Z scaling, fractality, and principles of relativity in the interactions of hadrons and nucl

ˇz-TH-06/2000Zscaling,fractality,ofhadronsandprinciplesandnucleiofatrelativityhighenergies

intheinteractions

I.Zborovsk´y1

NuclearPhysicsInstitute,AcademyofSciencesoftheReˇ

ˇCzechRepublic,z,CzechRepublicAbstract

Theformationlengthofparticlesproducedintherelativisticcollisionsofhadronsand

nucleihasrelevancetofundamentalprinciplesofphysicsatsmallinteractiondistances.Therelationisexpressedbyazscalingobservedinthedifferentialcrosssectionsfortheinclusivereactionsathighenergies.Thescalingvariablereflectsthelengthoftheelementaryparticletrajectoryintermsofafractalmeasure.Characterizingthefractalapproach,wedemonstratetherelativityprinciplesinspacewithbrokenisotropy.Wederiverelativistictransformationsaccountingforasymmetryofspace-timeexpressedbydifferentanomalousfractaldimensionsoftheinteractingobjects.

1Introduction

Observationofparticleswithlargetransversemomentaproducedinhighenergycollisionsofhadronsandnucleiprovidesuniqueinformationaboutthepropertiesofquarkandgluoninteractions.Asfollowsfromnumerousstudiesinrelativisticphysics(seee.g.[1,2,3]),acommonfeatureoftheprocessesisthelocalcharacterofthehadroninteractions.Thisleadstoaconclusionaboutdimensionlessconstituentsparticipatinginthecollisions.Factthattheinteractionislocalmanifestsnaturallyinascale-invarianceoftheinteractioncrosssections.Theinvarianceisaspecialcaseofthe‘automodelity’principlewhichisanexpressionofself-similarity[4,5].Thispropertyenablestopredictandstudyvariousphenomenologicalregularitiesreflectingthepoint-likenatureoftheunderlyinginteractions.Aspecialroleplaythedeepinelasticprocesseswhichconfirmtheideaofhadronsascompositeextendedobjectswithinternaldegreesoffreedom.Inthecollisionswithnucleithestructurewasrevealedatthesub-nucleonlevel.

Theideaswereimplementedintotheformulationofthezscaling[6]fordescriptionoftheinclusiveparticleproductionathighenergies.Theconceptofself-similarityoftheinteractionswascomplementedbyconsiderationsaboutthefractalcharacteroftheobjectsundergoingthecollisions[7].Thegoalofthepaperistofocusonthegeneralpremissesofthezscalinginviewoffundamentalprinciplesofphysicsatsmallinteractiondistances.Itconcernsthescaledependenceofphysicallawsgraduallyemerginginvariousexperimentalandtheoret-icalinvestigations.Suchextensionofphysicsisintrinsicallylinkedtotheevolutionoftheconceptofspace-time.Ithasbeenproposed[8,9]thatthetopologyofspace-timebecomescomplicatedwithdecreasingscales.Intheextremecase,whenapproachingthePlankscale,thestructureofspace-timeissupposedtobeextremelyirregular(foam-like).Thestructureischaracterizedbyexplicitlyscaledependentmetricpotentials.Askingquestionsaboutthemetricsleadsonetoquestiontherelativity.Thereexistssuggestions[10]thattheconven-tionalspecial-relativisticrelationbetweenmomentumandenergymightbemodifiedreflecting‘recoil’backgroundmetricchangesduringthehighenergyinteractions.Thisinturncoulddissolvesomecutoffproblemsintheultra-relativisticregion.

Thebasicassumptiontackledinthepaperistheprincipleofrelativitywhich,besidesmotion,appliesalsotothelawsofscale.Thereisonemathematicalconceptexpressingtheself-similarityandcomplyingwiththescalerelativisticapproach,so-calledfractals[11].Thegeometricalobjectsmodeltheinternalpartonstructureofhadronsandnucleirevealedintheirinteractionsathighenergies.ThedescriptionofthepartoninteractionsasexpressedbytheconstructionofthezscalingispresentedinSec.II.SomeaspectsofthefractalityinviewofthescalingvariablearediscussedinSec.III.InSec.IV.,wepresentaformalismresultingfromapplicationoftherelativityprinciplestothespacewithbrokenisotropywhichisacharacteristicfeatureofamorefundamentalconceptoffractalspace-time.Oneoftheconsequencesoftheapproachisadispersionlawconnectingtheenergyandmomentumofaparticlehavingimplicationstothenon-standardrelationsofthequantitiestotheparticle’svelocity.InSec.V.,weapplytheresultstothekinematicvariablesdeterminedbythestartingassumptionsinthedefinitionofthescalingvariablez.Asaresultweobtaintherelationbetweenthecoefficientexpressingthespace-timeanisotropyinducedintheinteractionandtheratiooftheanomalous(fractal)dimensionsoftheinteractingfractalobjects.

