nonlocal distance (A19) is indeed an

intrinsic way of describing fractal since it not only shows thedimension but also reflects the connection between local andglobal segments of fractal

Using the nonlocal distance (A19) we can propose adefinition for fractal measure

Let 119882 be a nonempty subset of 119899-dimensional Euclideanspace 119877

119899 We consider a countable set 119865119894 of subsets of

diameter at most 120576 that covers 119882 that is

119882 sub

infin

⋃

119894=1

119865119894 diam119863 (119865

119894) le 120576 forall119894 (A21)

where diam119863(119865119894) defined by using the nonlocal distance

(A19) denotes the diameter of 119865119894 that is

diam119863 (119865119894) = sup

1003816100381610038161003816Δ119863 [119909 119910]1003816100381610038161003816 119909 119910 isin 119865

119894 (A22)

Fractal Measure For a positive 119863 and each 120576 gt 0 we considercovers of 119882 by countable families 119865

119894 of (arbitrary) sets 119865

119894

with diameter less than 120576 and take the infimum of the sum ofdiam119863(119865

119894) Then we have

Π119863

120576(119882)

= inf

infin

sum

119894=1

diam119863 (119865119894) 119882 sub

infin

⋃

119894=1

119865119894 diam119863 (119865

119894) le 120576

(A23)

If the following limit exists

Π119863

(119882) = lim120576rarr0

Π119863

120576(119882) = finite (A24)

then the value Π119863

(119882) is called the 119863-dimensional fractalmeasure meanwhile the dimension of 119882 equals 119863

Journal of Applied Mathematics 7

A4 Definition of Fractal Derivative Obviously to describethe analytic properties of fractal we need the correspondingcalculus theory

Before proceeding to introduce the definition of fractalderivative let us consider a 120596-dimensional fractal curve120573120596(119897) (see Figure 3) which is determined by an independent

characteristic parameter 119897 (eg the fill parameter of Peanorsquoscurve) filling up a 120596-dimensional region Assume that thelength of the fractal curve is specified by a 120596-dimensionalvolume 119909(119897) then the (nonlocal) length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) in Figure 3 should be denoted by

Π119863

([119886 119887]) = limΔ119897rarr0

inf

119899

sum

119894=1

1003816100381610038161003816Δ120596 [119909 (119897119894minus1

) 119909 (119897119894)]

1003816100381610038161003816 (A25)

where we have used the fractal measure (A23)It is carefully noted that the length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) can not bemeasured by Euclidean scale

see Figure 3As such we can present a definition for fractal derivative

as followsFractal Derivative For any differentiable function 119910 = 119891(119909)if 119909 = 119909(119897) is not only a 120596-dimensional volume but alsodescribes the length of a 120596-dimensional fractal curve 120574

120596(119897)

then the fractal derivative of 119910 = 119891(119909) with respect to thefractal curve 120574

120596(119897) is defined as

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (A26)

where 119891(119897) = 119891[119909(119897)]Clearly if120596 = 1 then the formula (A26)will return to the

Newton-Leibniz derivative andmeanwhile 120574120596=1

(119897) is restoredto a 1-dimensional coordinate axis

In general the fractional derivative of order 120596 of anydifferentiable function 119891(119897) is defined in the form [13]

119889120596

119891 (119897)

119889119897120596

= limΔ119897rarr0

(

infin

sum

119895=0

120596 (120596 minus 1) sdot sdot sdot (120596 minus 119895 + 1) (minus1)119895

119895119891 (119897 minus 119895Δ119897)

times (Δ119897)minus120596

)

(A27)

Comparing (A19) and (A27) we have

119889120596

119891 (119897)

119889119897120596= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

(Δ119897)120596

(A28)

Using formula (A28) the formula (A26) can be rewrittenas

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)]sdot limΔ119897rarr0

1(Δ119897)120596

1(Δ119897)120596

=

limΔ119897rarr0

(Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)] (Δ119897)120596

)

limΔ119897rarr0

(Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (Δ119897)120596

)

=119889120596

119891 (119897) 119889119897120596

119889120596119909 (119897) 119889119897120596

(A29)

Formula (A29) indicates that we can compute the fractalderivative using the fractional derivative

B Computation Examples

In Appendix A we have noted that the fractal derivative canbe computed using the formula (A29) In this appendix wepresent two computing examples

Example 1 If 119909(119897) = 120596(119898)119897119898 describes the length of a 119898-

dimensional fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119897119863119898

119862

119897119863119898

119909=

119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909 (B1)

where 120596(119898) is a constant that depends only on 119898

Proof The fractional derivatives of order 119898 of the constant 119862

and the power function 119910(119897) = 119886119897119899 are respectively as follows

[8]

119889119898

119862

119889119897119898=

119862

Γ (1 minus 119898)119897minus119898

(B2)

119889119898

119910 (119897)

119889119897119898= 119886

Γ (119899 + 1)

Γ (119899 minus 119898 + 1)119897119899minus119898

(B3)

Using formulas (A29) (B2) and (B3) the fractal deriva-tive119897119863119898

119891(119909)119897119863119898

119909 can be computed as follows

119897119863119898

119862

119897119863119898

119909=

119889119898

119862119889119897119898

119889119898 [120596 (119898) 119897119898] 119889119897119898=

(119862Γ (1 minus 119898)) 119897minus119898

120596 (119898) Γ (1 + 119898)

=119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909

(B4)

The proof is complete

Example 2 If 119909 is the characteristic parameter of a 119898-dimensional fractal curve andmeanwhile it also describes thelength of this fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B5)

8 Journal of Applied Mathematics

Fractal curve

Nonlocal scale

Euclidean scale

120573120596(l1) 120573120596(l3)

120573120596(l2) 120573120596(l4)a = 120573120596(l0)120573120596(li) 120573120596(li+2)

)

120573120596(li+1)

120573120596(lnminus1)

b = 120573120596(ln

120573120596(l)

x(l0) = 0 x(l1) x(l2) x(l3) x(lnminus3) x(lnminus2) x(lnminus1) x(ln)

x(l)

l0 = 0 l1 l2 l3

l

lmlmminus3 lmminus2lmminus1

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middotmiddot middot middot

Figure 3 The fractal curve 120573120596(119897) consists of the union ⋃

119899

119894=1[120573120596(119897119894minus1

) 120573120596(119897119894)] where 119899 = 119899(119898) The dimension of fractal curve 120573

120596(119897) equals

120596 = lim119898rarrinfin

(ln 119899(119898) ln119898) Since 120573120596(119897) is a fractal curve the distance between points 120573

120596(1198970) and 120573

120596(119897119899) cannot be measured using the

Euclidean scale (ruler) 119897 otherwise we will have 119889[120573120596(1198970) 120573120596(119897119899)] = lim

119898rarrinfin119899(119898)((119897

119898minus 1198970)119898) = infin or =0 However the fractal curve 120573

120596(119897)

can be measured using the nonlocal scale (ruler) 119909(119897) for the way of measure see the formula (A25)

Proof Using formulas (A29) (B2) and (B3) we have

119909119863119898

119891 (119909)

119909119863119898

119909=

119889119898

119862119889119909119898

119889119898119909119889119909119898=

(119862Γ (1 minus 119898)) 119909minus119898

(Γ (2) Γ (2 minus 119898)) 1199091minus119898

= 119862Γ (2 minus 119898)

Γ (1 minus 119898)sdot

1

119909

(B6)

Considering the property of Gamma function

Γ (119909 + 1) = 119909Γ (119909) (B7)

we have

1 minus 119898 =Γ (2 minus 119898)

Γ (1 minus 119898) (B8)

Substituting (B8) into (B6) we arrive at

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B9)

The proof is complete

C Quantum Behavior and Nonlocal Distance

Now we investigate the connection between quantum behav-iour and nonlocal distance For simplicity we still con-sider the Cantor set in Figure 1 where the nonlocal dis-tance between points 119909

(119899)

2and 119909

(119899)

1is equal to

Π (119909(119899)

2 119909(119899)

1) =

10038161003816100381610038161003816Δ119898=119863

[119909(119899)

2 119909(119899)

1]10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2119899

sum

119895=1

119863 (119863 minus 1) sdot sdot sdot (119863 minus 119895 + 1) (minus1)119895

119895119909(119899)

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(C1)

Clearly the output value of nonlocal distance (C1)depends on the value of each element in the set 119909

(119899)

119894

where 119894 runs from 1 to 2119899 If 119899 rarr infin we will have

Π(119909(119899)

2 119909(119899)

1) rarr 0 however the number of elements in the

set 119909(119899)

119894 would tend to infinity too This means that if we

want to precisely measure the distance of smaller scale(eg lim

119899≫1Π(119909(119899)

2 119909(119899)

1)) correspondingly we will need to

collect the points 119909(119899)

119894 As a result if we want to precisely

measure the nonlocal distance between points lim119899rarrinfin

119909(119899)

2

and lim119899rarrinfin

119909(119899)

1 we will need to collect a set of infinite

points that is 119909(infin)119894

infin

119894=1 Unfortunately we must fail to arrive

Journal of Applied Mathematics 9

at this purpose on an actual measurement In other words wecannot precisely measure the distance of microscopic scalethis fact is consistent with the ldquoHeisenberg Uncertainty Prin-ciplerdquo (In fact the connection between quantum mechanicsand fractal has been noticed in some earlier papers [14ndash17])

On the other hand the nonlocal distance (C1) betweenpoints119909

(119899)

2and119909(119899)

1depends clearly on the position of119909(119899)

119894(119894 =

1 2 2119899

) for example any displacement of point 119909(119899)

2119899

would influence the output value of nonlocal distance (C1) Itis a clearly nonlocal correlation (correlations span arbitrarilydistances) and similar to quantum entanglement

The above two facts imply that the nonlocal descriptionmay be an intrinsic way of describing quantum behavior (Inreferences [18 19] we have shown that the local descriptionis not a way of completely describing physical reality)

Acknowledgment

This project is supported by the Scholarship Award for Excel-lent Doctoral Student Granted by the Ministry of Educationof China (2012) (Grant no 0903005109081-019)

References

[1] G rsquot Hooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquoNuclear Physics B vol 44 no 1 pp 189ndash2131972

[2] K Svozil ldquoQuantum field theory on fractal spacetime a newregularisation methodrdquo Journal of Physics A vol 20 no 12 pp3861ndash3875 1987

[3] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 15 pages 2009

[4] P Horava ldquoSpectral dimension of the universe in quantumgravity at a Lifshitz pointrdquo Physical Review Letters vol 102 no16 Article ID 161301 4 pages 2009

[5] G Calcagni ldquoQuantum field theory gravity and cosmology in afractal universerdquo Journal ofHigh Energy Physics vol 2010 article120 2010

[6] G Calcagni ldquoFractal universe and quantum gravityrdquo PhysicalReview Letters vol 104 no 25 Article ID 251301 2010

[7] D Benedetti ldquoFractal properties of quantum spacetimerdquo Phys-ical Review Letters vol 102 no 11 Article ID 111303 4 pages2009

[8] V E Tarasov Fractional Dynamics Applications of FractionalCalculus to Dynamics of Particles Field and Media HigherEducation Press Beijing China 2010

[9] T C Halsey M H Jensen L P Kadanoff I Procaccia andB I Shraiman ldquoFractal measures and their singularities thecharacterization of strange setsrdquo Physical Review A vol 33 no2 pp 1141ndash1151 1986

[10] H Sagan Space-Filling Curves Springer New York NY USA1994

[11] BMandelbrot ldquoHow long is the coast of Britain Statistical self-similarity and fractional dimensionrdquo Science vol 156 no 3775pp 636ndash638 1967

[12] M Giona ldquoFractal calculus on [0 1]rdquo Chaos Solitons andFractals vol 5 no 6 pp 987ndash1000 1995

[13] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[14] M S El Naschie ldquoQuantum mechanics and the possibility of aCantorian space-timerdquo Chaos Solitons and Fractals vol 1 no5 pp 485ndash487 1991

[15] M S El Naschie ldquoA note on Heisenbergrsquos uncertainty principleand Cantorian space-timerdquo Chaos Solitons and Fractals vol 2no 4 pp 437ndash439 1992

[16] M S El Naschie ldquoOn certain infinite-dimensional Cantor setsand the Schrodinger waverdquo Chaos Solitons and Fractals vol 3no 1 pp 89ndash98 1993

[17] L Marek-Crnjac ldquoA short history of fractal-Cantorian space-timerdquo Chaos Solitons and Fractals vol 41 no 5 pp 2697ndash27052009

[18] Y Tao ldquoNecessity of integral formalismrdquo Communications inTheoretical Physics vol 56 no 4 pp 648ndash654 2011

[19] Y Tao ldquoSufficient condition for validity of quantum adiabatictheoremrdquo Communications in Theoretical Physics vol 57 no 3pp 343ndash347 2012

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Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

the form [2]

1198641198941119894119899

= (1199091 119909

119899) isin Ω 119909

119895= (119894119895

minus 1) Δ119909119895

+ 120572119895

0 le 120572119895

le Δ119909119895 119895 = 1 119899

(1)

Based on such a set of parallelepipeds Svozil [2] conjec-tured that the local Hausdorff measure 119889120583

119867(Ω) of Ω should

yield the following form

119889120583119867

(Ω) = lim119889(1198641198941 119894119899)rarr0

119899

prod

119895=1

(Δ119909119895)119863119899

=

119899

prod

119895=1

119889119863119899

119909119895 (2)

where119889(1198641198941119894119899

)denotes the diameter of parallelepiped1198641198941119894119899

and 119889119863119899 denotes the differential operator of order 119863119899

With these preparations above Svozil [2] proved that ifformula (2) holds then the integral of a spherically symmetricfunction 119891(119903) on Ω can be written as

intΩ

119891 (119903) 119889120583119867

=21205871198632

Γ (1198632)int

119877

0

119891 (119903) 119903119863minus1

119889119903 (3)

The integral formula (3) is the starting point of QFTFStherefore we must pay much attention to the validity offormula (2) Nevertheless formula (2) does not always holdwhenever 119863 = 119899 for example using the fractional derivative[8] it is easy to check that 119889

119863119899

119909119895(119889119909119895)119863119899

= 1 if 119863 = 119899 Thismeans that the integral formula (3) does not always hold inthe framework of Hausdorff measure

In fact Hausdorff measure is not an ideal mathematicalframework for describing fractal Next we will see thatHausdorff measure indeed determines the dimension of afractal curve but does not describe its analytic properties forexample the self-similarity between local and global shapesof a fractal curve To realize this fact we attempt to check thecase of the Cantor set see Figure 1 [8 9]

As shown by Figure 1 the Cantor set is a fractal Using theHausdorff measure (A8) (see Appendix A) we can computethe dimension of the Cantor set as [9]

119863 =ln 2

ln 3= 06309 sdot sdot sdot (4)

Nevertheless for the Cantor set we do not realize anycorrelation between its local and global segments (ie self-similarity) via the Hausdorff measure For instance theHausdorff distance between points 119909

(3)

2and 119909

(3)

1is denoted

by

119867119863

(119909(3)

2 119909(3)

1) =

10038161003816100381610038161003816119909(3)

2minus 119909(3)

1

10038161003816100381610038161003816

119863

(5)

Obviously Hausdorff distance (5) is independent of thevalues of points 119909

(3)

119894 where 119894 runs from 3 to 8 Nevertheless

because of the self-similarity between parts of the Cantor setany displacement of point 119909

(3)

119894(119894 = 3 4 8) should influ-

ence the distance between 119909(3)

2and 119909

(3)

1This is undoubtedly a

nonlocal property Unfortunately Hausdorff distance (5) failsto show this property In the next section we will construct anew fractal measure so as to exhibit such a nonlocal property

x(1)1

x(1)2

x(2)4 x(2)

3x(2)

2 x(2)1

x(3)1

x(3)2x(3)

3x(3)4

x(3)5x(3)

6x(3)7x(3)

8

x(n)2119899 x(n)

1middot middot middot

∙∙

Figure 1 The Cantor ternary set is defined by repeatedly removingthe middle thirds of line segments [8 9] (a) One starts by removingthe middle third from the interval [119909

(1)

2 119909(1)

1] leaving [119909

(2)

4 119909(2)

3] and

[119909(2)

2 119909(2)

1] (b) Next the ldquomiddle thirdrdquo of all remaining intervals

is removed (c) This process is continued ad infinitum Finally theCantor ternary set consists of all points in the interval [119909(1)

2 119909(1)

1] that

are not removed at any step in this infinite process

3 Fractal Measure

In Section 2 we have noted that the key point of guaranteeingthe validity of the integral formula (3) is that the Hausdorffmeasure is compelled to equal some differences of fractionalorder that is formula (2) holds Such a fact reminds usthat the differences of fractional order itself may be a typeof measure An interesting thought is that whether or notthe differences of fractional order can describe the nonlocalproperty of a fractal curve To this end we attempt to check a119898-dimensional volume

119909 (119897) = 120596 (119898) 119897119898

(6)

where 120596(119898) is a constant factor that depends only on thedimension 119898 and 119898 may be a fraction

The fractional derivatives of order 119898 of 119909(119897) give [8]

119889119898

119909 (119897) = Γ (119898 + 1) (119889119897)119898

sim (119889119897)119898

(7)

Obviously (119889119897)119898 as a 119898-dimensional volume is a

119898-dimensional Hausdorff measure therefore formula (7)implies that the differences of order 119898 119889

119898

119909(119897) can bealso thought of as a measure for describing the length ofa 119898-dimensional fractal curve In this case the order ofdifferences 119889

119898

119909(119897) represents the Hausdorff dimension 119898Using the differences of order 119898 we define a new

distancemdashcall it the ldquononlocal distancerdquomdashin the form (see(A19) in Appendix A)

1003816100381610038161003816Δ119898 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119909 (119897 minus 119895Δ119897)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(8)

where |Δ119898

[119909(119897) 119909(119897 minus Δ119897)]| denotes the nonlocal distancebetween points 119909(119897) and 119909(119897 minus Δ119897)

Journal of Applied Mathematics 3

It is carefully noted that every 119909(119897minus119895Δ119897) represents a pointon a 119898-dimensional fractal curve where 119895 = 0 1 2

Clearly according to (8) the distance between points 119909(119897)

and 119909(119897minusΔ119897) would depend on all the points 119909(119897minus119895Δ119897) where119895 = 0 1 2

Moreover itrsquos easy to check that |Δ119898=1

[119909(119897) 119909(119897 minus Δ119897)]| =

|119909(119897) minus 119909(119897 minus Δ119897)| This means that the Euclidean distance is aspecial case of nonlocal distance whenever the dimension ofthe fractal curve 119898 equals 1

If we use the nonlocal distance (8) tomeasure the distancebetween points 119909

(3)

2and 119909

(3)

1(see Figure 1) then we will

surprisingly find that the nonlocal distance

10038161003816100381610038161003816Δ119898=119863

[119909(3)

2 119909(3)

1]10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

8

sum

119895=1

119863 (119863 minus 1) sdot sdot sdot (119863 minus 119895 + 1) (minus1)119895

119895119909(3)

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(9)

which remarkably differs from the Hausdorff distance (5)would depend on the values of points 119909

(3)

119894(119894 = 3 4 8)

This means that any displacement of point 119909(3)

119894(119894 = 3 4

8) would change the nonlocal distance between points119909(3)

2and 119909

(3)

1 Consequently nonlocal distance (8) is indeed

an intrinsic way of describing self-similar fractal since itnot only determines the dimension of a fractal curve (egCantor ternary set) but also reflects the correlation betweenits parts Interestingly the nonlocal distance seems to havesome connection with quantum behavior for details seeAppendix C

Using the nonlocal distance we have given a definition forfractal measure in Appendix A (see (A23))

To study the analytic properties of a fractal curve wedefine the fractal derivative (see (A26) in Appendix A) in theform

119897119863120596119891 (119909)

119897119863120596119909

= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (10)

where 119891(119897) = 119891[119909(119897)] is a differentiable function with respectto coordinate 119897 119897 is a parameter (eg the single parameter ofPeanorsquos curve [10] for details see Appendix A) which com-pletely determines the generation of a 120596-dimensional fractalcurve and 119909(119897) denotes the length of the correspondingfractal curve (We introduce a simple way of understandingthe fractal derivative (10) For the case of the Newton-Leibnizderivative of 119910 = 119891(119909) 119909 is a 1-dimensional coordinate axisand hence can bemeasured by a Euclidean scale (ruler)Thusthe differential element of 119909 is a 1-dimensional Euclideanlength 119889119909 which gives rise to the Newton-Leibniz derivative119889119891(119909)119889119909 Nevertheless if 119909 is a 120596-dimensional fractal curvethen it can not be measured by the Euclidean scale (ruler)In this case the differential element of 119909 should be a 120596-dimensional volume

119897119863120596119909 which gives rise to the fractal

derivative119897119863120596119891(119909)

119897119863120596119909 For details see Appendix A and

Figure 3)In particular the fractal derivative (10) will return to the

well-known Newton-Leibniz derivative whenever 120596 = 1

Using the formula of fractional derivative [8] the frac-tal derivative can be rewritten as (see (A27)ndash(A29) inAppendix A)

119897119863120596119891 (119909)

119897119863120596119909

=119889120596

119891 (119897) 119889119897120596

119889120596119909 (119897) 119889119897120596 (11)

By formula (11) we can easily compute the fractal deriva-tive of any differentiable function using the fractional deriva-tive for concrete examples see Appendix B

4 Fractal Integral

In Section 3 we have proposed a definition for fractal deriva-tive Correspondingly we can now present a convenientdefinition for fractal integral as follow

Definition 1 If119897119863119898

119891(119909)119897119863119898

119909 = 119892(119909) then the fractalintegral of 119892(119909) on a 119898-dimensional fractal curve 120573

119898(119897) is

defined in the form

int119897119863119898

119891 (119909) = int120573119898(119897)

119892 (119909)119897119863119898

119909

= int119882

119892 (119909)119889119898

119909

119889119897119898(119889119897)119898

(12)

where119882denotes the definitional domain of the characteristicparameter 119897 also the parameter 119897 completely determines thegeneration of the 119898-dimensional fractal curve 120573

119898(119897)

Using such a definition of fractal integral we can prove thefollowing proposition

Proposition 2 If 119909(119903) = 120596(119898)119903119898 not only describes the

length of a 119898-dimensional fractal curve 120573119898

(119903) but also denotesthe volume of a 119898-dimensional sphere (for example the 119898-dimensional fractal curve 120573

119898(119903) fills up the entire sphere Ω) Ω

and if119891(119909) = 119891(119903) is a spherically symmetric function then thefractal integration of 119891(119909) on the 119898-dimensional fractal curve120573119898

(119903) equals

int120573119898(119903)

119891 (119909)119903119863119898

119909 =21205871198982

Γ (1198982)int

119877

0

119891 (119903) 119903119898minus1

119889119903 (13)

where 120596(119898) = 1205871198982

Γ((1198982) + 1)

Proof The Riemann-Liouville fractional integrals of order 119898

of 119891(119897) are defined in the form [8]

int 119891 (119897) (119889119897)119898

=1

Γ (119898)int

119877

119910

(119897 minus 119910)119898minus1

119891 (119897) 119889119897 (14)

Using (A29) and (B3) we have

119903119863119898

119909 =119889119898

119909

119889119903119898(119889119903)119898

= 120596 (119898) Γ (1 + 119898) (119889119903)119898

(15)

Using (12) and (15) we arrive at

int120573119898(119903)

119891 (119909)119903119863119898

119909 = intΩ

119891 (119903) 120596 (119898) Γ (1 + 119898) (119889119903)119898

(16)

4 Journal of Applied Mathematics

Inserting (14) into (16) leads to

int120573119898(119903)

119891 (119909)119903119863119898

119909 = 120596 (119898)Γ (1 + 119898)

Γ (119898)

times int

119877

119910

(119903 minus 119910)119898minus1

119891 (119903) 119889119903

(17)

Using 120596(119898) = 1205871198982

Γ((1198982) + 1) and the formula (B7)(17) can be rewritten as

int120573119898(119903)

119891 (119909)119903119863119898

119909 =1205871198982

119898

Γ ((1198982) + 1)int

119877

119910

(119903 minus 119910)119898minus1

119891 (119903) 119889119903

(18)

For 119910 = 0 (18) yields

int120573119898(119903)

119891 (119909)119903119863119898

119909 =1205871198982

119898

Γ ((1198982) + 1)int

119877

0

119903119898minus1

119891 (119903) 119889119903 (19)

Using the formula (B7) we have119898

2Γ (

119898

2) = Γ (

119898

2+ 1) (20)

Substituting (20) into (19) leads to

int120573119898(119903)

119891 (119909)119903119863119898

119909 =21205871198982

Γ (1198982)int

119877

0

119891 (119903) 119903119898minus1

119889119903 (21)

The proof is complete

Clearly for 119891(119909) = 1 formula (13) gives the volume of a119898-dimensional sphere 120587

1198982

Γ((1198982) + 1)119877119898

Before proceeding to arrive at the main result of thispaper let us consider three measurable sets 119882

119894with the

dimension 119898119894 where 119894 = 1 2 3 According to Fubinirsquos the-

orem the Cartesian product of the sets 119882119894can produce a set

119882 with the dimension 119898 = 1198981

+ 1198982

+ 1198983 where 119882 =

1198821

otimes 1198822

otimes 1198823 The integration over a function 119891(119903

1 1199032 1199033)

on 119882 can be written in the form

int119882

119891 (1199031 1199032 1199033) 119889120583 (119882)

= int1198821

int1198822

int1198823

119891 (1199031 1199032 1199033) 119889120583 (119882

1) sdot 119889120583 (119882

2) sdot 119889120583 (119882

3)

(22)

Then we have the following lemma

Lemma 3 If 119889120583(119882119894) is denoted by

119889120583 (119882119894) =

21205871198981198942

Γ (1198981198942)

119903119898119894minus1

119894119889119903119894

(119894 = 1 2 3) (23)

and if 119891(1199031 1199032 1199033) is specified by a spherically symmetric

function 119891(119903) with 1199032

= 1199032

1+ 1199032

2+ 1199032

3 then we have

int1198821

int1198822

int1198823

119891 (1199031 1199032 1199033) 119889120583 (119882

1) sdot 119889120583 (119882

2) sdot 119889120583 (119882

3)

=21205871198982

Γ (1198982)int

119877

0

119891 (119903) 119903119898minus1

119889119903

(24)

The proof of Lemma 3 can be found in page 28 in theliterature [8]

With the preparations above we can introduce the mainresult of this paper as follow

Proposition 4 If 119909119894(119903119894) = 120596(119863

119894)119903119863119894

119894(119894 = 1 2 3) not only

describes the length of a 119863119894-dimensional fractal curve 120573

119863119894

(119903119894)

but also denotes the volume of a 119863119894-dimensional sphere Ω

119894and

if 119891(119903) = 119891(1199031 1199032 1199033) is a spherically symmetric function then

the fractal integration of 119891(119903) on the fractal graph 1205731198631

(1199031) otimes

1205731198632

(1199032) otimes 1205731198633

(1199033) equals

int1205731198631(1199031)

int1205731198632(1199032)

int1205731198633(1199033)

119891 (1199031 1199032 1199033)

3

prod

119894=1

119903119894119863119863119894

119909119894

=21205871198632

Γ (1198632)int

119877

0

119891 (119903) 119903119863minus1

119889119903

(25)

where 120596(119863119894) = 120587

1198631198942

Γ((1198631198942) + 1) 119903

2

= 1199032

1+ 1199032

2+ 1199032

3 and

119863 = 1198631

+ 1198632

+ 1198633

Proof Using (13) and (24) we easily arrive at (25)The proof is complete

Proposition 4 shows that we have arrived at the con-jectural integral formula (3) through the fractal measure(A23) In particular if 119877 rarr infin then the integral for-mula (25) will exhibit linearity translational invariance andscaling property These properties are natural and necessaryin applying dimensional regularization to quantum fieldtheory [2] It is worth mentioning that Proposition 4 canbe generalized so that 119894 might take any positive integer forexample 119894 = 1 2 infin If 119894 = 1 2 3 4 then 120573

119863119894

(119903119894) may

denote coordinate axiswith the dimension119863119894and so120573

1198631

(1199031)otimes

1205731198632

(1199032) otimes 1205731198633

(1199033) otimes 1205731198634

(1199034) may denote spacetime

5 Conclusion

Hausdorff measure is not an ideal mathematical frameworkfor describing fractal since it fails to describe the nonlocalproperty of fractal (eg self-similarity) However the fractalmeasure constructed by this paper not only shows the dimen-sion of a fractal but also describes its analytic properties (egnonlocal property) Not only so using this fractal measurewe can derive Svozilrsquos conjectural integral formula (3) whichis the starting point of quantum field theory on fractalspacetimeTherefore our fractal measure may be regarded asa possible mathematical basis of establishing quantum fieldtheory on fractal spacetime

Appendices

A Mathematical Preparations

In Euclidean geometry the dimension of a geometric graphis determined by the number of independent variables (iethe number of degrees of freedom) For example everypoint on a plane can be represented by 2-tuples real number(1199091 1199092) then the dimension of the plane is denoted by

Journal of Applied Mathematics 5

2 Nevertheless the existence of Peanorsquos curve powerfullyrefutes this viewpoint Peanorsquos curve which is determined byan independent characteristic parameter (ie fill parameter)would fill up the entire plane [10]Therefore mathematicianshave to reconsider the definition of dimension The mostfamous one of all definitions of dimension is the Hausdorffdimension which is defined through the Hausdorff measure[8]

A1 Hausdorff Measure and Hausdorff Dimension In orderto bring the definition of Hausdorff dimension we firstlyintroduce the Hausdorff measure [8]

Let 119882 be a nonempty subset of 119899-dimensional Euclideanspace 119877

119899 the diameter of 119882 is defined as

diam (119882) = sup 119889 (119909 119910) 119909 119910 isin 119882 (A1)

where 119889(119909 119910) which is the distance between points 119909 and 119910is a real-valued function on 119882 otimes 119882 such that the followingfour conditions are satisfied

119889 (119909 119910) ge 0 forall119909 119910 isin 119882 (A2)

119889 (119909 119910) = 0 iff 119909 = 119910 (A3)

119889 (119909 119910) = 119889 (119910 119909) forall119909 119910 isin 119882 (A4)

119889 (119909 119911) le 119889 (119909 119910) + 119889 (119910 119911) forall119909 119910 119911 isin 119882 (A5)

For example the distance of 119899-dimensional Euclideanspace 119877

119899 can be defined as

119889119864

(119909 119910) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 = (

119899

sum

119894=1

1003816100381610038161003816119909119894 minus 119910119894

1003816100381610038161003816

2

)

12

(A6)

Then it is easy to check that (A6) satisfies conditions(A2)ndash(A5)

Now let us consider a countable set 119864119894 of subsets of

diameter at most 120576 that covers 119882 that is

119882 sub

infin

⋃

119894=1

119864119894 diam (119864

119894) le 120576 forall119894 (A7)

For a positive 119863 and each 120576 gt 0 we consider covers of 119882

by countable families 119864119894 of (arbitrary) sets 119864

119894with diameter

less than 120576 and take the infimum of the sum of [diam(119864119894)]119863

Then we have

119867119863

120576(119882)

= inf

infin

sum

119894=1

[diam (119864119894)]119863

119882 sub

infin

⋃

119894=1

119864119894 diam (119864

119894) le 120576

(A8)

If the following limit exists

119867119863

(119882) = lim120576rarr0

119867119863

120576(119882) = finite (A9)

then the value119867119863

(119882) is called the119863-dimensional Hausdorffmeasure

A2 Shortcoming of Hausdorff Measure In general 119863 maybe a fraction In 1967 Mandelbrot realized that [11] the lengthof coastline can be measured using Hausdorff measure (A8)rather than Euclideanmeasure (A6) and then the dimensionof coastline is a fraction Mandelbrot called such geometricgraphs the ldquofractalrdquo

The fractal is self-similar between its local and globalshapes Unfortunately Hausdorff measure can determine thedimension of fractal but not reflect the connection (eg self-similarity) among the parts of the corresponding fractal Tosee this we consider Kochrsquos curve in Figure 2

Clearly the congruent triangle Δ119909111990921199093is similar to

Δ119909411990951199096 If we use the Hausdorff measure (A8) to measure

the local distance of Kochrsquos curve (eg the distance betweenpoints 119909

1and 119909

3) then we have

119867119863

(1199091 1199093) =

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816

119863

+10038161003816100381610038161199092 minus 119909

3

1003816100381610038161003816

119863

(A10)

where 119863 is the dimension of theKoch curveEquation (A10) shows that the Hausdorff distance

between points 1199091and 119909

3depends only on the positions of

points 119909119894

(119894 = 1 2 3) and is thereby independent of thepositions of points 119909

119895(119895 = 4 5 6) Nevertheless because

of the self-similarity of Kochrsquos curve any displacements ofpoints 119909

119895(119895 = 4 5 6) would influence the positions of 119909

119894(119894 =

1 2 3) and hence change the distance between points 1199091and

1199093 That is to say the local shape (eg Δ119909

111990921199093) is closely

related to the global shape (eg Δ119909411990951199096) Unfortunately

the Hausdorff distance (A10) undoubtedly fails to reflect thisfact Therefore we need to find a new measure of describingthe analytic properties of fractal

A3 Definition of Fractal Measure Hausdorff measure (A8)does not reflect the self-similarity of fractal so we cannotestablish the calculus theory of fractal using the Hausdorffmeasure In general people often use the fractional calculusto approximately describe the analytic properties of fractal[8 12]

The fractional calculus is a theory of integrals and deriva-tives of any arbitrary real order For example the fractionalderivatives of order 119898 of the function 119910(119897) = 119888119897

119899 equal [8]

119889119898

119910 (119897)

119889119897119898= 119888

Γ (119899 + 1)

Γ (119899 minus 119898 + 1)119897119899minus119898

(A11)

where Γ(119909) denotes the Gamma function and 119898 is an arbi-trary real number

Now let us consider a 119898-dimensional volume

119909 (119897) = 120596 (119898) 119897119898

(A12)

where 120596(119898) is a constant which depends only on the dimen-sion 119898

Using formula (A11) the fractional derivatives of order119898

of (A12) equal

119889119898

119909 (119897)

119889119897119898= Γ (119898 + 1) 120596 (119898) (A13)

6 Journal of Applied Mathematics

x1

x2

x4

x3

x5

x6

Figure 2 Kochrsquos curve which is similar to the generation of Cantorternary set (see Figure 1) is defined by repeatedly adding themiddlethirds of line segments [11]

Equation (A13) can be written as

Δ119898

119909 (119897) = Γ (119898 + 1) 120596 (119898) (Δ119897)119898

+ 119900 [(Δ119897)119898

] (A14)

whereΔ119898

119909(119897) denotes the differences of order119898 and 119900[(Δ119897)119898

]

denotes the infinitesimal terms of higher order compared to(Δ119897)119898Equation (A14) implies that

Δ119898

119909 (119897) sim (Δ119897)119898

(A15)

and thereby

119873

sum

119894=1

Δ119898

119909 (119897119894) sim

119873

sum

119894=1

(Δ119897)119898

(A16)

Obviously (Δ119897)119898 is a 119898-dimensional Hausdorff measure

which can describe the length of a 119898-dimensional fractalcurve Consequently (A15) and (A16) together imply thatΔ119898

119909(119897) can be also thought of as a 119898-dimensional measureIn this case the order of differences Δ

119898

119909(119897) represents theHausdorff dimension119898 Because of this fact we next attemptto use the differences of order 119898 to define a new measure

Let us consider the left-shift operator with stepΔ119897 and theidentity operator as follows

119871Δ119897

119909 (119897) = 119909 (119897 minus Δ119897)

1198710119909 (119897) = 119909 (119897)

(A17)

Using the left-shift operator 119871Δ119897and the identity operator

1198710 we can define the difference operator of order 119898 in the

form

(1198710

minus 119871Δ119897

)119898

=

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119871119895Δ119897

(A18)

Using (A18) we define a new distance between points119909(119897) and 119909(119897 minus Δ119897) in the form

1003816100381610038161003816Δ119898 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119909 (119897 minus 119895Δ119897)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(A19)

We call (A19) the ldquononlocal distancerdquo which describesthe length of a 119898-dimensional fractal curve

When 119898 = 1 the nonlocal distance (A19) returns to theEuclidean distance that is

1003816100381610038161003816Δ119898=1 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816 = |119909 (119897) minus 119909 (119897 minus Δ119897)| (A20)

In general the nonlocal distance (A19) does not satisfythe general properties of distance (A4)-(A5) but reflects theconnection between local and global segments of fractal (Forexample the nonlocal distance between points 119909(119897) and 119909(119897 +

Δ119897) that is |Δ119898

[119909(119897) 119909(119897+Δ119897)]| needs to be defined using theright-shift operator119877

Δ119897which leads to119877

Δ119897119909(119897) = 119909(119897+Δ119897)The

corresponding difference operator reads (119877Δ119897

minus 1198770)119898 Then

the nonlocal distance would not satisfy condition (A4))To understand the latter we need to realize that the outputvalue of (A19) would depend on the values of all points119909(119897 minus 119895Δ119897) (119895 = 0 1 2 ) rather than only on points 119909(119897) and119909(119897 minus Δ119897)

For instance in Figure 2 the nonlocal distance |Δ119863

[1199095

1199093]| between points 119909

3and 119909

5would depend on the positions

of points 119909119894

(119894 = 1 2 3 4 5) rather than only on points 1199093

and 1199095 Therefore the nonlocal distance (A19) is indeed an

intrinsic way of describing fractal since it not only shows thedimension but also reflects the connection between local andglobal segments of fractal

Using the nonlocal distance (A19) we can propose adefinition for fractal measure

Let 119882 be a nonempty subset of 119899-dimensional Euclideanspace 119877

119899 We consider a countable set 119865119894 of subsets of

diameter at most 120576 that covers 119882 that is

119882 sub

infin

⋃

119894=1

119865119894 diam119863 (119865

119894) le 120576 forall119894 (A21)

where diam119863(119865119894) defined by using the nonlocal distance

(A19) denotes the diameter of 119865119894 that is

diam119863 (119865119894) = sup

1003816100381610038161003816Δ119863 [119909 119910]1003816100381610038161003816 119909 119910 isin 119865

119894 (A22)

Fractal Measure For a positive 119863 and each 120576 gt 0 we considercovers of 119882 by countable families 119865

119894 of (arbitrary) sets 119865

119894

with diameter less than 120576 and take the infimum of the sum ofdiam119863(119865

119894) Then we have

Π119863

120576(119882)

= inf

infin

sum

119894=1

diam119863 (119865119894) 119882 sub

infin

⋃

119894=1

119865119894 diam119863 (119865

119894) le 120576

(A23)

If the following limit exists

Π119863

(119882) = lim120576rarr0

Π119863

120576(119882) = finite (A24)

then the value Π119863

(119882) is called the 119863-dimensional fractalmeasure meanwhile the dimension of 119882 equals 119863

Journal of Applied Mathematics 7

A4 Definition of Fractal Derivative Obviously to describethe analytic properties of fractal we need the correspondingcalculus theory

Before proceeding to introduce the definition of fractalderivative let us consider a 120596-dimensional fractal curve120573120596(119897) (see Figure 3) which is determined by an independent

characteristic parameter 119897 (eg the fill parameter of Peanorsquoscurve) filling up a 120596-dimensional region Assume that thelength of the fractal curve is specified by a 120596-dimensionalvolume 119909(119897) then the (nonlocal) length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) in Figure 3 should be denoted by

Π119863

([119886 119887]) = limΔ119897rarr0

inf

119899

sum

119894=1

1003816100381610038161003816Δ120596 [119909 (119897119894minus1

) 119909 (119897119894)]

1003816100381610038161003816 (A25)

where we have used the fractal measure (A23)It is carefully noted that the length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) can not bemeasured by Euclidean scale

see Figure 3As such we can present a definition for fractal derivative

as followsFractal Derivative For any differentiable function 119910 = 119891(119909)if 119909 = 119909(119897) is not only a 120596-dimensional volume but alsodescribes the length of a 120596-dimensional fractal curve 120574

120596(119897)

then the fractal derivative of 119910 = 119891(119909) with respect to thefractal curve 120574

120596(119897) is defined as

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (A26)

where 119891(119897) = 119891[119909(119897)]Clearly if120596 = 1 then the formula (A26)will return to the

Newton-Leibniz derivative andmeanwhile 120574120596=1

(119897) is restoredto a 1-dimensional coordinate axis

In general the fractional derivative of order 120596 of anydifferentiable function 119891(119897) is defined in the form [13]

119889120596

119891 (119897)

119889119897120596

= limΔ119897rarr0

(

infin

sum

119895=0

120596 (120596 minus 1) sdot sdot sdot (120596 minus 119895 + 1) (minus1)119895

119895119891 (119897 minus 119895Δ119897)

times (Δ119897)minus120596

)

(A27)

Comparing (A19) and (A27) we have

119889120596

119891 (119897)

119889119897120596= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

(Δ119897)120596

(A28)

Using formula (A28) the formula (A26) can be rewrittenas

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)]sdot limΔ119897rarr0

1(Δ119897)120596

1(Δ119897)120596

=

limΔ119897rarr0

(Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)] (Δ119897)120596

)

limΔ119897rarr0

(Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (Δ119897)120596

)

=119889120596

119891 (119897) 119889119897120596

119889120596119909 (119897) 119889119897120596

(A29)

Formula (A29) indicates that we can compute the fractalderivative using the fractional derivative

B Computation Examples

In Appendix A we have noted that the fractal derivative canbe computed using the formula (A29) In this appendix wepresent two computing examples

Example 1 If 119909(119897) = 120596(119898)119897119898 describes the length of a 119898-

dimensional fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119897119863119898

119862

119897119863119898

119909=

119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909 (B1)

where 120596(119898) is a constant that depends only on 119898

Proof The fractional derivatives of order 119898 of the constant 119862

and the power function 119910(119897) = 119886119897119899 are respectively as follows

[8]

119889119898

119862

119889119897119898=

119862

Γ (1 minus 119898)119897minus119898

(B2)

119889119898

119910 (119897)

119889119897119898= 119886

Γ (119899 + 1)