2

2Zscalingandconstituentinteractions

Weconsiderthehighenergycollisionsofhadronsandnucleiasanensembleofindivid-ualinteractionsoftheirconstituents.TheconstituentscanberegardedaspartonsinthepartonmodelorquarksandgluonswhicharebuildingblocksinthetheoryofQCD.Thepresentedapproachissubstantiallybasedonapremiseaboutfractalcharacterconcerningtheparton(quarkandgluon)contentofthecompositestructuresinvolved.Theinteractionsoftheconstituentsarelocalrelativetotheresolutionwhichisafunctionofthekinematicalchar-acteristicsoftheparticlesproducedinthecollisions.Thekinematicalvariablescharacterizetheresolutionintermsoftheunderlyingprocesses.Theyaredeterminedinawayaccountingformaximalrelativenumberofinitialpartonconfigurationswhichcanleadtotheproduc-tionoftheobservedsecondaries.Inthecaseofthesingle-particleinclusiveproduction,theconstructionwaspresentedinRef.[7].Itreflectstheprinciplesoflocality,self-similarityandfractalitywhichgovernthecollisionsofhadronsandnucleiathighenergies.

Thelocalityprincipleiswidelyacceptedandfollowsfromnumerousexperimentalandtheoreticalinvestigations.Inaccordancewiththisprinciple,ithasbeensuggested[12]thatgrossfeaturesofthesingle-inclusiveparticledistributionsforthereaction

M1+M2→m1+X

(1)

canbedescribedintermsofthecorrespondingkinematicalcharacteristicsoftheinteraction

(x1M1)+(x2M2)→m1+(x1M1+x2M2+m2).

(2)

TheM1andM2aremassesofthecollidinghadrons(ornuclei)andm1isthemassoftheinclusiveparticle.Theparameterm2isusedinconnectionwithinternalconservationlaws(forisospin,baryonnumber,andstrangeness).Thex1andx2arethescale-invariantfractionsoftheincomingfour-momentaP1andP2ofthecollidingobjects.Consideringtheprocess(2)asalocalcollisionoftheconstituents,wehaveexploitedthecoefficient

Ω(x1,x2)=m2(1−x1)δ1(1−x2)δ2

(3)

whereqisthefour-momentumoftheinclusiveparticlewiththemassm1.ThefractionsresultingfromtheserequirementswerepresentedinRef.[7].Theyhavetheform

x1=λ1+χ1,

3

x2=λ2+χ2,

asaquantityproportionaltothenumberofallinitialconstituentconfigurationswhichcanleadtotheproductionoftheinclusiveparticlem1.Heremisamassconstant(itstypicalvaluebeingthenucleonmass).Thecoefficientconnectsthekinematicalcharacteristicsoftheelementaryinteractionwithglobalparametersofthecollidinghadrons(ornuclei).Itreflectsthefractalcharacterofthestructuralobjectsrevealedbytheinteraction.Theparametersδ1andδ2relatetotheiranomalous(fractal)dimensions.

Themomentumfractionsx1andx2aredeterminedinthewaytomaximizethevalueofΩ,accountingsimultaneouslyfortheminimumrecoilmasshypothesis[12]intheconstituentinteraction.Thehypothesisstatesthattheinteractionisabinarycollisionsubjectedtothecondition

(x1P1+x2P2−q)2=(x1M1+x2M2+m2)2,(4)

(5)

where

λ1=

(P2q)+M2m2

(P1P2)−M1M2

.(6)

Theχicanbeexpressedasfollows

where

χ1=

󰀁

µ22

1)

1=αλ

(1−λλ2

2

λ1λ2+λ0,

µ22+ω2

2+ω2,

(7)

(1−λ2)

α

λ2

2α

λ.5(m20=

02−m2

1)

particlem1.Therelativeuncertaintyinthedeterminationoftheunderlyingsubprocesscanbereducedtosmallersubsystems,ifthemomentumfractionsarenottoolarge.Specially,fortheinteractionsinvolvingonlysinglenucleons,oneusuallyintroducesthemomentumfractionsoftheinteractingnucleiexpressedinunitsofthenucleonmass,x¯i=Aixi.Inthesinglenucleoninteractionregime,therelativeuncertaintycanbeapproximatedasfollows

ε(x1,x2)≡(1−x¯1/A1)A1(1−x¯2/A2)A2∼(1−x¯1)(1−x¯2).