Γ (119899 minus 119898 + 1)119897119899minus119898

(B3)

Using formulas (A29) (B2) and (B3) the fractal deriva-tive119897119863119898

119891(119909)119897119863119898

119909 can be computed as follows

119897119863119898

119862

119897119863119898

119909=

119889119898

119862119889119897119898

119889119898 [120596 (119898) 119897119898] 119889119897119898=

(119862Γ (1 minus 119898)) 119897minus119898

120596 (119898) Γ (1 + 119898)

=119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909

(B4)

The proof is complete

Example 2 If 119909 is the characteristic parameter of a 119898-dimensional fractal curve andmeanwhile it also describes thelength of this fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B5)

8 Journal of Applied Mathematics

Fractal curve

Nonlocal scale

Euclidean scale

120573120596(l1) 120573120596(l3)

120573120596(l2) 120573120596(l4)a = 120573120596(l0)120573120596(li) 120573120596(li+2)

)

120573120596(li+1)

120573120596(lnminus1)

b = 120573120596(ln

120573120596(l)

x(l0) = 0 x(l1) x(l2) x(l3) x(lnminus3) x(lnminus2) x(lnminus1) x(ln)

x(l)

l0 = 0 l1 l2 l3

l

lmlmminus3 lmminus2lmminus1

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middotmiddot middot middot

Figure 3 The fractal curve 120573120596(119897) consists of the union ⋃

119899

119894=1[120573120596(119897119894minus1

) 120573120596(119897119894)] where 119899 = 119899(119898) The dimension of fractal curve 120573

120596(119897) equals

120596 = lim119898rarrinfin

(ln 119899(119898) ln119898) Since 120573120596(119897) is a fractal curve the distance between points 120573

120596(1198970) and 120573

120596(119897119899) cannot be measured using the

Euclidean scale (ruler) 119897 otherwise we will have 119889[120573120596(1198970) 120573120596(119897119899)] = lim

119898rarrinfin119899(119898)((119897

119898minus 1198970)119898) = infin or =0 However the fractal curve 120573

120596(119897)

can be measured using the nonlocal scale (ruler) 119909(119897) for the way of measure see the formula (A25)

Proof Using formulas (A29) (B2) and (B3) we have

119909119863119898

119891 (119909)

119909119863119898

119909=

119889119898

119862119889119909119898

119889119898119909119889119909119898=

(119862Γ (1 minus 119898)) 119909minus119898

(Γ (2) Γ (2 minus 119898)) 1199091minus119898

= 119862Γ (2 minus 119898)

Γ (1 minus 119898)sdot

1

119909

(B6)

Considering the property of Gamma function

Γ (119909 + 1) = 119909Γ (119909) (B7)

we have

1 minus 119898 =Γ (2 minus 119898)

Γ (1 minus 119898) (B8)

Substituting (B8) into (B6) we arrive at

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B9)

The proof is complete

C Quantum Behavior and Nonlocal Distance

Now we investigate the connection between quantum behav-iour and nonlocal distance For simplicity we still con-sider the Cantor set in Figure 1 where the nonlocal dis-tance between points 119909

(119899)

2and 119909

(119899)

1is equal to

Π (119909(119899)

2 119909(119899)

1) =

10038161003816100381610038161003816Δ119898=119863

[119909(119899)

2 119909(119899)

1]10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2119899

sum

119895=1

119863 (119863 minus 1) sdot sdot sdot (119863 minus 119895 + 1) (minus1)119895

119895119909(119899)

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(C1)

Clearly the output value of nonlocal distance (C1)depends on the value of each element in the set 119909

(119899)

119894

where 119894 runs from 1 to 2119899 If 119899 rarr infin we will have

Π(119909(119899)

2 119909(119899)

1) rarr 0 however the number of elements in the

set 119909(119899)

119894 would tend to infinity too This means that if we

want to precisely measure the distance of smaller scale(eg lim

119899≫1Π(119909(119899)

2 119909(119899)

1)) correspondingly we will need to

collect the points 119909(119899)

119894 As a result if we want to precisely

measure the nonlocal distance between points lim119899rarrinfin

119909(119899)

2

and lim119899rarrinfin

119909(119899)

1 we will need to collect a set of infinite

points that is 119909(infin)119894

infin

119894=1 Unfortunately we must fail to arrive

Journal of Applied Mathematics 9

at this purpose on an actual measurement In other words wecannot precisely measure the distance of microscopic scalethis fact is consistent with the ldquoHeisenberg Uncertainty Prin-ciplerdquo (In fact the connection between quantum mechanicsand fractal has been noticed in some earlier papers [14ndash17])

On the other hand the nonlocal distance (C1) betweenpoints119909

(119899)

2and119909(119899)

1depends clearly on the position of119909(119899)

119894(119894 =

1 2 2119899

) for example any displacement of point 119909(119899)

2119899

would influence the output value of nonlocal distance (C1) Itis a clearly nonlocal correlation (correlations span arbitrarilydistances) and similar to quantum entanglement

The above two facts imply that the nonlocal descriptionmay be an intrinsic way of describing quantum behavior (Inreferences [18 19] we have shown that the local descriptionis not a way of completely describing physical reality)

Acknowledgment

This project is supported by the Scholarship Award for Excel-lent Doctoral Student Granted by the Ministry of Educationof China (2012) (Grant no 0903005109081-019)

References

[1] G rsquot Hooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquoNuclear Physics B vol 44 no 1 pp 189ndash2131972

[2] K Svozil ldquoQuantum field theory on fractal spacetime a newregularisation methodrdquo Journal of Physics A vol 20 no 12 pp3861ndash3875 1987

[3] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 15 pages 2009

[4] P Horava ldquoSpectral dimension of the universe in quantumgravity at a Lifshitz pointrdquo Physical Review Letters vol 102 no16 Article ID 161301 4 pages 2009

[5] G Calcagni ldquoQuantum field theory gravity and cosmology in afractal universerdquo Journal ofHigh Energy Physics vol 2010 article120 2010

[6] G Calcagni ldquoFractal universe and quantum gravityrdquo PhysicalReview Letters vol 104 no 25 Article ID 251301 2010

[7] D Benedetti ldquoFractal properties of quantum spacetimerdquo Phys-ical Review Letters vol 102 no 11 Article ID 111303 4 pages2009

[8] V E Tarasov Fractional Dynamics Applications of FractionalCalculus to Dynamics of Particles Field and Media HigherEducation Press Beijing China 2010

[9] T C Halsey M H Jensen L P Kadanoff I Procaccia andB I Shraiman ldquoFractal measures and their singularities thecharacterization of strange setsrdquo Physical Review A vol 33 no2 pp 1141ndash1151 1986

[10] H Sagan Space-Filling Curves Springer New York NY USA1994

[11] BMandelbrot ldquoHow long is the coast of Britain Statistical self-similarity and fractional dimensionrdquo Science vol 156 no 3775pp 636ndash638 1967

[12] M Giona ldquoFractal calculus on [0 1]rdquo Chaos Solitons andFractals vol 5 no 6 pp 987ndash1000 1995

[13] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[14] M S El Naschie ldquoQuantum mechanics and the possibility of aCantorian space-timerdquo Chaos Solitons and Fractals vol 1 no5 pp 485ndash487 1991

[15] M S El Naschie ldquoA note on Heisenbergrsquos uncertainty principleand Cantorian space-timerdquo Chaos Solitons and Fractals vol 2no 4 pp 437ndash439 1992

[16] M S El Naschie ldquoOn certain infinite-dimensional Cantor setsand the Schrodinger waverdquo Chaos Solitons and Fractals vol 3no 1 pp 89ndash98 1993

[17] L Marek-Crnjac ldquoA short history of fractal-Cantorian space-timerdquo Chaos Solitons and Fractals vol 41 no 5 pp 2697ndash27052009

[18] Y Tao ldquoNecessity of integral formalismrdquo Communications inTheoretical Physics vol 56 no 4 pp 648ndash654 2011

[19] Y Tao ldquoSufficient condition for validity of quantum adiabatictheoremrdquo Communications in Theoretical Physics vol 57 no 3pp 343ndash347 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

It is carefully noted that every 119909(119897minus119895Δ119897) represents a pointon a 119898-dimensional fractal curve where 119895 = 0 1 2

Clearly according to (8) the distance between points 119909(119897)

and 119909(119897minusΔ119897) would depend on all the points 119909(119897minus119895Δ119897) where119895 = 0 1 2

Moreover itrsquos easy to check that |Δ119898=1

[119909(119897) 119909(119897 minus Δ119897)]| =

|119909(119897) minus 119909(119897 minus Δ119897)| This means that the Euclidean distance is aspecial case of nonlocal distance whenever the dimension ofthe fractal curve 119898 equals 1

If we use the nonlocal distance (8) tomeasure the distancebetween points 119909

(3)

2and 119909

(3)

1(see Figure 1) then we will

surprisingly find that the nonlocal distance

10038161003816100381610038161003816Δ119898=119863

[119909(3)

2 119909(3)

1]10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

8

sum

119895=1

119863 (119863 minus 1) sdot sdot sdot (119863 minus 119895 + 1) (minus1)119895

119895119909(3)

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(9)

which remarkably differs from the Hausdorff distance (5)would depend on the values of points 119909

(3)

119894(119894 = 3 4 8)

This means that any displacement of point 119909(3)

119894(119894 = 3 4

8) would change the nonlocal distance between points119909(3)

2and 119909

(3)

1 Consequently nonlocal distance (8) is indeed

an intrinsic way of describing self-similar fractal since itnot only determines the dimension of a fractal curve (egCantor ternary set) but also reflects the correlation betweenits parts Interestingly the nonlocal distance seems to havesome connection with quantum behavior for details seeAppendix C

Using the nonlocal distance we have given a definition forfractal measure in Appendix A (see (A23))

To study the analytic properties of a fractal curve wedefine the fractal derivative (see (A26) in Appendix A) in theform

119897119863120596119891 (119909)

119897119863120596119909

= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (10)

where 119891(119897) = 119891[119909(119897)] is a differentiable function with respectto coordinate 119897 119897 is a parameter (eg the single parameter ofPeanorsquos curve [10] for details see Appendix A) which com-pletely determines the generation of a 120596-dimensional fractalcurve and 119909(119897) denotes the length of the correspondingfractal curve (We introduce a simple way of understandingthe fractal derivative (10) For the case of the Newton-Leibnizderivative of 119910 = 119891(119909) 119909 is a 1-dimensional coordinate axisand hence can bemeasured by a Euclidean scale (ruler)Thusthe differential element of 119909 is a 1-dimensional Euclideanlength 119889119909 which gives rise to the Newton-Leibniz derivative119889119891(119909)119889119909 Nevertheless if 119909 is a 120596-dimensional fractal curvethen it can not be measured by the Euclidean scale (ruler)In this case the differential element of 119909 should be a 120596-dimensional volume

119897119863120596119909 which gives rise to the fractal

derivative119897119863120596119891(119909)

119897119863120596119909 For details see Appendix A and

Figure 3)In particular the fractal derivative (10) will return to the

well-known Newton-Leibniz derivative whenever 120596 = 1

Using the formula of fractional derivative [8] the frac-tal derivative can be rewritten as (see (A27)ndash(A29) inAppendix A)

119897119863120596119891 (119909)

119897119863120596119909

=119889120596

119891 (119897) 119889119897120596

119889120596119909 (119897) 119889119897120596 (11)

By formula (11) we can easily compute the fractal deriva-tive of any differentiable function using the fractional deriva-tive for concrete examples see Appendix B

4 Fractal Integral

In Section 3 we have proposed a definition for fractal deriva-tive Correspondingly we can now present a convenientdefinition for fractal integral as follow

Definition 1 If119897119863119898

119891(119909)119897119863119898

119909 = 119892(119909) then the fractalintegral of 119892(119909) on a 119898-dimensional fractal curve 120573

119898(119897) is

defined in the form

int119897119863119898

119891 (119909) = int120573119898(119897)

119892 (119909)119897119863119898

119909

= int119882

119892 (119909)119889119898

119909

119889119897119898(119889119897)119898

(12)

where119882denotes the definitional domain of the characteristicparameter 119897 also the parameter 119897 completely determines thegeneration of the 119898-dimensional fractal curve 120573

119898(119897)

Using such a definition of fractal integral we can prove thefollowing proposition

Proposition 2 If 119909(119903) = 120596(119898)119903119898 not only describes the

length of a 119898-dimensional fractal curve 120573119898

(119903) but also denotesthe volume of a 119898-dimensional sphere (for example the 119898-dimensional fractal curve 120573

119898(119903) fills up the entire sphere Ω) Ω

and if119891(119909) = 119891(119903) is a spherically symmetric function then thefractal integration of 119891(119909) on the 119898-dimensional fractal curve120573119898

(119903) equals

int120573119898(119903)

119891 (119909)119903119863119898

119909 =21205871198982

Γ (1198982)int

119877

0

119891 (119903) 119903119898minus1

119889119903 (13)

where 120596(119898) = 1205871198982

Γ((1198982) + 1)

Proof The Riemann-Liouville fractional integrals of order 119898

of 119891(119897) are defined in the form [8]

int 119891 (119897) (119889119897)119898

=1

Γ (119898)int

119877

119910

(119897 minus 119910)119898minus1

119891 (119897) 119889119897 (14)

Using (A29) and (B3) we have

119903119863119898

119909 =119889119898

119909

119889119903119898(119889119903)119898

= 120596 (119898) Γ (1 + 119898) (119889119903)119898

(15)

Using (12) and (15) we arrive at

int120573119898(119903)

119891 (119909)119903119863119898

119909 = intΩ

119891 (119903) 120596 (119898) Γ (1 + 119898) (119889119903)119898

(16)

4 Journal of Applied Mathematics

Inserting (14) into (16) leads to

int120573119898(119903)

119891 (119909)119903119863119898

119909 = 120596 (119898)Γ (1 + 119898)

Γ (119898)

times int

119877

119910

(119903 minus 119910)119898minus1

119891 (119903) 119889119903

(17)

Using 120596(119898) = 1205871198982

Γ((1198982) + 1) and the formula (B7)(17) can be rewritten as

int120573119898(119903)

119891 (119909)119903119863119898

119909 =1205871198982

119898

Γ ((1198982) + 1)int

119877

119910

(119903 minus 119910)119898minus1

119891 (119903) 119889119903

(18)

For 119910 = 0 (18) yields

int120573119898(119903)

119891 (119909)119903119863119898

119909 =1205871198982

119898

Γ ((1198982) + 1)int

119877

0

119903119898minus1

119891 (119903) 119889119903 (19)

Using the formula (B7) we have119898

2Γ (

119898

2) = Γ (

119898

2+ 1) (20)

Substituting (20) into (19) leads to

int120573119898(119903)

119891 (119909)119903119863119898

119909 =21205871198982

Γ (1198982)int

119877

0

119891 (119903) 119903119898minus1

119889119903 (21)

The proof is complete

Clearly for 119891(119909) = 1 formula (13) gives the volume of a119898-dimensional sphere 120587

1198982

Γ((1198982) + 1)119877119898

Before proceeding to arrive at the main result of thispaper let us consider three measurable sets 119882

119894with the

dimension 119898119894 where 119894 = 1 2 3 According to Fubinirsquos the-

orem the Cartesian product of the sets 119882119894can produce a set

119882 with the dimension 119898 = 1198981

+ 1198982

+ 1198983 where 119882 =

1198821

otimes 1198822

otimes 1198823 The integration over a function 119891(119903

1 1199032 1199033)

on 119882 can be written in the form

int119882

119891 (1199031 1199032 1199033) 119889120583 (119882)

= int1198821

int1198822

int1198823

119891 (1199031 1199032 1199033) 119889120583 (119882

1) sdot 119889120583 (119882

2) sdot 119889120583 (119882

3)

(22)

Then we have the following lemma

Lemma 3 If 119889120583(119882119894) is denoted by

119889120583 (119882119894) =

21205871198981198942

Γ (1198981198942)

119903119898119894minus1

119894119889119903119894

(119894 = 1 2 3) (23)

and if 119891(1199031 1199032 1199033) is specified by a spherically symmetric

function 119891(119903) with 1199032

= 1199032

1+ 1199032

2+ 1199032

3 then we have

int1198821

int1198822

int1198823

119891 (1199031 1199032 1199033) 119889120583 (119882

1) sdot 119889120583 (119882

2) sdot 119889120583 (119882

3)

=21205871198982

Γ (1198982)int

119877

0

119891 (119903) 119903119898minus1

119889119903

(24)

The proof of Lemma 3 can be found in page 28 in theliterature [8]

With the preparations above we can introduce the mainresult of this paper as follow

Proposition 4 If 119909119894(119903119894) = 120596(119863

119894)119903119863119894

119894(119894 = 1 2 3) not only

describes the length of a 119863119894-dimensional fractal curve 120573

119863119894

(119903119894)

but also denotes the volume of a 119863119894-dimensional sphere Ω

119894and

if 119891(119903) = 119891(1199031 1199032 1199033) is a spherically symmetric function then

the fractal integration of 119891(119903) on the fractal graph 1205731198631

(1199031) otimes

1205731198632

(1199032) otimes 1205731198633

(1199033) equals

int1205731198631(1199031)

int1205731198632(1199032)

int1205731198633(1199033)

119891 (1199031 1199032 1199033)

3

prod

119894=1

119903119894119863119863119894

119909119894

=21205871198632

Γ (1198632)int

119877

0

119891 (119903) 119903119863minus1

119889119903

(25)

where 120596(119863119894) = 120587

1198631198942

Γ((1198631198942) + 1) 119903

2

= 1199032

1+ 1199032

2+ 1199032

3 and

119863 = 1198631

+ 1198632

+ 1198633

Proof Using (13) and (24) we easily arrive at (25)The proof is complete

Proposition 4 shows that we have arrived at the con-jectural integral formula (3) through the fractal measure(A23) In particular if 119877 rarr infin then the integral for-mula (25) will exhibit linearity translational invariance andscaling property These properties are natural and necessaryin applying dimensional regularization to quantum fieldtheory [2] It is worth mentioning that Proposition 4 canbe generalized so that 119894 might take any positive integer forexample 119894 = 1 2 infin If 119894 = 1 2 3 4 then 120573

119863119894

(119903119894) may

denote coordinate axiswith the dimension119863119894and so120573

1198631

(1199031)otimes

1205731198632

(1199032) otimes 1205731198633

(1199033) otimes 1205731198634

(1199034) may denote spacetime

5 Conclusion

Hausdorff measure is not an ideal mathematical frameworkfor describing fractal since it fails to describe the nonlocalproperty of fractal (eg self-similarity) However the fractalmeasure constructed by this paper not only shows the dimen-sion of a fractal but also describes its analytic properties (egnonlocal property) Not only so using this fractal measurewe can derive Svozilrsquos conjectural integral formula (3) whichis the starting point of quantum field theory on fractalspacetimeTherefore our fractal measure may be regarded asa possible mathematical basis of establishing quantum fieldtheory on fractal spacetime

Appendices

A Mathematical Preparations

In Euclidean geometry the dimension of a geometric graphis determined by the number of independent variables (iethe number of degrees of freedom) For example everypoint on a plane can be represented by 2-tuples real number(1199091 1199092) then the dimension of the plane is denoted by

Journal of Applied Mathematics 5

2 Nevertheless the existence of Peanorsquos curve powerfullyrefutes this viewpoint Peanorsquos curve which is determined byan independent characteristic parameter (ie fill parameter)would fill up the entire plane [10]Therefore mathematicianshave to reconsider the definition of dimension The mostfamous one of all definitions of dimension is the Hausdorffdimension which is defined through the Hausdorff measure[8]

A1 Hausdorff Measure and Hausdorff Dimension In orderto bring the definition of Hausdorff dimension we firstlyintroduce the Hausdorff measure [8]

Let 119882 be a nonempty subset of 119899-dimensional Euclideanspace 119877

119899 the diameter of 119882 is defined as

diam (119882) = sup 119889 (119909 119910) 119909 119910 isin 119882 (A1)

where 119889(119909 119910) which is the distance between points 119909 and 119910is a real-valued function on 119882 otimes 119882 such that the followingfour conditions are satisfied

119889 (119909 119910) ge 0 forall119909 119910 isin 119882 (A2)

119889 (119909 119910) = 0 iff 119909 = 119910 (A3)

119889 (119909 119910) = 119889 (119910 119909) forall119909 119910 isin 119882 (A4)

119889 (119909 119911) le 119889 (119909 119910) + 119889 (119910 119911) forall119909 119910 119911 isin 119882 (A5)

For example the distance of 119899-dimensional Euclideanspace 119877

119899 can be defined as

119889119864

(119909 119910) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 = (

119899

sum

119894=1

1003816100381610038161003816119909119894 minus 119910119894

1003816100381610038161003816

2

)

12

(A6)

Then it is easy to check that (A6) satisfies conditions(A2)ndash(A5)

Now let us consider a countable set 119864119894 of subsets of

diameter at most 120576 that covers 119882 that is

119882 sub

infin

⋃

119894=1

119864119894 diam (119864

119894) le 120576 forall119894 (A7)

For a positive 119863 and each 120576 gt 0 we consider covers of 119882

by countable families 119864119894 of (arbitrary) sets 119864

119894with diameter

less than 120576 and take the infimum of the sum of [diam(119864119894)]119863

Then we have

119867119863

120576(119882)

= inf

infin

sum

119894=1

[diam (119864119894)]119863

119882 sub

infin

⋃

119894=1

119864119894 diam (119864

119894) le 120576

(A8)

If the following limit exists

119867119863

(119882) = lim120576rarr0

119867119863

120576(119882) = finite (A9)

then the value119867119863

(119882) is called the119863-dimensional Hausdorffmeasure

A2 Shortcoming of Hausdorff Measure In general 119863 maybe a fraction In 1967 Mandelbrot realized that [11] the lengthof coastline can be measured using Hausdorff measure (A8)rather than Euclideanmeasure (A6) and then the dimensionof coastline is a fraction Mandelbrot called such geometricgraphs the ldquofractalrdquo

The fractal is self-similar between its local and globalshapes Unfortunately Hausdorff measure can determine thedimension of fractal but not reflect the connection (eg self-similarity) among the parts of the corresponding fractal Tosee this we consider Kochrsquos curve in Figure 2

Clearly the congruent triangle Δ119909111990921199093is similar to

Δ119909411990951199096 If we use the Hausdorff measure (A8) to measure

the local distance of Kochrsquos curve (eg the distance betweenpoints 119909

1and 119909

3) then we have

119867119863

(1199091 1199093) =

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816

119863

+10038161003816100381610038161199092 minus 119909

3

1003816100381610038161003816

119863

(A10)

where 119863 is the dimension of theKoch curveEquation (A10) shows that the Hausdorff distance

between points 1199091and 119909

3depends only on the positions of

points 119909119894

(119894 = 1 2 3) and is thereby independent of thepositions of points 119909

119895(119895 = 4 5 6) Nevertheless because

of the self-similarity of Kochrsquos curve any displacements ofpoints 119909

119895(119895 = 4 5 6) would influence the positions of 119909

119894(119894 =

1 2 3) and hence change the distance between points 1199091and

1199093 That is to say the local shape (eg Δ119909

111990921199093) is closely

related to the global shape (eg Δ119909411990951199096) Unfortunately

the Hausdorff distance (A10) undoubtedly fails to reflect thisfact Therefore we need to find a new measure of describingthe analytic properties of fractal

A3 Definition of Fractal Measure Hausdorff measure (A8)does not reflect the self-similarity of fractal so we cannotestablish the calculus theory of fractal using the Hausdorffmeasure In general people often use the fractional calculusto approximately describe the analytic properties of fractal[8 12]

The fractional calculus is a theory of integrals and deriva-tives of any arbitrary real order For example the fractionalderivatives of order 119898 of the function 119910(119897) = 119888119897

119899 equal [8]

119889119898

119910 (119897)

119889119897119898= 119888

Γ (119899 + 1)

Γ (119899 minus 119898 + 1)119897119899minus119898

(A11)

where Γ(119909) denotes the Gamma function and 119898 is an arbi-trary real number

Now let us consider a 119898-dimensional volume

119909 (119897) = 120596 (119898) 119897119898

(A12)

where 120596(119898) is a constant which depends only on the dimen-sion 119898

Using formula (A11) the fractional derivatives of order119898

of (A12) equal

119889119898

119909 (119897)

119889119897119898= Γ (119898 + 1) 120596 (119898) (A13)

6 Journal of Applied Mathematics

x1

x2

x4

x3

x5

x6

Figure 2 Kochrsquos curve which is similar to the generation of Cantorternary set (see Figure 1) is defined by repeatedly adding themiddlethirds of line segments [11]

Equation (A13) can be written as

Δ119898

119909 (119897) = Γ (119898 + 1) 120596 (119898) (Δ119897)119898

+ 119900 [(Δ119897)119898

] (A14)

whereΔ119898

119909(119897) denotes the differences of order119898 and 119900[(Δ119897)119898

]

denotes the infinitesimal terms of higher order compared to(Δ119897)119898Equation (A14) implies that

Δ119898

119909 (119897) sim (Δ119897)119898

(A15)

and thereby

119873

sum

119894=1

Δ119898

119909 (119897119894) sim

119873

sum

119894=1

(Δ119897)119898

(A16)

Obviously (Δ119897)119898 is a 119898-dimensional Hausdorff measure

which can describe the length of a 119898-dimensional fractalcurve Consequently (A15) and (A16) together imply thatΔ119898

119909(119897) can be also thought of as a 119898-dimensional measureIn this case the order of differences Δ

119898

119909(119897) represents theHausdorff dimension119898 Because of this fact we next attemptto use the differences of order 119898 to define a new measure

Let us consider the left-shift operator with stepΔ119897 and theidentity operator as follows

119871Δ119897

119909 (119897) = 119909 (119897 minus Δ119897)

1198710119909 (119897) = 119909 (119897)

(A17)

Using the left-shift operator 119871Δ119897and the identity operator

1198710 we can define the difference operator of order 119898 in the

form

(1198710

minus 119871Δ119897

)119898

=

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119871119895Δ119897

(A18)

Using (A18) we define a new distance between points119909(119897) and 119909(119897 minus Δ119897) in the form

1003816100381610038161003816Δ119898 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119909 (119897 minus 119895Δ119897)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(A19)

We call (A19) the ldquononlocal distancerdquo which describesthe length of a 119898-dimensional fractal curve

When 119898 = 1 the nonlocal distance (A19) returns to theEuclidean distance that is

1003816100381610038161003816Δ119898=1 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816 = |119909 (119897) minus 119909 (119897 minus Δ119897)| (A20)

In general the nonlocal distance (A19) does not satisfythe general properties of distance (A4)-(A5) but reflects theconnection between local and global segments of fractal (Forexample the nonlocal distance between points 119909(119897) and 119909(119897 +

Δ119897) that is |Δ119898

[119909(119897) 119909(119897+Δ119897)]| needs to be defined using theright-shift operator119877

Δ119897which leads to119877

Δ119897119909(119897) = 119909(119897+Δ119897)The

corresponding difference operator reads (119877Δ119897

minus 1198770)119898 Then

the nonlocal distance would not satisfy condition (A4))To understand the latter we need to realize that the outputvalue of (A19) would depend on the values of all points119909(119897 minus 119895Δ119897) (119895 = 0 1 2 ) rather than only on points 119909(119897) and119909(119897 minus Δ119897)

For instance in Figure 2 the nonlocal distance |Δ119863

[1199095

1199093]| between points 119909

3and 119909

5would depend on the positions

of points 119909119894

(119894 = 1 2 3 4 5) rather than only on points 1199093

and 1199095 Therefore the nonlocal distance (A19) is indeed an

intrinsic way of describing fractal since it not only shows thedimension but also reflects the connection between local andglobal segments of fractal

Using the nonlocal distance (A19) we can propose adefinition for fractal measure

Let 119882 be a nonempty subset of 119899-dimensional Euclideanspace 119877

119899 We consider a countable set 119865119894 of subsets of

diameter at most 120576 that covers 119882 that is

119882 sub

infin

⋃

119894=1

119865119894 diam119863 (119865

119894) le 120576 forall119894 (A21)

where diam119863(119865119894) defined by using the nonlocal distance

(A19) denotes the diameter of 119865119894 that is

diam119863 (119865119894) = sup

1003816100381610038161003816Δ119863 [119909 119910]1003816100381610038161003816 119909 119910 isin 119865

119894 (A22)

Fractal Measure For a positive 119863 and each 120576 gt 0 we considercovers of 119882 by countable families 119865

119894 of (arbitrary) sets 119865

119894

with diameter less than 120576 and take the infimum of the sum ofdiam119863(119865

119894) Then we have

Π119863

120576(119882)

= inf

infin

sum

119894=1

diam119863 (119865119894) 119882 sub

infin

⋃

119894=1

119865119894 diam119863 (119865

119894) le 120576

(A23)

If the following limit exists

Π119863

(119882) = lim120576rarr0

Π119863

120576(119882) = finite (A24)

then the value Π119863

(119882) is called the 119863-dimensional fractalmeasure meanwhile the dimension of 119882 equals 119863

Journal of Applied Mathematics 7

A4 Definition of Fractal Derivative Obviously to describethe analytic properties of fractal we need the correspondingcalculus theory

Before proceeding to introduce the definition of fractalderivative let us consider a 120596-dimensional fractal curve120573120596(119897) (see Figure 3) which is determined by an independent

characteristic parameter 119897 (eg the fill parameter of Peanorsquoscurve) filling up a 120596-dimensional region Assume that thelength of the fractal curve is specified by a 120596-dimensionalvolume 119909(119897) then the (nonlocal) length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) in Figure 3 should be denoted by

Π119863

([119886 119887]) = limΔ119897rarr0

inf

119899

sum

119894=1

1003816100381610038161003816Δ120596 [119909 (119897119894minus1

) 119909 (119897119894)]

1003816100381610038161003816 (A25)

where we have used the fractal measure (A23)It is carefully noted that the length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) can not bemeasured by Euclidean scale

see Figure 3As such we can present a definition for fractal derivative

as followsFractal Derivative For any differentiable function 119910 = 119891(119909)if 119909 = 119909(119897) is not only a 120596-dimensional volume but alsodescribes the length of a 120596-dimensional fractal curve 120574

120596(119897)

then the fractal derivative of 119910 = 119891(119909) with respect to thefractal curve 120574

120596(119897) is defined as

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (A26)

where 119891(119897) = 119891[119909(119897)]Clearly if120596 = 1 then the formula (A26)will return to the

Newton-Leibniz derivative andmeanwhile 120574120596=1

(119897) is restoredto a 1-dimensional coordinate axis

In general the fractional derivative of order 120596 of anydifferentiable function 119891(119897) is defined in the form [13]

119889120596

119891 (119897)

119889119897120596

= limΔ119897rarr0

(

infin

sum

119895=0

120596 (120596 minus 1) sdot sdot sdot (120596 minus 119895 + 1) (minus1)119895

119895119891 (119897 minus 119895Δ119897)

times (Δ119897)minus120596

)

(A27)

Comparing (A19) and (A27) we have

119889120596

119891 (119897)

119889119897120596= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

(Δ119897)120596

(A28)

Using formula (A28) the formula (A26) can be rewrittenas

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)]sdot limΔ119897rarr0

1(Δ119897)120596

1(Δ119897)120596

=

limΔ119897rarr0

(Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)] (Δ119897)120596

)

limΔ119897rarr0

(Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (Δ119897)120596

)

=119889120596

119891 (119897) 119889119897120596

119889120596119909 (119897) 119889119897120596

(A29)

Formula (A29) indicates that we can compute the fractalderivative using the fractional derivative

B Computation Examples

In Appendix A we have noted that the fractal derivative canbe computed using the formula (A29) In this appendix wepresent two computing examples

Example 1 If 119909(119897) = 120596(119898)119897119898 describes the length of a 119898-

dimensional fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119897119863119898

119862

119897119863119898

119909=

119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909 (B1)

where 120596(119898) is a constant that depends only on 119898

Proof The fractional derivatives of order 119898 of the constant 119862

and the power function 119910(119897) = 119886119897119899 are respectively as follows

[8]

119889119898

119862

119889119897119898=

119862

Γ (1 minus 119898)119897minus119898

(B2)

119889119898

119910 (119897)

119889119897119898= 119886

Γ (119899 + 1)

Γ (119899 minus 119898 + 1)119897119899minus119898

(B3)

Using formulas (A29) (B2) and (B3) the fractal deriva-tive119897119863119898

119891(119909)119897119863119898

119909 can be computed as follows

119897119863119898

119862

119897119863119898

119909=

119889119898

119862119889119897119898

119889119898 [120596 (119898) 119897119898] 119889119897119898=

(119862Γ (1 minus 119898)) 119897minus119898

120596 (119898) Γ (1 + 119898)

=119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909

(B4)

The proof is complete

Example 2 If 119909 is the characteristic parameter of a 119898-dimensional fractal curve andmeanwhile it also describes thelength of this fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B5)

8 Journal of Applied Mathematics

Fractal curve

Nonlocal scale

Euclidean scale

120573120596(l1) 120573120596(l3)

120573120596(l2) 120573120596(l4)a = 120573120596(l0)120573120596(li) 120573120596(li+2)

)

120573120596(li+1)

120573120596(lnminus1)

b = 120573120596(ln

120573120596(l)

x(l0) = 0 x(l1) x(l2) x(l3) x(lnminus3) x(lnminus2) x(lnminus1) x(ln)

x(l)

l0 = 0 l1 l2 l3

l

lmlmminus3 lmminus2lmminus1

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middotmiddot middot middot

Figure 3 The fractal curve 120573120596(119897) consists of the union ⋃

119899

119894=1[120573120596(119897119894minus1

) 120573120596(119897119894)] where 119899 = 119899(119898) The dimension of fractal curve 120573

120596(119897) equals

120596 = lim119898rarrinfin

(ln 119899(119898) ln119898) Since 120573120596(119897) is a fractal curve the distance between points 120573

120596(1198970) and 120573

120596(119897119899) cannot be measured using the

Euclidean scale (ruler) 119897 otherwise we will have 119889[120573120596(1198970) 120573120596(119897119899)] = lim

119898rarrinfin119899(119898)((119897

119898minus 1198970)119898) = infin or =0 However the fractal curve 120573

120596(119897)

can be measured using the nonlocal scale (ruler) 119909(119897) for the way of measure see the formula (A25)

Proof Using formulas (A29) (B2) and (B3) we have

119909119863119898

119891 (119909)

119909119863119898

119909=

119889119898

119862119889119909119898

119889119898119909119889119909119898=

(119862Γ (1 minus 119898)) 119909minus119898

(Γ (2) Γ (2 minus 119898)) 1199091minus119898

= 119862Γ (2 minus 119898)

Γ (1 minus 119898)sdot

1

119909

(B6)

Considering the property of Gamma function

Γ (119909 + 1) = 119909Γ (119909) (B7)

we have

1 minus 119898 =Γ (2 minus 119898)

Γ (1 minus 119898) (B8)

Substituting (B8) into (B6) we arrive at

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B9)

The proof is complete

C Quantum Behavior and Nonlocal Distance

Now we investigate the connection between quantum behav-iour and nonlocal distance For simplicity we still con-sider the Cantor set in Figure 1 where the nonlocal dis-tance between points 119909

(119899)

2and 119909

(119899)

1is equal to

Π (119909(119899)

2 119909(119899)

1) =

10038161003816100381610038161003816Δ119898=119863

[119909(119899)

2 119909(119899)

1]10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2119899

sum

119895=1

119863 (119863 minus 1) sdot sdot sdot (119863 minus 119895 + 1) (minus1)119895

119895119909(119899)

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(C1)

Clearly the output value of nonlocal distance (C1)depends on the value of each element in the set 119909

(119899)

119894

where 119894 runs from 1 to 2119899 If 119899 rarr infin we will have

Π(119909(119899)

2 119909(119899)

1) rarr 0 however the number of elements in the

set 119909(119899)

119894 would tend to infinity too This means that if we

want to precisely measure the distance of smaller scale(eg lim

119899≫1Π(119909(119899)

2 119909(119899)

1)) correspondingly we will need to

collect the points 119909(119899)

119894 As a result if we want to precisely

measure the nonlocal distance between points lim119899rarrinfin

119909(119899)

2

and lim119899rarrinfin

119909(119899)

1 we will need to collect a set of infinite

points that is 119909(infin)119894

infin

119894=1 Unfortunately we must fail to arrive

Journal of Applied Mathematics 9

at this purpose on an actual measurement In other words wecannot precisely measure the distance of microscopic scalethis fact is consistent with the ldquoHeisenberg Uncertainty Prin-ciplerdquo (In fact the connection between quantum mechanicsand fractal has been noticed in some earlier papers [14ndash17])

On the other hand the nonlocal distance (C1) betweenpoints119909

(119899)

2and119909(119899)

1depends clearly on the position of119909(119899)

119894(119894 =

1 2 2119899

) for example any displacement of point 119909(119899)

2119899

would influence the output value of nonlocal distance (C1) Itis a clearly nonlocal correlation (correlations span arbitrarilydistances) and similar to quantum entanglement

The above two facts imply that the nonlocal descriptionmay be an intrinsic way of describing quantum behavior (Inreferences [18 19] we have shown that the local descriptionis not a way of completely describing physical reality)

Acknowledgment

This project is supported by the Scholarship Award for Excel-lent Doctoral Student Granted by the Ministry of Educationof China (2012) (Grant no 0903005109081-019)

References

[1] G rsquot Hooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquoNuclear Physics B vol 44 no 1 pp 189ndash2131972

[2] K Svozil ldquoQuantum field theory on fractal spacetime a newregularisation methodrdquo Journal of Physics A vol 20 no 12 pp3861ndash3875 1987

[3] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 15 pages 2009

[4] P Horava ldquoSpectral dimension of the universe in quantumgravity at a Lifshitz pointrdquo Physical Review Letters vol 102 no16 Article ID 161301 4 pages 2009

[5] G Calcagni ldquoQuantum field theory gravity and cosmology in afractal universerdquo Journal ofHigh Energy Physics vol 2010 article120 2010

[6] G Calcagni ldquoFractal universe and quantum gravityrdquo PhysicalReview Letters vol 104 no 25 Article ID 251301 2010

[7] D Benedetti ldquoFractal properties of quantum spacetimerdquo Phys-ical Review Letters vol 102 no 11 Article ID 111303 4 pages2009

[8] V E Tarasov Fractional Dynamics Applications of FractionalCalculus to Dynamics of Particles Field and Media HigherEducation Press Beijing China 2010

[9] T C Halsey M H Jensen L P Kadanoff I Procaccia andB I Shraiman ldquoFractal measures and their singularities thecharacterization of strange setsrdquo Physical Review A vol 33 no2 pp 1141ndash1151 1986

[10] H Sagan Space-Filling Curves Springer New York NY USA1994

[11] BMandelbrot ldquoHow long is the coast of Britain Statistical self-similarity and fractional dimensionrdquo Science vol 156 no 3775pp 636ndash638 1967

[12] M Giona ldquoFractal calculus on [0 1]rdquo Chaos Solitons andFractals vol 5 no 6 pp 987ndash1000 1995

[13] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[14] M S El Naschie ldquoQuantum mechanics and the possibility of aCantorian space-timerdquo Chaos Solitons and Fractals vol 1 no5 pp 485ndash487 1991

[15] M S El Naschie ldquoA note on Heisenbergrsquos uncertainty principleand Cantorian space-timerdquo Chaos Solitons and Fractals vol 2no 4 pp 437ndash439 1992

[16] M S El Naschie ldquoOn certain infinite-dimensional Cantor setsand the Schrodinger waverdquo Chaos Solitons and Fractals vol 3no 1 pp 89ndash98 1993

[17] L Marek-Crnjac ldquoA short history of fractal-Cantorian space-timerdquo Chaos Solitons and Fractals vol 41 no 5 pp 2697ndash27052009

[18] Y Tao ldquoNecessity of integral formalismrdquo Communications inTheoretical Physics vol 56 no 4 pp 648ndash654 2011

[19] Y Tao ldquoSufficient condition for validity of quantum adiabatictheoremrdquo Communications in Theoretical Physics vol 57 no 3pp 343ndash347 2012

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

Inserting (14) into (16) leads to

int120573119898(119903)

119891 (119909)119903119863119898

119909 = 120596 (119898)Γ (1 + 119898)

Γ (119898)

times int

119877

119910

(119903 minus 119910)119898minus1

119891 (119903) 119889119903

(17)

Using 120596(119898) = 1205871198982

Γ((1198982) + 1) and the formula (B7)(17) can be rewritten as

int120573119898(119903)

119891 (119909)119903119863119898

119909 =1205871198982

119898

Γ ((1198982) + 1)int

119877

119910

(119903 minus 119910)119898minus1

119891 (119903) 119889119903

(18)

For 119910 = 0 (18) yields

int120573119898(119903)

119891 (119909)119903119863119898

119909 =1205871198982

119898

Γ ((1198982) + 1)int

119877

0

119903119898minus1

119891 (119903) 119889119903 (19)

Using the formula (B7) we have119898

2Γ (

119898

2) = Γ (

119898

2+ 1) (20)

Substituting (20) into (19) leads to

int120573119898(119903)

119891 (119909)119903119863119898

119909 =21205871198982

Γ (1198982)int

119877

0

119891 (119903) 119903119898minus1

119889119903 (21)

The proof is complete

Clearly for 119891(119909) = 1 formula (13) gives the volume of a119898-dimensional sphere 120587

1198982

Γ((1198982) + 1)119877119898

Before proceeding to arrive at the main result of thispaper let us consider three measurable sets 119882

119894with the

dimension 119898119894 where 119894 = 1 2 3 According to Fubinirsquos the-

orem the Cartesian product of the sets 119882119894can produce a set

119882 with the dimension 119898 = 1198981

+ 1198982

+ 1198983 where 119882 =

1198821

otimes 1198822

otimes 1198823 The integration over a function 119891(119903

1 1199032 1199033)

on 119882 can be written in the form

int119882

119891 (1199031 1199032 1199033) 119889120583 (119882)

= int1198821

int1198822

int1198823

119891 (1199031 1199032 1199033) 119889120583 (119882

1) sdot 119889120583 (119882

2) sdot 119889120583 (119882

3)