(13)

Ontheotherhand,thefactorizationdoesnotapplyfortheprocessesinwhichx¯i>1.They

areknownascumulativeprocesses[4,12,13]andcorrespondtothejoiningofpartonsfromdifferentnucleonsofnuclei.Thisregionisinterestingfromthepointofviewoffractalityatsmallscales.Herewefocusonsmallinteractiondistancesundertheconditionthatsimultane-ouslylargeamountofenergyisdepositedinit.Inthesingle-particleinclusivereactions,thecumulativeprocesscorrespondstolargerspatialresolutionsdeterminedbythedimensionsoftheinteractionregionoftheconstituentcollision.Theresolutionincreaseswiththemomen-tumtransferintheunderlyingsubprocessesandisinfluencedbythetypesofthecollidingobjects.WediscussthisconnectioninAppendixAinmoredetail.

Thesecondbasicingredientofthezscalingschemeistheself-similarityprinciple.Inhighenergyphysics,theprincipleisreflectedbythepropertythattheunderlyingproductionprocessesaresimilar.Itresultsindroppingofcertainquantitiesorparametersoutofthephysicalpictureoftheinteractions.Theparticleproductionfromself-similarprocessescanbedescribedintermsofindependentvariables.Constructionofthevariablesdependsonthetypeoftheinclusivereaction.Forsingle-particleinclusiveproductionathighenergies,thescalingfunction[7]

H(z)≡

ψ(z)

ρtotσinel

󰀇

z∂λ1

+

z∂λ2

󰀈−1

E

d3σ

m2ρ(s)

1/2

,

ε(x1,x2)≡(1−x1)A1(1−x2)A2.

(16)

Thefinitepartz0ischaracterizedbythetransversekineticenergy

/2

sˆ⊥kin=sˆλ+sˆ1χ−m1−(M1x1+M2x2+m2)

1/2

(17)

5

ofthesubprocess(2).Itincludestheterms

1/2sˆλ

=

representingthetransverseenergyoftheinclusiveparticleanditsrecoil,respectively.The

factorρ(s)≡d󰀄n󰀅/dη|η=0istheaveragemultiplicitydensityofthechargedparticlesproducedinthecentralregionofthecorrespondingnucleon-nucleoninteraction.Themultiplicitydensityρ(s)dependsonthetotalcenter-of-massenergyandincludesthedynamicalingredientofthescaling.Thedivergentpartofthevariablezisgivenasaδpoweroftheresolutionε−1.Theparameterδrepresentsanomalous(fractal)dimensionofthetrajectoriesfollowedbytheinclusiveparticlesproducedinthehighenergycollisionsofhadronsandnuclei.Weconsidertheanomalousdimensionδtobeacharacteristicreflectingtheintimatestructureofparticlemotionatsmallscalesandthusrelatedtothefractalpropertiesofspace-time.