(22)

Then we have the following lemma

Lemma 3 If 119889120583(119882119894) is denoted by

119889120583 (119882119894) =

21205871198981198942

Γ (1198981198942)

119903119898119894minus1

119894119889119903119894

(119894 = 1 2 3) (23)

and if 119891(1199031 1199032 1199033) is specified by a spherically symmetric

function 119891(119903) with 1199032

= 1199032

1+ 1199032

2+ 1199032

3 then we have

int1198821

int1198822

int1198823

119891 (1199031 1199032 1199033) 119889120583 (119882

1) sdot 119889120583 (119882

2) sdot 119889120583 (119882

3)

=21205871198982

Γ (1198982)int

119877

0

119891 (119903) 119903119898minus1

119889119903

(24)

The proof of Lemma 3 can be found in page 28 in theliterature [8]

With the preparations above we can introduce the mainresult of this paper as follow

Proposition 4 If 119909119894(119903119894) = 120596(119863

119894)119903119863119894

119894(119894 = 1 2 3) not only

describes the length of a 119863119894-dimensional fractal curve 120573

119863119894

(119903119894)

but also denotes the volume of a 119863119894-dimensional sphere Ω

119894and

if 119891(119903) = 119891(1199031 1199032 1199033) is a spherically symmetric function then

the fractal integration of 119891(119903) on the fractal graph 1205731198631

(1199031) otimes

1205731198632

(1199032) otimes 1205731198633

(1199033) equals

int1205731198631(1199031)

int1205731198632(1199032)

int1205731198633(1199033)

119891 (1199031 1199032 1199033)

3

prod

119894=1

119903119894119863119863119894

119909119894

=21205871198632

Γ (1198632)int

119877

0

119891 (119903) 119903119863minus1

119889119903

(25)

where 120596(119863119894) = 120587

1198631198942

Γ((1198631198942) + 1) 119903

2

= 1199032

1+ 1199032

2+ 1199032

3 and

119863 = 1198631

+ 1198632

+ 1198633

Proof Using (13) and (24) we easily arrive at (25)The proof is complete

Proposition 4 shows that we have arrived at the con-jectural integral formula (3) through the fractal measure(A23) In particular if 119877 rarr infin then the integral for-mula (25) will exhibit linearity translational invariance andscaling property These properties are natural and necessaryin applying dimensional regularization to quantum fieldtheory [2] It is worth mentioning that Proposition 4 canbe generalized so that 119894 might take any positive integer forexample 119894 = 1 2 infin If 119894 = 1 2 3 4 then 120573

119863119894

(119903119894) may

denote coordinate axiswith the dimension119863119894and so120573

1198631

(1199031)otimes

1205731198632

(1199032) otimes 1205731198633

(1199033) otimes 1205731198634

(1199034) may denote spacetime

5 Conclusion

Hausdorff measure is not an ideal mathematical frameworkfor describing fractal since it fails to describe the nonlocalproperty of fractal (eg self-similarity) However the fractalmeasure constructed by this paper not only shows the dimen-sion of a fractal but also describes its analytic properties (egnonlocal property) Not only so using this fractal measurewe can derive Svozilrsquos conjectural integral formula (3) whichis the starting point of quantum field theory on fractalspacetimeTherefore our fractal measure may be regarded asa possible mathematical basis of establishing quantum fieldtheory on fractal spacetime

Appendices

A Mathematical Preparations

In Euclidean geometry the dimension of a geometric graphis determined by the number of independent variables (iethe number of degrees of freedom) For example everypoint on a plane can be represented by 2-tuples real number(1199091 1199092) then the dimension of the plane is denoted by

Journal of Applied Mathematics 5

2 Nevertheless the existence of Peanorsquos curve powerfullyrefutes this viewpoint Peanorsquos curve which is determined byan independent characteristic parameter (ie fill parameter)would fill up the entire plane [10]Therefore mathematicianshave to reconsider the definition of dimension The mostfamous one of all definitions of dimension is the Hausdorffdimension which is defined through the Hausdorff measure[8]

A1 Hausdorff Measure and Hausdorff Dimension In orderto bring the definition of Hausdorff dimension we firstlyintroduce the Hausdorff measure [8]

Let 119882 be a nonempty subset of 119899-dimensional Euclideanspace 119877

119899 the diameter of 119882 is defined as

diam (119882) = sup 119889 (119909 119910) 119909 119910 isin 119882 (A1)

where 119889(119909 119910) which is the distance between points 119909 and 119910is a real-valued function on 119882 otimes 119882 such that the followingfour conditions are satisfied

119889 (119909 119910) ge 0 forall119909 119910 isin 119882 (A2)

119889 (119909 119910) = 0 iff 119909 = 119910 (A3)

119889 (119909 119910) = 119889 (119910 119909) forall119909 119910 isin 119882 (A4)

119889 (119909 119911) le 119889 (119909 119910) + 119889 (119910 119911) forall119909 119910 119911 isin 119882 (A5)

For example the distance of 119899-dimensional Euclideanspace 119877

119899 can be defined as

119889119864

(119909 119910) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 = (

119899

sum

119894=1

1003816100381610038161003816119909119894 minus 119910119894

1003816100381610038161003816

2

)

12

(A6)

Then it is easy to check that (A6) satisfies conditions(A2)ndash(A5)

Now let us consider a countable set 119864119894 of subsets of

diameter at most 120576 that covers 119882 that is

119882 sub

infin

⋃

119894=1

119864119894 diam (119864

119894) le 120576 forall119894 (A7)

For a positive 119863 and each 120576 gt 0 we consider covers of 119882

by countable families 119864119894 of (arbitrary) sets 119864

119894with diameter

less than 120576 and take the infimum of the sum of [diam(119864119894)]119863

Then we have

119867119863

120576(119882)

= inf

infin

sum

119894=1

[diam (119864119894)]119863

119882 sub

infin

⋃

119894=1

119864119894 diam (119864

119894) le 120576

(A8)

If the following limit exists

119867119863

(119882) = lim120576rarr0

119867119863

120576(119882) = finite (A9)

then the value119867119863

(119882) is called the119863-dimensional Hausdorffmeasure

A2 Shortcoming of Hausdorff Measure In general 119863 maybe a fraction In 1967 Mandelbrot realized that [11] the lengthof coastline can be measured using Hausdorff measure (A8)rather than Euclideanmeasure (A6) and then the dimensionof coastline is a fraction Mandelbrot called such geometricgraphs the ldquofractalrdquo

The fractal is self-similar between its local and globalshapes Unfortunately Hausdorff measure can determine thedimension of fractal but not reflect the connection (eg self-similarity) among the parts of the corresponding fractal Tosee this we consider Kochrsquos curve in Figure 2

Clearly the congruent triangle Δ119909111990921199093is similar to

Δ119909411990951199096 If we use the Hausdorff measure (A8) to measure

the local distance of Kochrsquos curve (eg the distance betweenpoints 119909

1and 119909

3) then we have

119867119863

(1199091 1199093) =

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816

119863

+10038161003816100381610038161199092 minus 119909

3

1003816100381610038161003816

119863

(A10)

where 119863 is the dimension of theKoch curveEquation (A10) shows that the Hausdorff distance

between points 1199091and 119909

3depends only on the positions of

points 119909119894

(119894 = 1 2 3) and is thereby independent of thepositions of points 119909

119895(119895 = 4 5 6) Nevertheless because

of the self-similarity of Kochrsquos curve any displacements ofpoints 119909

119895(119895 = 4 5 6) would influence the positions of 119909

119894(119894 =

1 2 3) and hence change the distance between points 1199091and

1199093 That is to say the local shape (eg Δ119909

111990921199093) is closely

related to the global shape (eg Δ119909411990951199096) Unfortunately

the Hausdorff distance (A10) undoubtedly fails to reflect thisfact Therefore we need to find a new measure of describingthe analytic properties of fractal

A3 Definition of Fractal Measure Hausdorff measure (A8)does not reflect the self-similarity of fractal so we cannotestablish the calculus theory of fractal using the Hausdorffmeasure In general people often use the fractional calculusto approximately describe the analytic properties of fractal[8 12]

The fractional calculus is a theory of integrals and deriva-tives of any arbitrary real order For example the fractionalderivatives of order 119898 of the function 119910(119897) = 119888119897

119899 equal [8]

119889119898

119910 (119897)

119889119897119898= 119888

Γ (119899 + 1)

Γ (119899 minus 119898 + 1)119897119899minus119898

(A11)

where Γ(119909) denotes the Gamma function and 119898 is an arbi-trary real number

Now let us consider a 119898-dimensional volume

119909 (119897) = 120596 (119898) 119897119898

(A12)

where 120596(119898) is a constant which depends only on the dimen-sion 119898

Using formula (A11) the fractional derivatives of order119898

of (A12) equal

119889119898

119909 (119897)

119889119897119898= Γ (119898 + 1) 120596 (119898) (A13)

6 Journal of Applied Mathematics

x1

x2

x4

x3

x5

x6

Figure 2 Kochrsquos curve which is similar to the generation of Cantorternary set (see Figure 1) is defined by repeatedly adding themiddlethirds of line segments [11]

Equation (A13) can be written as

Δ119898

119909 (119897) = Γ (119898 + 1) 120596 (119898) (Δ119897)119898

+ 119900 [(Δ119897)119898

] (A14)

whereΔ119898

119909(119897) denotes the differences of order119898 and 119900[(Δ119897)119898

]

denotes the infinitesimal terms of higher order compared to(Δ119897)119898Equation (A14) implies that

Δ119898

119909 (119897) sim (Δ119897)119898

(A15)

and thereby

119873

sum

119894=1

Δ119898

119909 (119897119894) sim

119873

sum

119894=1

(Δ119897)119898

(A16)

Obviously (Δ119897)119898 is a 119898-dimensional Hausdorff measure

which can describe the length of a 119898-dimensional fractalcurve Consequently (A15) and (A16) together imply thatΔ119898

119909(119897) can be also thought of as a 119898-dimensional measureIn this case the order of differences Δ

119898

119909(119897) represents theHausdorff dimension119898 Because of this fact we next attemptto use the differences of order 119898 to define a new measure

Let us consider the left-shift operator with stepΔ119897 and theidentity operator as follows

119871Δ119897

119909 (119897) = 119909 (119897 minus Δ119897)

1198710119909 (119897) = 119909 (119897)

(A17)

Using the left-shift operator 119871Δ119897and the identity operator

1198710 we can define the difference operator of order 119898 in the

form

(1198710

minus 119871Δ119897

)119898

=

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119871119895Δ119897

(A18)

Using (A18) we define a new distance between points119909(119897) and 119909(119897 minus Δ119897) in the form

1003816100381610038161003816Δ119898 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119909 (119897 minus 119895Δ119897)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(A19)

We call (A19) the ldquononlocal distancerdquo which describesthe length of a 119898-dimensional fractal curve

When 119898 = 1 the nonlocal distance (A19) returns to theEuclidean distance that is

1003816100381610038161003816Δ119898=1 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816 = |119909 (119897) minus 119909 (119897 minus Δ119897)| (A20)

In general the nonlocal distance (A19) does not satisfythe general properties of distance (A4)-(A5) but reflects theconnection between local and global segments of fractal (Forexample the nonlocal distance between points 119909(119897) and 119909(119897 +

Δ119897) that is |Δ119898

[119909(119897) 119909(119897+Δ119897)]| needs to be defined using theright-shift operator119877

Δ119897which leads to119877

Δ119897119909(119897) = 119909(119897+Δ119897)The

corresponding difference operator reads (119877Δ119897

minus 1198770)119898 Then

the nonlocal distance would not satisfy condition (A4))To understand the latter we need to realize that the outputvalue of (A19) would depend on the values of all points119909(119897 minus 119895Δ119897) (119895 = 0 1 2 ) rather than only on points 119909(119897) and119909(119897 minus Δ119897)

For instance in Figure 2 the nonlocal distance |Δ119863

[1199095

1199093]| between points 119909

3and 119909

5would depend on the positions

of points 119909119894

(119894 = 1 2 3 4 5) rather than only on points 1199093

and 1199095 Therefore the nonlocal distance (A19) is indeed an

intrinsic way of describing fractal since it not only shows thedimension but also reflects the connection between local andglobal segments of fractal

Using the nonlocal distance (A19) we can propose adefinition for fractal measure

Let 119882 be a nonempty subset of 119899-dimensional Euclideanspace 119877

119899 We consider a countable set 119865119894 of subsets of

diameter at most 120576 that covers 119882 that is

119882 sub

infin

⋃

119894=1

119865119894 diam119863 (119865

119894) le 120576 forall119894 (A21)

where diam119863(119865119894) defined by using the nonlocal distance

(A19) denotes the diameter of 119865119894 that is

diam119863 (119865119894) = sup

1003816100381610038161003816Δ119863 [119909 119910]1003816100381610038161003816 119909 119910 isin 119865

119894 (A22)

Fractal Measure For a positive 119863 and each 120576 gt 0 we considercovers of 119882 by countable families 119865

119894 of (arbitrary) sets 119865

119894

with diameter less than 120576 and take the infimum of the sum ofdiam119863(119865

119894) Then we have

Π119863

120576(119882)

= inf

infin

sum

119894=1

diam119863 (119865119894) 119882 sub

infin

⋃

119894=1

119865119894 diam119863 (119865

119894) le 120576

(A23)

If the following limit exists

Π119863

(119882) = lim120576rarr0

Π119863

120576(119882) = finite (A24)

then the value Π119863

(119882) is called the 119863-dimensional fractalmeasure meanwhile the dimension of 119882 equals 119863

Journal of Applied Mathematics 7

A4 Definition of Fractal Derivative Obviously to describethe analytic properties of fractal we need the correspondingcalculus theory

Before proceeding to introduce the definition of fractalderivative let us consider a 120596-dimensional fractal curve120573120596(119897) (see Figure 3) which is determined by an independent

characteristic parameter 119897 (eg the fill parameter of Peanorsquoscurve) filling up a 120596-dimensional region Assume that thelength of the fractal curve is specified by a 120596-dimensionalvolume 119909(119897) then the (nonlocal) length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) in Figure 3 should be denoted by

Π119863

([119886 119887]) = limΔ119897rarr0

inf

119899

sum

119894=1

1003816100381610038161003816Δ120596 [119909 (119897119894minus1

) 119909 (119897119894)]

1003816100381610038161003816 (A25)

where we have used the fractal measure (A23)It is carefully noted that the length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) can not bemeasured by Euclidean scale

see Figure 3As such we can present a definition for fractal derivative

as followsFractal Derivative For any differentiable function 119910 = 119891(119909)if 119909 = 119909(119897) is not only a 120596-dimensional volume but alsodescribes the length of a 120596-dimensional fractal curve 120574

120596(119897)

then the fractal derivative of 119910 = 119891(119909) with respect to thefractal curve 120574

120596(119897) is defined as

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (A26)

where 119891(119897) = 119891[119909(119897)]Clearly if120596 = 1 then the formula (A26)will return to the

Newton-Leibniz derivative andmeanwhile 120574120596=1

(119897) is restoredto a 1-dimensional coordinate axis

In general the fractional derivative of order 120596 of anydifferentiable function 119891(119897) is defined in the form [13]

119889120596

119891 (119897)

119889119897120596

= limΔ119897rarr0

(

infin

sum

119895=0

120596 (120596 minus 1) sdot sdot sdot (120596 minus 119895 + 1) (minus1)119895

119895119891 (119897 minus 119895Δ119897)

times (Δ119897)minus120596

)

(A27)

Comparing (A19) and (A27) we have

119889120596

119891 (119897)

119889119897120596= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

(Δ119897)120596

(A28)

Using formula (A28) the formula (A26) can be rewrittenas

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)]sdot limΔ119897rarr0

1(Δ119897)120596

1(Δ119897)120596

=

limΔ119897rarr0

(Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)] (Δ119897)120596

)

limΔ119897rarr0

(Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (Δ119897)120596

)

=119889120596

119891 (119897) 119889119897120596

119889120596119909 (119897) 119889119897120596

(A29)

Formula (A29) indicates that we can compute the fractalderivative using the fractional derivative

B Computation Examples

In Appendix A we have noted that the fractal derivative canbe computed using the formula (A29) In this appendix wepresent two computing examples

Example 1 If 119909(119897) = 120596(119898)119897119898 describes the length of a 119898-

dimensional fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119897119863119898

119862

119897119863119898

119909=

119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909 (B1)

where 120596(119898) is a constant that depends only on 119898

Proof The fractional derivatives of order 119898 of the constant 119862

and the power function 119910(119897) = 119886119897119899 are respectively as follows

[8]

119889119898

119862

119889119897119898=

119862

Γ (1 minus 119898)119897minus119898

(B2)

119889119898

119910 (119897)

119889119897119898= 119886

Γ (119899 + 1)

Γ (119899 minus 119898 + 1)119897119899minus119898

(B3)

Using formulas (A29) (B2) and (B3) the fractal deriva-tive119897119863119898

119891(119909)119897119863119898

119909 can be computed as follows

119897119863119898

119862

119897119863119898

119909=

119889119898

119862119889119897119898

119889119898 [120596 (119898) 119897119898] 119889119897119898=

(119862Γ (1 minus 119898)) 119897minus119898

120596 (119898) Γ (1 + 119898)

=119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909

(B4)

The proof is complete

Example 2 If 119909 is the characteristic parameter of a 119898-dimensional fractal curve andmeanwhile it also describes thelength of this fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B5)

8 Journal of Applied Mathematics

Fractal curve

Nonlocal scale

Euclidean scale

120573120596(l1) 120573120596(l3)

120573120596(l2) 120573120596(l4)a = 120573120596(l0)120573120596(li) 120573120596(li+2)

)

120573120596(li+1)

120573120596(lnminus1)

b = 120573120596(ln

120573120596(l)

x(l0) = 0 x(l1) x(l2) x(l3) x(lnminus3) x(lnminus2) x(lnminus1) x(ln)

x(l)

l0 = 0 l1 l2 l3

l

lmlmminus3 lmminus2lmminus1

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middotmiddot middot middot

Figure 3 The fractal curve 120573120596(119897) consists of the union ⋃

119899

119894=1[120573120596(119897119894minus1

) 120573120596(119897119894)] where 119899 = 119899(119898) The dimension of fractal curve 120573

120596(119897) equals

120596 = lim119898rarrinfin

(ln 119899(119898) ln119898) Since 120573120596(119897) is a fractal curve the distance between points 120573

120596(1198970) and 120573

120596(119897119899) cannot be measured using the

Euclidean scale (ruler) 119897 otherwise we will have 119889[120573120596(1198970) 120573120596(119897119899)] = lim

119898rarrinfin119899(119898)((119897

119898minus 1198970)119898) = infin or =0 However the fractal curve 120573

120596(119897)

can be measured using the nonlocal scale (ruler) 119909(119897) for the way of measure see the formula (A25)

Proof Using formulas (A29) (B2) and (B3) we have

119909119863119898

119891 (119909)

119909119863119898

119909=

119889119898

119862119889119909119898

119889119898119909119889119909119898=

(119862Γ (1 minus 119898)) 119909minus119898

(Γ (2) Γ (2 minus 119898)) 1199091minus119898

= 119862Γ (2 minus 119898)

Γ (1 minus 119898)sdot

1

119909

(B6)

Considering the property of Gamma function

Γ (119909 + 1) = 119909Γ (119909) (B7)

we have

1 minus 119898 =Γ (2 minus 119898)

Γ (1 minus 119898) (B8)

Substituting (B8) into (B6) we arrive at

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B9)

The proof is complete

C Quantum Behavior and Nonlocal Distance

Now we investigate the connection between quantum behav-iour and nonlocal distance For simplicity we still con-sider the Cantor set in Figure 1 where the nonlocal dis-tance between points 119909

(119899)

2and 119909

(119899)

1is equal to

Π (119909(119899)

2 119909(119899)

1) =

10038161003816100381610038161003816Δ119898=119863

[119909(119899)

2 119909(119899)

1]10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2119899

sum

119895=1

119863 (119863 minus 1) sdot sdot sdot (119863 minus 119895 + 1) (minus1)119895

119895119909(119899)

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(C1)

Clearly the output value of nonlocal distance (C1)depends on the value of each element in the set 119909

(119899)

119894

where 119894 runs from 1 to 2119899 If 119899 rarr infin we will have

Π(119909(119899)

2 119909(119899)

1) rarr 0 however the number of elements in the

set 119909(119899)

119894 would tend to infinity too This means that if we

want to precisely measure the distance of smaller scale(eg lim

119899≫1Π(119909(119899)

2 119909(119899)

1)) correspondingly we will need to

collect the points 119909(119899)

119894 As a result if we want to precisely

measure the nonlocal distance between points lim119899rarrinfin

119909(119899)

2

and lim119899rarrinfin

119909(119899)

1 we will need to collect a set of infinite

points that is 119909(infin)119894

infin

119894=1 Unfortunately we must fail to arrive

Journal of Applied Mathematics 9

at this purpose on an actual measurement In other words wecannot precisely measure the distance of microscopic scalethis fact is consistent with the ldquoHeisenberg Uncertainty Prin-ciplerdquo (In fact the connection between quantum mechanicsand fractal has been noticed in some earlier papers [14ndash17])

On the other hand the nonlocal distance (C1) betweenpoints119909

(119899)

2and119909(119899)

1depends clearly on the position of119909(119899)

119894(119894 =

1 2 2119899

) for example any displacement of point 119909(119899)

2119899

would influence the output value of nonlocal distance (C1) Itis a clearly nonlocal correlation (correlations span arbitrarilydistances) and similar to quantum entanglement

The above two facts imply that the nonlocal descriptionmay be an intrinsic way of describing quantum behavior (Inreferences [18 19] we have shown that the local descriptionis not a way of completely describing physical reality)

Acknowledgment

This project is supported by the Scholarship Award for Excel-lent Doctoral Student Granted by the Ministry of Educationof China (2012) (Grant no 0903005109081-019)

References

[1] G rsquot Hooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquoNuclear Physics B vol 44 no 1 pp 189ndash2131972

[2] K Svozil ldquoQuantum field theory on fractal spacetime a newregularisation methodrdquo Journal of Physics A vol 20 no 12 pp3861ndash3875 1987

[3] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 15 pages 2009

[4] P Horava ldquoSpectral dimension of the universe in quantumgravity at a Lifshitz pointrdquo Physical Review Letters vol 102 no16 Article ID 161301 4 pages 2009

[5] G Calcagni ldquoQuantum field theory gravity and cosmology in afractal universerdquo Journal ofHigh Energy Physics vol 2010 article120 2010

[6] G Calcagni ldquoFractal universe and quantum gravityrdquo PhysicalReview Letters vol 104 no 25 Article ID 251301 2010

[7] D Benedetti ldquoFractal properties of quantum spacetimerdquo Phys-ical Review Letters vol 102 no 11 Article ID 111303 4 pages2009

[8] V E Tarasov Fractional Dynamics Applications of FractionalCalculus to Dynamics of Particles Field and Media HigherEducation Press Beijing China 2010

[9] T C Halsey M H Jensen L P Kadanoff I Procaccia andB I Shraiman ldquoFractal measures and their singularities thecharacterization of strange setsrdquo Physical Review A vol 33 no2 pp 1141ndash1151 1986

[10] H Sagan Space-Filling Curves Springer New York NY USA1994

[11] BMandelbrot ldquoHow long is the coast of Britain Statistical self-similarity and fractional dimensionrdquo Science vol 156 no 3775pp 636ndash638 1967

[12] M Giona ldquoFractal calculus on [0 1]rdquo Chaos Solitons andFractals vol 5 no 6 pp 987ndash1000 1995

[13] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[14] M S El Naschie ldquoQuantum mechanics and the possibility of aCantorian space-timerdquo Chaos Solitons and Fractals vol 1 no5 pp 485ndash487 1991

[15] M S El Naschie ldquoA note on Heisenbergrsquos uncertainty principleand Cantorian space-timerdquo Chaos Solitons and Fractals vol 2no 4 pp 437ndash439 1992

[16] M S El Naschie ldquoOn certain infinite-dimensional Cantor setsand the Schrodinger waverdquo Chaos Solitons and Fractals vol 3no 1 pp 89ndash98 1993

[17] L Marek-Crnjac ldquoA short history of fractal-Cantorian space-timerdquo Chaos Solitons and Fractals vol 41 no 5 pp 2697ndash27052009

[18] Y Tao ldquoNecessity of integral formalismrdquo Communications inTheoretical Physics vol 56 no 4 pp 648ndash654 2011

[19] Y Tao ldquoSufficient condition for validity of quantum adiabatictheoremrdquo Communications in Theoretical Physics vol 57 no 3pp 343ndash347 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

2 Nevertheless the existence of Peanorsquos curve powerfullyrefutes this viewpoint Peanorsquos curve which is determined byan independent characteristic parameter (ie fill parameter)would fill up the entire plane [10]Therefore mathematicianshave to reconsider the definition of dimension The mostfamous one of all definitions of dimension is the Hausdorffdimension which is defined through the Hausdorff measure[8]

A1 Hausdorff Measure and Hausdorff Dimension In orderto bring the definition of Hausdorff dimension we firstlyintroduce the Hausdorff measure [8]

Let 119882 be a nonempty subset of 119899-dimensional Euclideanspace 119877

119899 the diameter of 119882 is defined as

diam (119882) = sup 119889 (119909 119910) 119909 119910 isin 119882 (A1)

where 119889(119909 119910) which is the distance between points 119909 and 119910is a real-valued function on 119882 otimes 119882 such that the followingfour conditions are satisfied

119889 (119909 119910) ge 0 forall119909 119910 isin 119882 (A2)

119889 (119909 119910) = 0 iff 119909 = 119910 (A3)

119889 (119909 119910) = 119889 (119910 119909) forall119909 119910 isin 119882 (A4)

119889 (119909 119911) le 119889 (119909 119910) + 119889 (119910 119911) forall119909 119910 119911 isin 119882 (A5)

For example the distance of 119899-dimensional Euclideanspace 119877

119899 can be defined as

119889119864

(119909 119910) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 = (

119899

sum

119894=1

1003816100381610038161003816119909119894 minus 119910119894

1003816100381610038161003816

2

)

12

(A6)

Then it is easy to check that (A6) satisfies conditions(A2)ndash(A5)

Now let us consider a countable set 119864119894 of subsets of

diameter at most 120576 that covers 119882 that is

119882 sub

infin

⋃

119894=1

119864119894 diam (119864

119894) le 120576 forall119894 (A7)

For a positive 119863 and each 120576 gt 0 we consider covers of 119882

by countable families 119864119894 of (arbitrary) sets 119864

119894with diameter

less than 120576 and take the infimum of the sum of [diam(119864119894)]119863

Then we have

119867119863

120576(119882)

= inf

infin

sum

119894=1

[diam (119864119894)]119863

119882 sub

infin

⋃

119894=1

119864119894 diam (119864

119894) le 120576

(A8)

If the following limit exists

119867119863

(119882) = lim120576rarr0

119867119863

120576(119882) = finite (A9)

then the value119867119863

(119882) is called the119863-dimensional Hausdorffmeasure

A2 Shortcoming of Hausdorff Measure In general 119863 maybe a fraction In 1967 Mandelbrot realized that [11] the lengthof coastline can be measured using Hausdorff measure (A8)rather than Euclideanmeasure (A6) and then the dimensionof coastline is a fraction Mandelbrot called such geometricgraphs the ldquofractalrdquo

The fractal is self-similar between its local and globalshapes Unfortunately Hausdorff measure can determine thedimension of fractal but not reflect the connection (eg self-similarity) among the parts of the corresponding fractal Tosee this we consider Kochrsquos curve in Figure 2

Clearly the congruent triangle Δ119909111990921199093is similar to

Δ119909411990951199096 If we use the Hausdorff measure (A8) to measure

the local distance of Kochrsquos curve (eg the distance betweenpoints 119909

1and 119909

3) then we have

119867119863

(1199091 1199093) =

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816

119863

+10038161003816100381610038161199092 minus 119909

3

1003816100381610038161003816

119863

(A10)

where 119863 is the dimension of theKoch curveEquation (A10) shows that the Hausdorff distance

between points 1199091and 119909

3depends only on the positions of

points 119909119894

(119894 = 1 2 3) and is thereby independent of thepositions of points 119909

119895(119895 = 4 5 6) Nevertheless because

of the self-similarity of Kochrsquos curve any displacements ofpoints 119909

119895(119895 = 4 5 6) would influence the positions of 119909

119894(119894 =

1 2 3) and hence change the distance between points 1199091and

1199093 That is to say the local shape (eg Δ119909

111990921199093) is closely

related to the global shape (eg Δ119909411990951199096) Unfortunately

the Hausdorff distance (A10) undoubtedly fails to reflect thisfact Therefore we need to find a new measure of describingthe analytic properties of fractal

A3 Definition of Fractal Measure Hausdorff measure (A8)does not reflect the self-similarity of fractal so we cannotestablish the calculus theory of fractal using the Hausdorffmeasure In general people often use the fractional calculusto approximately describe the analytic properties of fractal[8 12]

The fractional calculus is a theory of integrals and deriva-tives of any arbitrary real order For example the fractionalderivatives of order 119898 of the function 119910(119897) = 119888119897

119899 equal [8]

119889119898

119910 (119897)

119889119897119898= 119888

Γ (119899 + 1)

Γ (119899 minus 119898 + 1)119897119899minus119898

(A11)

where Γ(119909) denotes the Gamma function and 119898 is an arbi-trary real number

Now let us consider a 119898-dimensional volume

119909 (119897) = 120596 (119898) 119897119898

(A12)

where 120596(119898) is a constant which depends only on the dimen-sion 119898

Using formula (A11) the fractional derivatives of order119898

of (A12) equal

119889119898

119909 (119897)

119889119897119898= Γ (119898 + 1) 120596 (119898) (A13)

6 Journal of Applied Mathematics

x1

x2

x4

x3

x5

x6

Figure 2 Kochrsquos curve which is similar to the generation of Cantorternary set (see Figure 1) is defined by repeatedly adding themiddlethirds of line segments [11]

Equation (A13) can be written as

Δ119898

119909 (119897) = Γ (119898 + 1) 120596 (119898) (Δ119897)119898

+ 119900 [(Δ119897)119898

] (A14)

whereΔ119898

119909(119897) denotes the differences of order119898 and 119900[(Δ119897)119898

]

denotes the infinitesimal terms of higher order compared to(Δ119897)119898Equation (A14) implies that

Δ119898

119909 (119897) sim (Δ119897)119898

(A15)

and thereby

119873

sum

119894=1

Δ119898

119909 (119897119894) sim

119873

sum

119894=1

(Δ119897)119898

(A16)

Obviously (Δ119897)119898 is a 119898-dimensional Hausdorff measure

which can describe the length of a 119898-dimensional fractalcurve Consequently (A15) and (A16) together imply thatΔ119898

119909(119897) can be also thought of as a 119898-dimensional measureIn this case the order of differences Δ

119898

119909(119897) represents theHausdorff dimension119898 Because of this fact we next attemptto use the differences of order 119898 to define a new measure

Let us consider the left-shift operator with stepΔ119897 and theidentity operator as follows

119871Δ119897

119909 (119897) = 119909 (119897 minus Δ119897)

1198710119909 (119897) = 119909 (119897)

(A17)

Using the left-shift operator 119871Δ119897and the identity operator

1198710 we can define the difference operator of order 119898 in the

form

(1198710

minus 119871Δ119897

)119898

=

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119871119895Δ119897

(A18)

Using (A18) we define a new distance between points119909(119897) and 119909(119897 minus Δ119897) in the form

1003816100381610038161003816Δ119898 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119909 (119897 minus 119895Δ119897)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(A19)

We call (A19) the ldquononlocal distancerdquo which describesthe length of a 119898-dimensional fractal curve

When 119898 = 1 the nonlocal distance (A19) returns to theEuclidean distance that is

1003816100381610038161003816Δ119898=1 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816 = |119909 (119897) minus 119909 (119897 minus Δ119897)| (A20)

In general the nonlocal distance (A19) does not satisfythe general properties of distance (A4)-(A5) but reflects theconnection between local and global segments of fractal (Forexample the nonlocal distance between points 119909(119897) and 119909(119897 +

Δ119897) that is |Δ119898

[119909(119897) 119909(119897+Δ119897)]| needs to be defined using theright-shift operator119877

Δ119897which leads to119877

Δ119897119909(119897) = 119909(119897+Δ119897)The

corresponding difference operator reads (119877Δ119897

minus 1198770)119898 Then

the nonlocal distance would not satisfy condition (A4))To understand the latter we need to realize that the outputvalue of (A19) would depend on the values of all points119909(119897 minus 119895Δ119897) (119895 = 0 1 2 ) rather than only on points 119909(119897) and119909(119897 minus Δ119897)

For instance in Figure 2 the nonlocal distance |Δ119863

[1199095

1199093]| between points 119909

3and 119909

5would depend on the positions

of points 119909119894

(119894 = 1 2 3 4 5) rather than only on points 1199093

and 1199095 Therefore the nonlocal distance (A19) is indeed an

intrinsic way of describing fractal since it not only shows thedimension but also reflects the connection between local andglobal segments of fractal

Using the nonlocal distance (A19) we can propose adefinition for fractal measure

Let 119882 be a nonempty subset of 119899-dimensional Euclideanspace 119877

119899 We consider a countable set 119865119894 of subsets of

diameter at most 120576 that covers 119882 that is

119882 sub

infin

⋃

119894=1

119865119894 diam119863 (119865

119894) le 120576 forall119894 (A21)

where diam119863(119865119894) defined by using the nonlocal distance

(A19) denotes the diameter of 119865119894 that is

diam119863 (119865119894) = sup

1003816100381610038161003816Δ119863 [119909 119910]1003816100381610038161003816 119909 119910 isin 119865

119894 (A22)

Fractal Measure For a positive 119863 and each 120576 gt 0 we considercovers of 119882 by countable families 119865

119894 of (arbitrary) sets 119865

119894

with diameter less than 120576 and take the infimum of the sum ofdiam119863(119865

119894) Then we have

Π119863

120576(119882)

= inf

infin

sum

119894=1

diam119863 (119865119894) 119882 sub

infin

⋃

119894=1

119865119894 diam119863 (119865

119894) le 120576

(A23)

If the following limit exists

Π119863

(119882) = lim120576rarr0

Π119863

120576(119882) = finite (A24)

then the value Π119863

(119882) is called the 119863-dimensional fractalmeasure meanwhile the dimension of 119882 equals 119863

Journal of Applied Mathematics 7

A4 Definition of Fractal Derivative Obviously to describethe analytic properties of fractal we need the correspondingcalculus theory

Before proceeding to introduce the definition of fractalderivative let us consider a 120596-dimensional fractal curve120573120596(119897) (see Figure 3) which is determined by an independent

characteristic parameter 119897 (eg the fill parameter of Peanorsquoscurve) filling up a 120596-dimensional region Assume that thelength of the fractal curve is specified by a 120596-dimensionalvolume 119909(119897) then the (nonlocal) length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) in Figure 3 should be denoted by

Π119863

([119886 119887]) = limΔ119897rarr0

inf

119899

sum

119894=1

1003816100381610038161003816Δ120596 [119909 (119897119894minus1

) 119909 (119897119894)]

1003816100381610038161003816 (A25)

where we have used the fractal measure (A23)It is carefully noted that the length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) can not bemeasured by Euclidean scale

see Figure 3As such we can present a definition for fractal derivative

as followsFractal Derivative For any differentiable function 119910 = 119891(119909)if 119909 = 119909(119897) is not only a 120596-dimensional volume but alsodescribes the length of a 120596-dimensional fractal curve 120574

120596(119897)

then the fractal derivative of 119910 = 119891(119909) with respect to thefractal curve 120574

120596(119897) is defined as

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (A26)

where 119891(119897) = 119891[119909(119897)]Clearly if120596 = 1 then the formula (A26)will return to the

Newton-Leibniz derivative andmeanwhile 120574120596=1

(119897) is restoredto a 1-dimensional coordinate axis

In general the fractional derivative of order 120596 of anydifferentiable function 119891(119897) is defined in the form [13]

119889120596

119891 (119897)

119889119897120596

= limΔ119897rarr0

(

infin

sum

119895=0

120596 (120596 minus 1) sdot sdot sdot (120596 minus 119895 + 1) (minus1)119895

119895119891 (119897 minus 119895Δ119897)

times (Δ119897)minus120596

)

(A27)

Comparing (A19) and (A27) we have

119889120596

119891 (119897)

119889119897120596= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

(Δ119897)120596

(A28)

Using formula (A28) the formula (A26) can be rewrittenas

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)]sdot limΔ119897rarr0

1(Δ119897)120596

1(Δ119897)120596

=

limΔ119897rarr0

(Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)] (Δ119897)120596

)

limΔ119897rarr0

(Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (Δ119897)120596

)

=119889120596

119891 (119897) 119889119897120596

119889120596119909 (119897) 119889119897120596

(A29)

Formula (A29) indicates that we can compute the fractalderivative using the fractional derivative

B Computation Examples

In Appendix A we have noted that the fractal derivative canbe computed using the formula (A29) In this appendix wepresent two computing examples

Example 1 If 119909(119897) = 120596(119898)119897119898 describes the length of a 119898-

dimensional fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119897119863119898

119862

119897119863119898

119909=

119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909 (B1)

where 120596(119898) is a constant that depends only on 119898

Proof The fractional derivatives of order 119898 of the constant 119862

and the power function 119910(119897) = 119886119897119899 are respectively as follows

[8]

119889119898

119862

119889119897119898=

119862

Γ (1 minus 119898)119897minus119898

(B2)

119889119898

119910 (119897)

119889119897119898= 119886

Γ (119899 + 1)

Γ (119899 minus 119898 + 1)119897119899minus119898

(B3)

Using formulas (A29) (B2) and (B3) the fractal deriva-tive119897119863119898

119891(119909)119897119863119898

119909 can be computed as follows

119897119863119898

119862

119897119863119898

119909=

119889119898

119862119889119897119898

119889119898 [120596 (119898) 119897119898] 119889119897119898=

(119862Γ (1 minus 119898)) 119897minus119898

120596 (119898) Γ (1 + 119898)

=119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909

(B4)

The proof is complete

Example 2 If 119909 is the characteristic parameter of a 119898-dimensional fractal curve andmeanwhile it also describes thelength of this fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B5)

8 Journal of Applied Mathematics

Fractal curve

Nonlocal scale

Euclidean scale

120573120596(l1) 120573120596(l3)

120573120596(l2) 120573120596(l4)a = 120573120596(l0)120573120596(li) 120573120596(li+2)

)

120573120596(li+1)

120573120596(lnminus1)

b = 120573120596(ln

120573120596(l)

x(l0) = 0 x(l1) x(l2) x(l3) x(lnminus3) x(lnminus2) x(lnminus1) x(ln)

x(l)

l0 = 0 l1 l2 l3

l

lmlmminus3 lmminus2lmminus1

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middotmiddot middot middot

Figure 3 The fractal curve 120573120596(119897) consists of the union ⋃

119899

119894=1[120573120596(119897119894minus1

) 120573120596(119897119894)] where 119899 = 119899(119898) The dimension of fractal curve 120573

120596(119897) equals

120596 = lim119898rarrinfin

(ln 119899(119898) ln119898) Since 120573120596(119897) is a fractal curve the distance between points 120573

120596(1198970) and 120573

120596(119897119899) cannot be measured using the

Euclidean scale (ruler) 119897 otherwise we will have 119889[120573120596(1198970) 120573120596(119897119899)] = lim

119898rarrinfin119899(119898)((119897

119898minus 1198970)119898) = infin or =0 However the fractal curve 120573

120596(119897)

can be measured using the nonlocal scale (ruler) 119909(119897) for the way of measure see the formula (A25)

Proof Using formulas (A29) (B2) and (B3) we have

119909119863119898

119891 (119909)

119909119863119898

119909=

119889119898

119862119889119909119898

119889119898119909119889119909119898=

(119862Γ (1 minus 119898)) 119909minus119898

(Γ (2) Γ (2 minus 119898)) 1199091minus119898

= 119862Γ (2 minus 119898)

Γ (1 minus 119898)sdot

1

119909

(B6)

Considering the property of Gamma function

Γ (119909 + 1) = 119909Γ (119909) (B7)

we have

1 minus 119898 =Γ (2 minus 119898)

Γ (1 minus 119898) (B8)

Substituting (B8) into (B6) we arrive at

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B9)

The proof is complete

C Quantum Behavior and Nonlocal Distance

Now we investigate the connection between quantum behav-iour and nonlocal distance For simplicity we still con-sider the Cantor set in Figure 1 where the nonlocal dis-tance between points 119909

(119899)

2and 119909

(119899)

1is equal to

Π (119909(119899)

2 119909(119899)

1) =

10038161003816100381610038161003816Δ119898=119863

[119909(119899)

2 119909(119899)

1]10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2119899

sum

119895=1

119863 (119863 minus 1) sdot sdot sdot (119863 minus 119895 + 1) (minus1)119895

119895119909(119899)

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(C1)

Clearly the output value of nonlocal distance (C1)depends on the value of each element in the set 119909

(119899)

119894

where 119894 runs from 1 to 2119899 If 119899 rarr infin we will have

Π(119909(119899)

2 119909(119899)

1) rarr 0 however the number of elements in the

set 119909(119899)

119894 would tend to infinity too This means that if we

want to precisely measure the distance of smaller scale(eg lim

119899≫1Π(119909(119899)

2 119909(119899)

1)) correspondingly we will need to

collect the points 119909(119899)

119894 As a result if we want to precisely

measure the nonlocal distance between points lim119899rarrinfin

119909(119899)

2

and lim119899rarrinfin

119909(119899)

1 we will need to collect a set of infinite

points that is 119909(infin)119894

infin

119894=1 Unfortunately we must fail to arrive

Journal of Applied Mathematics 9

at this purpose on an actual measurement In other words wecannot precisely measure the distance of microscopic scalethis fact is consistent with the ldquoHeisenberg Uncertainty Prin-ciplerdquo (In fact the connection between quantum mechanicsand fractal has been noticed in some earlier papers [14ndash17])