Thecomparisonwithexperimentaldatashowsthatthezscalingrepresentsregularityvalidinnucleon-nucleonandproton-nucleuscollisionsinawiderangeofthecenter-of-massenergy,thedetectionangleθandtheatomicnumberA.Allparametersinthezscalingschemearegivenintermsofthemeasurablequantitiesexceptone,whichistheparameterδ.Restrictiononthisparameterisgivenbyexperimentaldata.Westressthatenergyandangularindependenceofthezscalingcanbeachievedsimultaneouslybythesamevalueofδ.Wehaveobtainedδ∼0.8[7]fortheinclusiveproductionofchargedparticles(pions).TheA-universalityofthezscalingwasdemonstratedwiththesamevalueaswell.Thesimultaneousenergyandangularindependenceofzscalingfortheinclusiveproductionsofjetsimplies[14]δ∼1.Itshouldbeemphasizedthattheproductionofjetsunderliethehardprocesseswithlargevaluesofz.Inthisregionthescalingfunctionmanifestaclearpowerlawψ(z)∼z−β.Similarbehaviourisseenfortheinclusiveproductionofsinglehadrons[7]inthetailofthespectrum.Thepowerlawofthescalingfunctionstressesthefractalattributesoftheprocesseswhicharepreferabletostudymainlyintheregionoflargez.Incontrasttothis,thescalingfunctionhasdifferentshapeforsmallvaluesofz.Itcorrespondstomediumandlowtransversemomentaq⊥.Theunderlyingpartoninteractionshavecharacterofthesoftprocesseshere.Thesoftinteractionsaredominantmainlyinbothfragmentationregions.Inthissense,thesoftandhardregimeoftheparticleproductionisdistinguishedbydifferentbehaviourofthescalingfunction.Thequantitativevaluesofthescalingfunctionshouldbereproducedusingatheoryofhadronicinteractions,whichwethrustisQCD.HoweverdifficultiesinapplyingtheQCDmethodsinthenon-perturbativeregimeandthelackoffundamentalunderstandingofhadronizationeveninanperturbativeapproximation,limitsustothephenomenology.Inourdescriptionoftheinclusivereactions,weaimatgraspingmainprinciplesthatinfluencetheparticleproductionatsmallscales.Weunderstandtheexistenceofthezscalingitselfasconfirmationofthehadroninteractionlocality,self-similarity,andfractalitywhichpossessanuniversalcharacter.

󰀁

(χ1P1+χ2P2)2,(18)

3Fractalityatsmalldistances

Besidestheself-similarityprinciple,theconstructionofthezscalingregardsalsothefrac-talityofthehadroninteractions.Thefractalpropertiesaremanifestedespeciallyatsmallinteractiondistances.Inthisregion,physicsbecomesscaledependentandpossessesitstypi-calpropertywhichisthedivergenceofelementaryquantitiessuchastheself-energy,charges

6

andsoon.Inthehighenergycollisions,oneoftherelevantphysicalquantitiesisthefor-mationlengthoftheproducedparticlesdivergentatsmallscales.Accordingtothezscalinghypothesis,theformationlengthisproportionaltothescalingvariablezandtheproductioncrosssectionsdependonitinanuniversal,energyindependentway[14,15,16].

Universalityplaysthecrucialrole.Thereexistsuggestions[17]thatuniversalpropertiesofmatterareattributedtothestructureofspace-timeitself.Inthetheoryofrelativitybothspecialandgeneral,thisconcernstheLorenztransformationandthecurvatureofthetrajec-toriesreflectingfundamentalprinciplesofphysics.Freeparticlesaremovingalongsmoothgeodesicallines,characteristicfortheclassical(curved)space-time.Thesituationbecomesdifferentatscalestypicalforthequantumworldoftheelementaryparticles.Thecommonpropertyistheunpredictabilityofmotionatsmalldistances.Inthisregiontheparticlesfollowirregulartrajectorieswhichbecomenon-differentiable.Thegeometryofthelinesofmotioncanbeattributedtothepropertiesoffractalswhichareextremelyirregularobjectsfragmentedatallscales.Asanexampleonecanmentionthequantum-mechanicalpathofaparticleinthesenseofFeynmantrajectories[18].

Oneofthemaincharacteristicsoffractalsisthedivergenceoftheirmeasuresintermsoftheincreasingresolution.WeillustratethispropertybythevonKochcurve[19]whichischaracterizedbythefractalmeasure

zε=z0·εDT−D

(19)

representingitslength.Thelengthofthecurveisafunctionoftheresolutionε−1(seeFig.1.).Relation(19)istypicalforvariousfractalsandstateshowthefractalmeasuredependsontheresolution.FractalobjectsarecharacterizedbythetopologicaldimensionsDTwhichareintegerquantitiesandbythefractaldimensionsDacquiringgenerallynon-integervalues.Theanomalousdimensionofafractal

δ≡D−DT>0(20)ispositive.Itisequivalenttopowerlawdivergenceofthemeasurezεwiththeincreasing

resolution.VonKochcurveilluminatestheprocessof‘fractalization’.Thecurvehasthelength

zn=z0(p/q)n,p=4,q=3(21)inthen-thapproximation.Itiscomposedofpnsegments,eachofthelengthz0q−n.The

measurecanberewrittentotheform

zn=z0(q−n)1−lnp/lnq,

(22)

whichgivestherelation(19)withDT=1,D=lnp/lnq,andε=q−n.TheanomalousdimensionofthevonKochcurveδispositiveandthus,withincreasingresolution,itslengthtendstoinfinity.