On the other hand the nonlocal distance (C1) betweenpoints119909

(119899)

2and119909(119899)

1depends clearly on the position of119909(119899)

119894(119894 =

1 2 2119899

) for example any displacement of point 119909(119899)

2119899

would influence the output value of nonlocal distance (C1) Itis a clearly nonlocal correlation (correlations span arbitrarilydistances) and similar to quantum entanglement

The above two facts imply that the nonlocal descriptionmay be an intrinsic way of describing quantum behavior (Inreferences [18 19] we have shown that the local descriptionis not a way of completely describing physical reality)

Acknowledgment

This project is supported by the Scholarship Award for Excel-lent Doctoral Student Granted by the Ministry of Educationof China (2012) (Grant no 0903005109081-019)

References

[1] G rsquot Hooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquoNuclear Physics B vol 44 no 1 pp 189ndash2131972

[2] K Svozil ldquoQuantum field theory on fractal spacetime a newregularisation methodrdquo Journal of Physics A vol 20 no 12 pp3861ndash3875 1987

[3] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 15 pages 2009

[4] P Horava ldquoSpectral dimension of the universe in quantumgravity at a Lifshitz pointrdquo Physical Review Letters vol 102 no16 Article ID 161301 4 pages 2009

[5] G Calcagni ldquoQuantum field theory gravity and cosmology in afractal universerdquo Journal ofHigh Energy Physics vol 2010 article120 2010

[6] G Calcagni ldquoFractal universe and quantum gravityrdquo PhysicalReview Letters vol 104 no 25 Article ID 251301 2010

[7] D Benedetti ldquoFractal properties of quantum spacetimerdquo Phys-ical Review Letters vol 102 no 11 Article ID 111303 4 pages2009

[8] V E Tarasov Fractional Dynamics Applications of FractionalCalculus to Dynamics of Particles Field and Media HigherEducation Press Beijing China 2010

[9] T C Halsey M H Jensen L P Kadanoff I Procaccia andB I Shraiman ldquoFractal measures and their singularities thecharacterization of strange setsrdquo Physical Review A vol 33 no2 pp 1141ndash1151 1986

[10] H Sagan Space-Filling Curves Springer New York NY USA1994

[11] BMandelbrot ldquoHow long is the coast of Britain Statistical self-similarity and fractional dimensionrdquo Science vol 156 no 3775pp 636ndash638 1967

[12] M Giona ldquoFractal calculus on [0 1]rdquo Chaos Solitons andFractals vol 5 no 6 pp 987ndash1000 1995

[13] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[14] M S El Naschie ldquoQuantum mechanics and the possibility of aCantorian space-timerdquo Chaos Solitons and Fractals vol 1 no5 pp 485ndash487 1991

[15] M S El Naschie ldquoA note on Heisenbergrsquos uncertainty principleand Cantorian space-timerdquo Chaos Solitons and Fractals vol 2no 4 pp 437ndash439 1992

[16] M S El Naschie ldquoOn certain infinite-dimensional Cantor setsand the Schrodinger waverdquo Chaos Solitons and Fractals vol 3no 1 pp 89ndash98 1993

[17] L Marek-Crnjac ldquoA short history of fractal-Cantorian space-timerdquo Chaos Solitons and Fractals vol 41 no 5 pp 2697ndash27052009

[18] Y Tao ldquoNecessity of integral formalismrdquo Communications inTheoretical Physics vol 56 no 4 pp 648ndash654 2011

[19] Y Tao ldquoSufficient condition for validity of quantum adiabatictheoremrdquo Communications in Theoretical Physics vol 57 no 3pp 343ndash347 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

x1

x2

x4

x3

x5

x6

Figure 2 Kochrsquos curve which is similar to the generation of Cantorternary set (see Figure 1) is defined by repeatedly adding themiddlethirds of line segments [11]

Equation (A13) can be written as

Δ119898

119909 (119897) = Γ (119898 + 1) 120596 (119898) (Δ119897)119898

+ 119900 [(Δ119897)119898

] (A14)

whereΔ119898

119909(119897) denotes the differences of order119898 and 119900[(Δ119897)119898

]

denotes the infinitesimal terms of higher order compared to(Δ119897)119898Equation (A14) implies that

Δ119898

119909 (119897) sim (Δ119897)119898

(A15)

and thereby

119873

sum

119894=1

Δ119898

119909 (119897119894) sim

119873

sum

119894=1

(Δ119897)119898

(A16)

Obviously (Δ119897)119898 is a 119898-dimensional Hausdorff measure

which can describe the length of a 119898-dimensional fractalcurve Consequently (A15) and (A16) together imply thatΔ119898

119909(119897) can be also thought of as a 119898-dimensional measureIn this case the order of differences Δ

119898

119909(119897) represents theHausdorff dimension119898 Because of this fact we next attemptto use the differences of order 119898 to define a new measure

Let us consider the left-shift operator with stepΔ119897 and theidentity operator as follows

119871Δ119897

119909 (119897) = 119909 (119897 minus Δ119897)

1198710119909 (119897) = 119909 (119897)

(A17)

Using the left-shift operator 119871Δ119897and the identity operator

1198710 we can define the difference operator of order 119898 in the

form

(1198710

minus 119871Δ119897

)119898

=

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119871119895Δ119897

(A18)

Using (A18) we define a new distance between points119909(119897) and 119909(119897 minus Δ119897) in the form

1003816100381610038161003816Δ119898 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=0

119898 (119898 minus 1) sdot sdot sdot (119898 minus 119895 + 1) (minus1)119895

119895119909 (119897 minus 119895Δ119897)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(A19)

We call (A19) the ldquononlocal distancerdquo which describesthe length of a 119898-dimensional fractal curve

When 119898 = 1 the nonlocal distance (A19) returns to theEuclidean distance that is

1003816100381610038161003816Δ119898=1 [119909 (119897) 119909 (119897 minus Δ119897)]1003816100381610038161003816 = |119909 (119897) minus 119909 (119897 minus Δ119897)| (A20)

In general the nonlocal distance (A19) does not satisfythe general properties of distance (A4)-(A5) but reflects theconnection between local and global segments of fractal (Forexample the nonlocal distance between points 119909(119897) and 119909(119897 +

Δ119897) that is |Δ119898

[119909(119897) 119909(119897+Δ119897)]| needs to be defined using theright-shift operator119877

Δ119897which leads to119877

Δ119897119909(119897) = 119909(119897+Δ119897)The

corresponding difference operator reads (119877Δ119897

minus 1198770)119898 Then

the nonlocal distance would not satisfy condition (A4))To understand the latter we need to realize that the outputvalue of (A19) would depend on the values of all points119909(119897 minus 119895Δ119897) (119895 = 0 1 2 ) rather than only on points 119909(119897) and119909(119897 minus Δ119897)

For instance in Figure 2 the nonlocal distance |Δ119863

[1199095

1199093]| between points 119909

3and 119909

5would depend on the positions

of points 119909119894

(119894 = 1 2 3 4 5) rather than only on points 1199093

and 1199095 Therefore the nonlocal distance (A19) is indeed an

intrinsic way of describing fractal since it not only shows thedimension but also reflects the connection between local andglobal segments of fractal

Using the nonlocal distance (A19) we can propose adefinition for fractal measure

Let 119882 be a nonempty subset of 119899-dimensional Euclideanspace 119877

119899 We consider a countable set 119865119894 of subsets of

diameter at most 120576 that covers 119882 that is

119882 sub

infin

⋃

119894=1

119865119894 diam119863 (119865

119894) le 120576 forall119894 (A21)

where diam119863(119865119894) defined by using the nonlocal distance

(A19) denotes the diameter of 119865119894 that is

diam119863 (119865119894) = sup

1003816100381610038161003816Δ119863 [119909 119910]1003816100381610038161003816 119909 119910 isin 119865

119894 (A22)

Fractal Measure For a positive 119863 and each 120576 gt 0 we considercovers of 119882 by countable families 119865

119894 of (arbitrary) sets 119865

119894

with diameter less than 120576 and take the infimum of the sum ofdiam119863(119865

119894) Then we have

Π119863

120576(119882)

= inf

infin

sum

119894=1

diam119863 (119865119894) 119882 sub

infin

⋃

119894=1

119865119894 diam119863 (119865

119894) le 120576

(A23)

If the following limit exists

Π119863

(119882) = lim120576rarr0

Π119863

120576(119882) = finite (A24)

then the value Π119863

(119882) is called the 119863-dimensional fractalmeasure meanwhile the dimension of 119882 equals 119863

Journal of Applied Mathematics 7

A4 Definition of Fractal Derivative Obviously to describethe analytic properties of fractal we need the correspondingcalculus theory

Before proceeding to introduce the definition of fractalderivative let us consider a 120596-dimensional fractal curve120573120596(119897) (see Figure 3) which is determined by an independent

characteristic parameter 119897 (eg the fill parameter of Peanorsquoscurve) filling up a 120596-dimensional region Assume that thelength of the fractal curve is specified by a 120596-dimensionalvolume 119909(119897) then the (nonlocal) length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) in Figure 3 should be denoted by

Π119863

([119886 119887]) = limΔ119897rarr0

inf

119899

sum

119894=1

1003816100381610038161003816Δ120596 [119909 (119897119894minus1

) 119909 (119897119894)]

1003816100381610038161003816 (A25)

where we have used the fractal measure (A23)It is carefully noted that the length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) can not bemeasured by Euclidean scale

see Figure 3As such we can present a definition for fractal derivative

as followsFractal Derivative For any differentiable function 119910 = 119891(119909)if 119909 = 119909(119897) is not only a 120596-dimensional volume but alsodescribes the length of a 120596-dimensional fractal curve 120574

120596(119897)

then the fractal derivative of 119910 = 119891(119909) with respect to thefractal curve 120574

120596(119897) is defined as

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (A26)

where 119891(119897) = 119891[119909(119897)]Clearly if120596 = 1 then the formula (A26)will return to the

Newton-Leibniz derivative andmeanwhile 120574120596=1

(119897) is restoredto a 1-dimensional coordinate axis

In general the fractional derivative of order 120596 of anydifferentiable function 119891(119897) is defined in the form [13]

119889120596

119891 (119897)

119889119897120596

= limΔ119897rarr0

(

infin

sum

119895=0

120596 (120596 minus 1) sdot sdot sdot (120596 minus 119895 + 1) (minus1)119895

119895119891 (119897 minus 119895Δ119897)

times (Δ119897)minus120596

)

(A27)

Comparing (A19) and (A27) we have

119889120596

119891 (119897)

119889119897120596= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

(Δ119897)120596

(A28)

Using formula (A28) the formula (A26) can be rewrittenas

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)]sdot limΔ119897rarr0

1(Δ119897)120596

1(Δ119897)120596

=

limΔ119897rarr0

(Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)] (Δ119897)120596

)

limΔ119897rarr0

(Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (Δ119897)120596

)

=119889120596

119891 (119897) 119889119897120596

119889120596119909 (119897) 119889119897120596

(A29)

Formula (A29) indicates that we can compute the fractalderivative using the fractional derivative

B Computation Examples

In Appendix A we have noted that the fractal derivative canbe computed using the formula (A29) In this appendix wepresent two computing examples

Example 1 If 119909(119897) = 120596(119898)119897119898 describes the length of a 119898-

dimensional fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119897119863119898

119862

119897119863119898

119909=

119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909 (B1)

where 120596(119898) is a constant that depends only on 119898

Proof The fractional derivatives of order 119898 of the constant 119862

and the power function 119910(119897) = 119886119897119899 are respectively as follows

[8]

119889119898

119862

119889119897119898=

119862

Γ (1 minus 119898)119897minus119898

(B2)

119889119898

119910 (119897)

119889119897119898= 119886

Γ (119899 + 1)

Γ (119899 minus 119898 + 1)119897119899minus119898

(B3)

Using formulas (A29) (B2) and (B3) the fractal deriva-tive119897119863119898

119891(119909)119897119863119898

119909 can be computed as follows

119897119863119898

119862

119897119863119898

119909=

119889119898

119862119889119897119898

119889119898 [120596 (119898) 119897119898] 119889119897119898=

(119862Γ (1 minus 119898)) 119897minus119898

120596 (119898) Γ (1 + 119898)

=119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909

(B4)

The proof is complete

Example 2 If 119909 is the characteristic parameter of a 119898-dimensional fractal curve andmeanwhile it also describes thelength of this fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B5)

8 Journal of Applied Mathematics

Fractal curve

Nonlocal scale

Euclidean scale

120573120596(l1) 120573120596(l3)

120573120596(l2) 120573120596(l4)a = 120573120596(l0)120573120596(li) 120573120596(li+2)

)

120573120596(li+1)

120573120596(lnminus1)

b = 120573120596(ln

120573120596(l)

x(l0) = 0 x(l1) x(l2) x(l3) x(lnminus3) x(lnminus2) x(lnminus1) x(ln)

x(l)

l0 = 0 l1 l2 l3

l

lmlmminus3 lmminus2lmminus1

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middotmiddot middot middot

Figure 3 The fractal curve 120573120596(119897) consists of the union ⋃

119899

119894=1[120573120596(119897119894minus1

) 120573120596(119897119894)] where 119899 = 119899(119898) The dimension of fractal curve 120573

120596(119897) equals

120596 = lim119898rarrinfin

(ln 119899(119898) ln119898) Since 120573120596(119897) is a fractal curve the distance between points 120573

120596(1198970) and 120573

120596(119897119899) cannot be measured using the

Euclidean scale (ruler) 119897 otherwise we will have 119889[120573120596(1198970) 120573120596(119897119899)] = lim

119898rarrinfin119899(119898)((119897

119898minus 1198970)119898) = infin or =0 However the fractal curve 120573

120596(119897)

can be measured using the nonlocal scale (ruler) 119909(119897) for the way of measure see the formula (A25)

Proof Using formulas (A29) (B2) and (B3) we have

119909119863119898

119891 (119909)

119909119863119898

119909=

119889119898

119862119889119909119898

119889119898119909119889119909119898=

(119862Γ (1 minus 119898)) 119909minus119898

(Γ (2) Γ (2 minus 119898)) 1199091minus119898

= 119862Γ (2 minus 119898)

Γ (1 minus 119898)sdot

1

119909

(B6)

Considering the property of Gamma function

Γ (119909 + 1) = 119909Γ (119909) (B7)

we have

1 minus 119898 =Γ (2 minus 119898)

Γ (1 minus 119898) (B8)

Substituting (B8) into (B6) we arrive at

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B9)

The proof is complete

C Quantum Behavior and Nonlocal Distance

Now we investigate the connection between quantum behav-iour and nonlocal distance For simplicity we still con-sider the Cantor set in Figure 1 where the nonlocal dis-tance between points 119909

(119899)

2and 119909

(119899)

1is equal to

Π (119909(119899)

2 119909(119899)

1) =

10038161003816100381610038161003816Δ119898=119863

[119909(119899)

2 119909(119899)

1]10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2119899

sum

119895=1

119863 (119863 minus 1) sdot sdot sdot (119863 minus 119895 + 1) (minus1)119895

119895119909(119899)

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(C1)

Clearly the output value of nonlocal distance (C1)depends on the value of each element in the set 119909

(119899)

119894

where 119894 runs from 1 to 2119899 If 119899 rarr infin we will have

Π(119909(119899)

2 119909(119899)

1) rarr 0 however the number of elements in the

set 119909(119899)

119894 would tend to infinity too This means that if we

want to precisely measure the distance of smaller scale(eg lim

119899≫1Π(119909(119899)

2 119909(119899)

1)) correspondingly we will need to

collect the points 119909(119899)

119894 As a result if we want to precisely

measure the nonlocal distance between points lim119899rarrinfin

119909(119899)

2

and lim119899rarrinfin

119909(119899)

1 we will need to collect a set of infinite

points that is 119909(infin)119894

infin

119894=1 Unfortunately we must fail to arrive

Journal of Applied Mathematics 9

at this purpose on an actual measurement In other words wecannot precisely measure the distance of microscopic scalethis fact is consistent with the ldquoHeisenberg Uncertainty Prin-ciplerdquo (In fact the connection between quantum mechanicsand fractal has been noticed in some earlier papers [14ndash17])

On the other hand the nonlocal distance (C1) betweenpoints119909

(119899)

2and119909(119899)

1depends clearly on the position of119909(119899)

119894(119894 =

1 2 2119899

) for example any displacement of point 119909(119899)

2119899

would influence the output value of nonlocal distance (C1) Itis a clearly nonlocal correlation (correlations span arbitrarilydistances) and similar to quantum entanglement

The above two facts imply that the nonlocal descriptionmay be an intrinsic way of describing quantum behavior (Inreferences [18 19] we have shown that the local descriptionis not a way of completely describing physical reality)

Acknowledgment

This project is supported by the Scholarship Award for Excel-lent Doctoral Student Granted by the Ministry of Educationof China (2012) (Grant no 0903005109081-019)

References

[1] G rsquot Hooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquoNuclear Physics B vol 44 no 1 pp 189ndash2131972

[2] K Svozil ldquoQuantum field theory on fractal spacetime a newregularisation methodrdquo Journal of Physics A vol 20 no 12 pp3861ndash3875 1987

[3] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 15 pages 2009

[4] P Horava ldquoSpectral dimension of the universe in quantumgravity at a Lifshitz pointrdquo Physical Review Letters vol 102 no16 Article ID 161301 4 pages 2009

[5] G Calcagni ldquoQuantum field theory gravity and cosmology in afractal universerdquo Journal ofHigh Energy Physics vol 2010 article120 2010

[6] G Calcagni ldquoFractal universe and quantum gravityrdquo PhysicalReview Letters vol 104 no 25 Article ID 251301 2010

[7] D Benedetti ldquoFractal properties of quantum spacetimerdquo Phys-ical Review Letters vol 102 no 11 Article ID 111303 4 pages2009

[8] V E Tarasov Fractional Dynamics Applications of FractionalCalculus to Dynamics of Particles Field and Media HigherEducation Press Beijing China 2010

[9] T C Halsey M H Jensen L P Kadanoff I Procaccia andB I Shraiman ldquoFractal measures and their singularities thecharacterization of strange setsrdquo Physical Review A vol 33 no2 pp 1141ndash1151 1986

[10] H Sagan Space-Filling Curves Springer New York NY USA1994

[11] BMandelbrot ldquoHow long is the coast of Britain Statistical self-similarity and fractional dimensionrdquo Science vol 156 no 3775pp 636ndash638 1967

[12] M Giona ldquoFractal calculus on [0 1]rdquo Chaos Solitons andFractals vol 5 no 6 pp 987ndash1000 1995

[13] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[14] M S El Naschie ldquoQuantum mechanics and the possibility of aCantorian space-timerdquo Chaos Solitons and Fractals vol 1 no5 pp 485ndash487 1991

[15] M S El Naschie ldquoA note on Heisenbergrsquos uncertainty principleand Cantorian space-timerdquo Chaos Solitons and Fractals vol 2no 4 pp 437ndash439 1992

[16] M S El Naschie ldquoOn certain infinite-dimensional Cantor setsand the Schrodinger waverdquo Chaos Solitons and Fractals vol 3no 1 pp 89ndash98 1993

[17] L Marek-Crnjac ldquoA short history of fractal-Cantorian space-timerdquo Chaos Solitons and Fractals vol 41 no 5 pp 2697ndash27052009

[18] Y Tao ldquoNecessity of integral formalismrdquo Communications inTheoretical Physics vol 56 no 4 pp 648ndash654 2011

[19] Y Tao ldquoSufficient condition for validity of quantum adiabatictheoremrdquo Communications in Theoretical Physics vol 57 no 3pp 343ndash347 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

A4 Definition of Fractal Derivative Obviously to describethe analytic properties of fractal we need the correspondingcalculus theory

Before proceeding to introduce the definition of fractalderivative let us consider a 120596-dimensional fractal curve120573120596(119897) (see Figure 3) which is determined by an independent

characteristic parameter 119897 (eg the fill parameter of Peanorsquoscurve) filling up a 120596-dimensional region Assume that thelength of the fractal curve is specified by a 120596-dimensionalvolume 119909(119897) then the (nonlocal) length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) in Figure 3 should be denoted by

Π119863

([119886 119887]) = limΔ119897rarr0

inf

119899

sum

119894=1

1003816100381610038161003816Δ120596 [119909 (119897119894minus1

) 119909 (119897119894)]

1003816100381610038161003816 (A25)

where we have used the fractal measure (A23)It is carefully noted that the length between points 119886 =

120573120596(1198970) and 119887 = 120573

120596(119897119899) can not bemeasured by Euclidean scale

see Figure 3As such we can present a definition for fractal derivative

as followsFractal Derivative For any differentiable function 119910 = 119891(119909)if 119909 = 119909(119897) is not only a 120596-dimensional volume but alsodescribes the length of a 120596-dimensional fractal curve 120574

120596(119897)

then the fractal derivative of 119910 = 119891(119909) with respect to thefractal curve 120574

120596(119897) is defined as

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (A26)

where 119891(119897) = 119891[119909(119897)]Clearly if120596 = 1 then the formula (A26)will return to the

Newton-Leibniz derivative andmeanwhile 120574120596=1

(119897) is restoredto a 1-dimensional coordinate axis

In general the fractional derivative of order 120596 of anydifferentiable function 119891(119897) is defined in the form [13]

119889120596

119891 (119897)

119889119897120596

= limΔ119897rarr0

(

infin

sum

119895=0

120596 (120596 minus 1) sdot sdot sdot (120596 minus 119895 + 1) (minus1)119895

119895119891 (119897 minus 119895Δ119897)

times (Δ119897)minus120596

)

(A27)

Comparing (A19) and (A27) we have

119889120596

119891 (119897)

119889119897120596= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

(Δ119897)120596

(A28)

Using formula (A28) the formula (A26) can be rewrittenas

119897119863120596

119891 (119909)

119897119863120596

119909= limΔ119897rarr0

Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)]

Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)]sdot limΔ119897rarr0

1(Δ119897)120596

1(Δ119897)120596

=

limΔ119897rarr0

(Δ120596

[119891 (119897) 119891 (119897 minus Δ119897)] (Δ119897)120596

)

limΔ119897rarr0

(Δ120596

[119909 (119897) 119909 (119897 minus Δ119897)] (Δ119897)120596

)

=119889120596

119891 (119897) 119889119897120596

119889120596119909 (119897) 119889119897120596

(A29)

Formula (A29) indicates that we can compute the fractalderivative using the fractional derivative

B Computation Examples

In Appendix A we have noted that the fractal derivative canbe computed using the formula (A29) In this appendix wepresent two computing examples

Example 1 If 119909(119897) = 120596(119898)119897119898 describes the length of a 119898-

dimensional fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119897119863119898

119862

119897119863119898

119909=

119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909 (B1)

where 120596(119898) is a constant that depends only on 119898

Proof The fractional derivatives of order 119898 of the constant 119862

and the power function 119910(119897) = 119886119897119899 are respectively as follows

[8]

119889119898

119862

119889119897119898=

119862

Γ (1 minus 119898)119897minus119898

(B2)

119889119898

119910 (119897)

119889119897119898= 119886

Γ (119899 + 1)

Γ (119899 minus 119898 + 1)119897119899minus119898

(B3)

Using formulas (A29) (B2) and (B3) the fractal deriva-tive119897119863119898

119891(119909)119897119863119898

119909 can be computed as follows

119897119863119898

119862

119897119863119898

119909=

119889119898

119862119889119897119898

119889119898 [120596 (119898) 119897119898] 119889119897119898=

(119862Γ (1 minus 119898)) 119897minus119898

120596 (119898) Γ (1 + 119898)

=119862

Γ (1 minus 119898) Γ (1 + 119898)sdot

1

119909

(B4)

The proof is complete

Example 2 If 119909 is the characteristic parameter of a 119898-dimensional fractal curve andmeanwhile it also describes thelength of this fractal curve then the fractal derivative of theconstant function 119891(119909) = 119862 with respect to 119909 equals

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B5)

8 Journal of Applied Mathematics

Fractal curve

Nonlocal scale

Euclidean scale

120573120596(l1) 120573120596(l3)

120573120596(l2) 120573120596(l4)a = 120573120596(l0)120573120596(li) 120573120596(li+2)

)

120573120596(li+1)

120573120596(lnminus1)

b = 120573120596(ln

120573120596(l)

x(l0) = 0 x(l1) x(l2) x(l3) x(lnminus3) x(lnminus2) x(lnminus1) x(ln)

x(l)

l0 = 0 l1 l2 l3

l

lmlmminus3 lmminus2lmminus1

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middotmiddot middot middot

Figure 3 The fractal curve 120573120596(119897) consists of the union ⋃

119899

119894=1[120573120596(119897119894minus1

) 120573120596(119897119894)] where 119899 = 119899(119898) The dimension of fractal curve 120573

120596(119897) equals

120596 = lim119898rarrinfin

(ln 119899(119898) ln119898) Since 120573120596(119897) is a fractal curve the distance between points 120573

120596(1198970) and 120573

120596(119897119899) cannot be measured using the

Euclidean scale (ruler) 119897 otherwise we will have 119889[120573120596(1198970) 120573120596(119897119899)] = lim

119898rarrinfin119899(119898)((119897

119898minus 1198970)119898) = infin or =0 However the fractal curve 120573

120596(119897)

can be measured using the nonlocal scale (ruler) 119909(119897) for the way of measure see the formula (A25)

Proof Using formulas (A29) (B2) and (B3) we have

119909119863119898

119891 (119909)

119909119863119898

119909=

119889119898

119862119889119909119898

119889119898119909119889119909119898=

(119862Γ (1 minus 119898)) 119909minus119898

(Γ (2) Γ (2 minus 119898)) 1199091minus119898

= 119862Γ (2 minus 119898)

Γ (1 minus 119898)sdot

1

119909

(B6)

Considering the property of Gamma function

Γ (119909 + 1) = 119909Γ (119909) (B7)

we have

1 minus 119898 =Γ (2 minus 119898)

Γ (1 minus 119898) (B8)

Substituting (B8) into (B6) we arrive at

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B9)

The proof is complete

C Quantum Behavior and Nonlocal Distance

Now we investigate the connection between quantum behav-iour and nonlocal distance For simplicity we still con-sider the Cantor set in Figure 1 where the nonlocal dis-tance between points 119909

(119899)

2and 119909

(119899)

1is equal to

Π (119909(119899)

2 119909(119899)

1) =

10038161003816100381610038161003816Δ119898=119863

[119909(119899)

2 119909(119899)

1]10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2119899

sum

119895=1

119863 (119863 minus 1) sdot sdot sdot (119863 minus 119895 + 1) (minus1)119895

119895119909(119899)

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(C1)

Clearly the output value of nonlocal distance (C1)depends on the value of each element in the set 119909

(119899)

119894

where 119894 runs from 1 to 2119899 If 119899 rarr infin we will have

Π(119909(119899)

2 119909(119899)

1) rarr 0 however the number of elements in the

set 119909(119899)

119894 would tend to infinity too This means that if we

want to precisely measure the distance of smaller scale(eg lim

119899≫1Π(119909(119899)

2 119909(119899)

1)) correspondingly we will need to

collect the points 119909(119899)

119894 As a result if we want to precisely

measure the nonlocal distance between points lim119899rarrinfin

119909(119899)

2

and lim119899rarrinfin

119909(119899)

1 we will need to collect a set of infinite

points that is 119909(infin)119894

infin

119894=1 Unfortunately we must fail to arrive

Journal of Applied Mathematics 9

at this purpose on an actual measurement In other words wecannot precisely measure the distance of microscopic scalethis fact is consistent with the ldquoHeisenberg Uncertainty Prin-ciplerdquo (In fact the connection between quantum mechanicsand fractal has been noticed in some earlier papers [14ndash17])

On the other hand the nonlocal distance (C1) betweenpoints119909

(119899)

2and119909(119899)

1depends clearly on the position of119909(119899)

119894(119894 =

1 2 2119899

) for example any displacement of point 119909(119899)

2119899

would influence the output value of nonlocal distance (C1) Itis a clearly nonlocal correlation (correlations span arbitrarilydistances) and similar to quantum entanglement

The above two facts imply that the nonlocal descriptionmay be an intrinsic way of describing quantum behavior (Inreferences [18 19] we have shown that the local descriptionis not a way of completely describing physical reality)

Acknowledgment

This project is supported by the Scholarship Award for Excel-lent Doctoral Student Granted by the Ministry of Educationof China (2012) (Grant no 0903005109081-019)

References

[1] G rsquot Hooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquoNuclear Physics B vol 44 no 1 pp 189ndash2131972

[2] K Svozil ldquoQuantum field theory on fractal spacetime a newregularisation methodrdquo Journal of Physics A vol 20 no 12 pp3861ndash3875 1987

[3] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 15 pages 2009

[4] P Horava ldquoSpectral dimension of the universe in quantumgravity at a Lifshitz pointrdquo Physical Review Letters vol 102 no16 Article ID 161301 4 pages 2009

[5] G Calcagni ldquoQuantum field theory gravity and cosmology in afractal universerdquo Journal ofHigh Energy Physics vol 2010 article120 2010

[6] G Calcagni ldquoFractal universe and quantum gravityrdquo PhysicalReview Letters vol 104 no 25 Article ID 251301 2010

[7] D Benedetti ldquoFractal properties of quantum spacetimerdquo Phys-ical Review Letters vol 102 no 11 Article ID 111303 4 pages2009

[8] V E Tarasov Fractional Dynamics Applications of FractionalCalculus to Dynamics of Particles Field and Media HigherEducation Press Beijing China 2010

[9] T C Halsey M H Jensen L P Kadanoff I Procaccia andB I Shraiman ldquoFractal measures and their singularities thecharacterization of strange setsrdquo Physical Review A vol 33 no2 pp 1141ndash1151 1986

[10] H Sagan Space-Filling Curves Springer New York NY USA1994

[11] BMandelbrot ldquoHow long is the coast of Britain Statistical self-similarity and fractional dimensionrdquo Science vol 156 no 3775pp 636ndash638 1967

[12] M Giona ldquoFractal calculus on [0 1]rdquo Chaos Solitons andFractals vol 5 no 6 pp 987ndash1000 1995

[13] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[14] M S El Naschie ldquoQuantum mechanics and the possibility of aCantorian space-timerdquo Chaos Solitons and Fractals vol 1 no5 pp 485ndash487 1991

[15] M S El Naschie ldquoA note on Heisenbergrsquos uncertainty principleand Cantorian space-timerdquo Chaos Solitons and Fractals vol 2no 4 pp 437ndash439 1992

[16] M S El Naschie ldquoOn certain infinite-dimensional Cantor setsand the Schrodinger waverdquo Chaos Solitons and Fractals vol 3no 1 pp 89ndash98 1993

[17] L Marek-Crnjac ldquoA short history of fractal-Cantorian space-timerdquo Chaos Solitons and Fractals vol 41 no 5 pp 2697ndash27052009

[18] Y Tao ldquoNecessity of integral formalismrdquo Communications inTheoretical Physics vol 56 no 4 pp 648ndash654 2011

[19] Y Tao ldquoSufficient condition for validity of quantum adiabatictheoremrdquo Communications in Theoretical Physics vol 57 no 3pp 343ndash347 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

Fractal curve

Nonlocal scale

Euclidean scale

120573120596(l1) 120573120596(l3)

120573120596(l2) 120573120596(l4)a = 120573120596(l0)120573120596(li) 120573120596(li+2)

)

120573120596(li+1)

120573120596(lnminus1)

b = 120573120596(ln

120573120596(l)

x(l0) = 0 x(l1) x(l2) x(l3) x(lnminus3) x(lnminus2) x(lnminus1) x(ln)

x(l)

l0 = 0 l1 l2 l3

l

lmlmminus3 lmminus2lmminus1

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middotmiddot middot middot

Figure 3 The fractal curve 120573120596(119897) consists of the union ⋃

119899

119894=1[120573120596(119897119894minus1

) 120573120596(119897119894)] where 119899 = 119899(119898) The dimension of fractal curve 120573

120596(119897) equals

120596 = lim119898rarrinfin

(ln 119899(119898) ln119898) Since 120573120596(119897) is a fractal curve the distance between points 120573

120596(1198970) and 120573

120596(119897119899) cannot be measured using the

Euclidean scale (ruler) 119897 otherwise we will have 119889[120573120596(1198970) 120573120596(119897119899)] = lim

119898rarrinfin119899(119898)((119897

119898minus 1198970)119898) = infin or =0 However the fractal curve 120573

120596(119897)

can be measured using the nonlocal scale (ruler) 119909(119897) for the way of measure see the formula (A25)

Proof Using formulas (A29) (B2) and (B3) we have

119909119863119898

119891 (119909)

119909119863119898

119909=

119889119898

119862119889119909119898

119889119898119909119889119909119898=

(119862Γ (1 minus 119898)) 119909minus119898

(Γ (2) Γ (2 minus 119898)) 1199091minus119898

= 119862Γ (2 minus 119898)

Γ (1 minus 119898)sdot

1

119909

(B6)

Considering the property of Gamma function

Γ (119909 + 1) = 119909Γ (119909) (B7)

we have

1 minus 119898 =Γ (2 minus 119898)

Γ (1 minus 119898) (B8)

Substituting (B8) into (B6) we arrive at

119909119863119898

119862

119909119863119898

119909= 119862 (1 minus 119898) sdot

1

119909 (B9)

The proof is complete

C Quantum Behavior and Nonlocal Distance

Now we investigate the connection between quantum behav-iour and nonlocal distance For simplicity we still con-sider the Cantor set in Figure 1 where the nonlocal dis-tance between points 119909

(119899)

2and 119909

(119899)

1is equal to

Π (119909(119899)

2 119909(119899)

1) =

10038161003816100381610038161003816Δ119898=119863

[119909(119899)

2 119909(119899)

1]10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2119899

sum

119895=1

119863 (119863 minus 1) sdot sdot sdot (119863 minus 119895 + 1) (minus1)119895

119895119909(119899)

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(C1)

Clearly the output value of nonlocal distance (C1)depends on the value of each element in the set 119909

(119899)

119894

where 119894 runs from 1 to 2119899 If 119899 rarr infin we will have

Π(119909(119899)

2 119909(119899)

1) rarr 0 however the number of elements in the

set 119909(119899)

119894 would tend to infinity too This means that if we

want to precisely measure the distance of smaller scale(eg lim

119899≫1Π(119909(119899)

2 119909(119899)

1)) correspondingly we will need to

collect the points 119909(119899)

119894 As a result if we want to precisely

measure the nonlocal distance between points lim119899rarrinfin

119909(119899)

2

and lim119899rarrinfin

119909(119899)

1 we will need to collect a set of infinite

points that is 119909(infin)119894

infin

119894=1 Unfortunately we must fail to arrive

Journal of Applied Mathematics 9

at this purpose on an actual measurement In other words wecannot precisely measure the distance of microscopic scalethis fact is consistent with the ldquoHeisenberg Uncertainty Prin-ciplerdquo (In fact the connection between quantum mechanicsand fractal has been noticed in some earlier papers [14ndash17])

On the other hand the nonlocal distance (C1) betweenpoints119909

(119899)

2and119909(119899)

1depends clearly on the position of119909(119899)

119894(119894 =

1 2 2119899

) for example any displacement of point 119909(119899)

2119899

would influence the output value of nonlocal distance (C1) Itis a clearly nonlocal correlation (correlations span arbitrarilydistances) and similar to quantum entanglement

The above two facts imply that the nonlocal descriptionmay be an intrinsic way of describing quantum behavior (Inreferences [18 19] we have shown that the local descriptionis not a way of completely describing physical reality)

Acknowledgment

This project is supported by the Scholarship Award for Excel-lent Doctoral Student Granted by the Ministry of Educationof China (2012) (Grant no 0903005109081-019)

References

[1] G rsquot Hooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquoNuclear Physics B vol 44 no 1 pp 189ndash2131972

[2] K Svozil ldquoQuantum field theory on fractal spacetime a newregularisation methodrdquo Journal of Physics A vol 20 no 12 pp3861ndash3875 1987

[3] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 15 pages 2009

[4] P Horava ldquoSpectral dimension of the universe in quantumgravity at a Lifshitz pointrdquo Physical Review Letters vol 102 no16 Article ID 161301 4 pages 2009

[5] G Calcagni ldquoQuantum field theory gravity and cosmology in afractal universerdquo Journal ofHigh Energy Physics vol 2010 article120 2010

[6] G Calcagni ldquoFractal universe and quantum gravityrdquo PhysicalReview Letters vol 104 no 25 Article ID 251301 2010

[7] D Benedetti ldquoFractal properties of quantum spacetimerdquo Phys-ical Review Letters vol 102 no 11 Article ID 111303 4 pages2009

[8] V E Tarasov Fractional Dynamics Applications of FractionalCalculus to Dynamics of Particles Field and Media HigherEducation Press Beijing China 2010

[9] T C Halsey M H Jensen L P Kadanoff I Procaccia andB I Shraiman ldquoFractal measures and their singularities thecharacterization of strange setsrdquo Physical Review A vol 33 no2 pp 1141ndash1151 1986

[10] H Sagan Space-Filling Curves Springer New York NY USA1994

[11] BMandelbrot ldquoHow long is the coast of Britain Statistical self-similarity and fractional dimensionrdquo Science vol 156 no 3775pp 636ndash638 1967

[12] M Giona ldquoFractal calculus on [0 1]rdquo Chaos Solitons andFractals vol 5 no 6 pp 987ndash1000 1995

[13] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[14] M S El Naschie ldquoQuantum mechanics and the possibility of aCantorian space-timerdquo Chaos Solitons and Fractals vol 1 no5 pp 485ndash487 1991

[15] M S El Naschie ldquoA note on Heisenbergrsquos uncertainty principleand Cantorian space-timerdquo Chaos Solitons and Fractals vol 2no 4 pp 437ndash439 1992

[16] M S El Naschie ldquoOn certain infinite-dimensional Cantor setsand the Schrodinger waverdquo Chaos Solitons and Fractals vol 3no 1 pp 89ndash98 1993

[17] L Marek-Crnjac ldquoA short history of fractal-Cantorian space-timerdquo Chaos Solitons and Fractals vol 41 no 5 pp 2697ndash27052009

[18] Y Tao ldquoNecessity of integral formalismrdquo Communications inTheoretical Physics vol 56 no 4 pp 648ndash654 2011

[19] Y Tao ldquoSufficient condition for validity of quantum adiabatictheoremrdquo Communications in Theoretical Physics vol 57 no 3pp 343ndash347 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

at this purpose on an actual measurement In other words wecannot precisely measure the distance of microscopic scalethis fact is consistent with the ldquoHeisenberg Uncertainty Prin-ciplerdquo (In fact the connection between quantum mechanicsand fractal has been noticed in some earlier papers [14ndash17])

On the other hand the nonlocal distance (C1) betweenpoints119909

(119899)

2and119909(119899)

1depends clearly on the position of119909(119899)

119894(119894 =

1 2 2119899

) for example any displacement of point 119909(119899)

2119899

would influence the output value of nonlocal distance (C1) Itis a clearly nonlocal correlation (correlations span arbitrarilydistances) and similar to quantum entanglement

The above two facts imply that the nonlocal descriptionmay be an intrinsic way of describing quantum behavior (Inreferences [18 19] we have shown that the local descriptionis not a way of completely describing physical reality)

Acknowledgment

This project is supported by the Scholarship Award for Excel-lent Doctoral Student Granted by the Ministry of Educationof China (2012) (Grant no 0903005109081-019)

References

[1] G rsquot Hooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquoNuclear Physics B vol 44 no 1 pp 189ndash2131972

[2] K Svozil ldquoQuantum field theory on fractal spacetime a newregularisation methodrdquo Journal of Physics A vol 20 no 12 pp3861ndash3875 1987

[3] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 15 pages 2009

[4] P Horava ldquoSpectral dimension of the universe in quantumgravity at a Lifshitz pointrdquo Physical Review Letters vol 102 no16 Article ID 161301 4 pages 2009

[5] G Calcagni ldquoQuantum field theory gravity and cosmology in afractal universerdquo Journal ofHigh Energy Physics vol 2010 article120 2010

[6] G Calcagni ldquoFractal universe and quantum gravityrdquo PhysicalReview Letters vol 104 no 25 Article ID 251301 2010

[7] D Benedetti ldquoFractal properties of quantum spacetimerdquo Phys-ical Review Letters vol 102 no 11 Article ID 111303 4 pages2009

[8] V E Tarasov Fractional Dynamics Applications of FractionalCalculus to Dynamics of Particles Field and Media HigherEducation Press Beijing China 2010

[9] T C Halsey M H Jensen L P Kadanoff I Procaccia andB I Shraiman ldquoFractal measures and their singularities thecharacterization of strange setsrdquo Physical Review A vol 33 no2 pp 1141ndash1151 1986

[10] H Sagan Space-Filling Curves Springer New York NY USA1994

[11] BMandelbrot ldquoHow long is the coast of Britain Statistical self-similarity and fractional dimensionrdquo Science vol 156 no 3775pp 636ndash638 1967

[12] M Giona ldquoFractal calculus on [0 1]rdquo Chaos Solitons andFractals vol 5 no 6 pp 987ndash1000 1995

[13] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[14] M S El Naschie ldquoQuantum mechanics and the possibility of aCantorian space-timerdquo Chaos Solitons and Fractals vol 1 no5 pp 485ndash487 1991

[15] M S El Naschie ldquoA note on Heisenbergrsquos uncertainty principleand Cantorian space-timerdquo Chaos Solitons and Fractals vol 2no 4 pp 437ndash439 1992

[16] M S El Naschie ldquoOn certain infinite-dimensional Cantor setsand the Schrodinger waverdquo Chaos Solitons and Fractals vol 3no 1 pp 89ndash98 1993

[17] L Marek-Crnjac ldquoA short history of fractal-Cantorian space-timerdquo Chaos Solitons and Fractals vol 41 no 5 pp 2697ndash27052009

[18] Y Tao ldquoNecessity of integral formalismrdquo Communications inTheoretical Physics vol 56 no 4 pp 648ndash654 2011

[19] Y Tao ldquoSufficient condition for validity of quantum adiabatictheoremrdquo Communications in Theoretical Physics vol 57 no 3pp 343ndash347 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of