Theseconceptsfindapplicationintheworldofphysicsatsmalldistances.Thefractalcharacterintheinitialstatereflectstheparton(quarkandgluon)compositionofhadronsandnucleiandrevealsitselfwithalargerresolutionathighenergies.Accordingtothispicture,thecollisionsoftheconstituentstakeplaceonthefractalbackgroundoftheinteractingobjects.Theessentialassumptionconcerningtheinterpretationoftheideasrepresentedbythezscalinghypothesisisexpressedinthefollowingstatement:Presenceoftheinteractingfractalobjectsdeformsthestructureofsurroundingspaceatsmalldistances.Asaconsequence,space-time

7

becomeslocallynon-differentiable,fractalwithgeodesicallinesacquiringanextremelyirregularscale-dependentshape.Thenotionoffractalspace-timewasusedinRefs.[17,20]anditspropertieshavebeenstudiedbyothers[22].Distortionofthesurroundingspace-timeintheinteractionswithrecoilinduceanon-trivialoff-diagonaltermsinthemetricchangingtherelationsbetweenenergyandmomentum[10].Theissuesconcernmodellingofvariousaspectsofquantumfluctuationswhichinfluenceparticleproductionandtheirinteractionsatsmallscales.

Thesecondarypartons,producedinfractalspace-timefollowerraticandscale-dependentgeodesicsstartingfromtheregionstheyhavebeencreated.Duringtheinitialphaseofthemotion,onecanimaginethetrajectoriesasfractalcurvessimilartothevonKochcurvedepictedinFig.1.Formationofaparticlefromthebarepartonrealizesalongthetrajectorycharacterizedbyitslength.Theproducedparton,ancestorofthesecondaryparticle,interactswiththevacuumorthesurroundingmatterfieldacquiringsimultaneouslysometypeoftheparton‘coat’.Thenumberoftheenvelopingpartonsformingthecoatresultsinaneffectiveincreaseoftheancestor’sdimensionssimultaneouslychangingitsmass.Duringtheparticleformationprocess,thepathoftheleadingpartonbecomesgraduallysmootherincomparisontothatfollowingimmediatelytheinstantoftheconstituentinteraction.Inthefinalstageoftheprocess,therelativelengthoftheparticle’spathisnegligiblewithrespecttoitsvalueintheverybeginningofthemotion.Itisaconsequenceofthetransitionfromsmallscales,characterizedbyanextremelyirregularfractal-likecharacterofthetrajectory,toscaleslargerthan,say,thecorrespondingdeBroglielength.Thus,inthecollisionsofthestructuralobjectssuchashadronsornuclei,thelengthofthetrajectoryofaproducedparticlecanincreaseinfinitelywiththedecreasingscaleatwhichtheparticlewascreated.

Letuslookattheprobleminviewofthemodelsdescribingtheinteractionofthecon-stituentsasaparton-partoncollisionwithsubsequentformationofastringstretchedbetweenthetwopartsinthefinalstateoftheprocess(2).Followingthesamegeodesicaltrajectory,thestringisafractalobjectwiththescaledependentproperties.Inthispicture,thevalueofΩ(3)canbeconsideredasaquantityreflectingthetensionofthestring.Thestringtensioncoefficientischaracterizedbythedensityofthestraight-likesegmentsalongitslength.Forallresolutions,theproductofthecoefficientandthelengthofthestringzεrepresentsthefinitequantity

1/2sˆh=Ω·zε,(23)whichistheenergyofthestring.Theformofthetensioncoefficient,aspresentedbyEq.

(3),accountsfortheshapeofthestringdeformedinconsequenceofthefractalstructureofspace-time.Withtheincreasingresolution,thestringismoreandmorefragmentedandthedeformationsresultinthediminishingofitstension.Physically,thedeformationsofthestringcanbecausedbyfluctuationsoftheQCDvacuumdisturbedinpresenceoftheinteractingfractalobjectsatsmalldistances.Thestringtension

Ω∼εδ

(24)

isafunctionoftheresolutionε−1andischaracterizedbytheanomalous(fractal)dimensionδofthestring.Theresolutioncorrespondstoacharacteristicsizeofthespace-timeregionoftheinteractionanddependsonthemomentumfractionscarriedbytheinteractingconstituents.Inthedeterminationofthefractions,weconsideranoptimizationmethoddealingwiththefractaltrajectoriesofparticlesatanyscale.Theoptimaltrajectoryisdefinedbytheconditionthat,

8

fortheunderlyingcollisionoftheconstituents,thetensionofthefractal-likestring,stretchedoutbytheproducedparton,shouldbymaximal.Thisisequivalenttotheextremumoftheexpression(3)underthecondition(4).Themaximaltensionofthestringisthusgivenbytheminimalvalueoftheresolutionandcorrespondstothegeodesicswhichareinasense‘optimal’curves.

Theenergyofthestringconnectingthetwoobjectsinthefinalstateoftheprocess(2)isgivenbytheenergyofthecollidingconstituents.Thestringevolvesfurtherandsplitsintopieces.Theresultantnumberofthestringpiecesisproportionaltothenumberordensityofthefinalhadronsmeasuredinexperiment.Asknownfromvariousexperimentalandtheoret-icalstudiesconcerningthemultipleproduction,theproducedmultiplicityisproportionaltotheexcitationoftransversedegreesoffreedom.Therefore,thestringtransverseenergyisameasureofmultiplicity.Suchideasallowustointerprettheratio

sˆh≡sˆ⊥kin/ρ(s)

1/2

1/2

(25)

asaquantityproportionaltotheenergyofthestringpiece,whichdoesnotsplitalready,

butduringtheformationprocessconvertsintotheobservedsecondary.Thestringsplittingisself-similarinthesensethattheleadingpieceofthestringforgetsthestringhistoryanditsformationdoesnotdependonthenumberandbehaviourofotherpieces.Wewritetheenergyofthesinglepieceofthestringintheform(23).Thisisequivalenttothedefinitionofthevariablez≡zεasafractalmeasureproportionaltothelengthofthestringpiece,ortheformationlength,onwhichtheinclusiveparticleisformed.ThecorrespondingscalingfunctionH(z)reflectstheevolutionoftheformationprocessoftheinclusiveparticlealongitsfractal-liketrajectoryofthelengthz.WeaddherethatthereexistsalsocomplementaryinterpretationofthefactorΩ.AccordingtotheideaspresentedinRef.[7],Ωreflectstherelativenumberofallinitialconfigurationscontainingtheconstituentswhichcarrythemomentumfractionsx1andx2.Thenumberoftheconfigurationsinonecollidingobjectisgivenbythepowerlawcharacteristicforfractals.Infractaldynamicstheresolutionε−1isgivenbythemaximalnumberoftheinitialconfigurationswhichcanleadtotheproductionoftheparticlem1.Generally,thefractalapproachtothehighenergycollisionsofhadronsandnucleineedsmoreprofoundunderstanding.Itconcernsthedeformationofspace-timeatsmallscalesandattributesadditionalmeaningtothephysicalquantitiessuchasthemomentum,mass,energyorvelocity.Theymaybedefinedfromparametersofthefractalobjectsintermsofthefractalgeometry[17].Thisincludesextensionoftherelativityprinciplestotherelativityofscalesaswellastothemorecomprehensivescale-motionrelativisticconcepts.

4

Breakdownofthereflectioninvariance,thewayto-wardsscale-motionrelativity

GeneralsolutiontothetheoryofthespecialrelativityistheLorenztransformation.AsdemonstratedbyNottale,itcanbeobtainedunderminimalnumberofthreesuccessivecon-straints.Theyare(i)homogeneityofspace-timetranslatedasthelinearityofthetransfor-mation,(ii)thegroupstructuredefinedbytheinternalcompositionlawand(iii)isotropyofspace-timeexpressedasthereflectioninvariance.Letusconsidertherelativisticboostalongthex-axis.Thetransformationconcernsthevariablesxandtwhichrefernotonlytotheco-ordinateandtime,butalsotoanyquantitieshavingthemathematicalpropertiesconsidered.

9

Withoutanylossofgenerality,thelinearityofthetransformationcanbeexpressedintheform[21]

x′=γ(u)[x−ut],(26)

t′=γ(u)[A(u)t−B(u)x],

(27)

whereγ,A,andBarefunctionsofaparameteru.Theparameterrepresentsusualvelocity

(inunitsofthevelocityoflightc)inthemotionrelativityorthe‘scalevelocity’used,e.g.,intheconceptofthescalerelativityconcerningfractaldimensionsandfractalmeasures[17].Letuscomposethetransformationwiththesuccessiveone

x′′=γ(v′)[x′−v′t′],

(28)t′′=γ(v′)[A(v′)t′−B(v′)x′].

(29)

Theresultcanbewrittenintheform

x′′

=γ(u)γ(v′

)[1+B(u)v′

]󰀉

x−

u+A(u)v′

A(u)A(v′)+B(v′)u

x󰀊

.(31)

Theprincipleofrelativityisexpressedbytheconstraint(ii)whichisthegroupstructureofthetransformations.TheconditiontellsusthatEqs.(30)and(31)keepthesameformastheinitialonesintermsofthecomposedvelocity

v=

u+A(u)v′

(v′)B(u)+B(v′)

A(v)

=

Astructureswhicharefractalsofvariousfractaldimensions.ThecorrespondingLorenz-typescaletransformationsrelatethephysicalquantitiesexpressedintermsofonefractalstructurewiththequantitiesgivenwithrespecttotheotherone.Singlefractalstructureshavedifferentanomalousfractaldimensionsandplayanalogousroleastheinertialsystemsinthemotionrelativity.

Thesecondapproach,thescale-motionrelativity,preservesrelationsoftheenergyandmomentumtothevelocity.Thisappliesalsotosmallscaleswhereweassumespace-timetopossessanintrinsic(fractal-like)structure.Ourcompletechangeofviewofaparticleinthecorrespondingfractalspace-timeconcernsthedivergenceofthefractalmeasurerepresentingthelengthoftheparticle’strajectory.Thedegreeofrevelationofthestructuresdependsontheresolution.Foranygivenresolutionε−1,thenon-differentiablefractalspace-timeFcanbeapproximatedbyaRiemannspaceRεdefinedwithinadifferentiablegeometry.AspointedinRef.[17],thefamilyoftheRiemannspacesischaracterizedbymetrictensorscurvaturesofwhichareexpectedtofluctuateinachaoticway.Thefluctuationsincreasewiththedecreasingscale.Forthehighresolutionsε−1theapproximationstothefractalmeasuretendtoinfinity.Thecorrespondinglengthzεoftheparticletrajectorycanbearbitrarylarge.Thepropagationofaphysicalsignalalongsuchatrajectoryrequiresthevelocitiesexceedingthevalueofc,thespeedoflightinsymmetricspace-time.Therefore,theapplicationoftheprinciplesofrelativitytospace-timewithfractalpropertiesshouldbetreatedcarefully.Thesignificantcharacteristicsofthefractalspacesistheirfluctuatingandirregularnature.Thecorrespondinggeodesicallinesareextremelyunpredictableandfragmentedatanyscale.Asaconsequence,theisotropyofspace-timeisclearlybroken.Thisisconnectedwiththebreakingofthereflectioninvarianceattheinfinitesimallevel[22].Theapplicationoftheideastospace-timeatsmallscalesleedsustoleaveouttheconstraint(iii)ofthereflectioninvariance,whenconsideringtherelativistictransformations(26)and(27).Inthatcasetheunknownfunctionsγ,A,andBdonotobeytheparityrelationsresultingfromtheisotropyrequirement.LetuscombineEqs.(32),(33),and(34)intotheexpression

A

Itssolutionhastheform

A(u)=1+2au,

(37)

providedB(u)v′=B(v′)u.TheconditiongivesthefunctionB(u)=uwiththenormalizationconstantcincludedalreadyinthedefinitionofthevariableu.ThesolutionsatisfiesEq.(35)aswell.Theviolationofthespace-timereflectioninvarianceisexpressedbyanon-zerovalueofa.Intermsoftheparametera,thecomposedvelocity(32)canbewrittenasfollows

v=

v′+u+2auv′

󰀇

u+A(u)v′

1+B(u)v′

.(36)

1+uv′

󰀈

=γ(u)γ(v′)(1+uv′).

(39)

Itssolution,whichfora=0isgivenbythestandardγfactor,hastheform

γ(u)=

1

.

1+2au−u211

(40)

4.1Space-timeasymmetryin3+1dimensions

LetusdescribeapointPintwoCartesianreferencesystemsSandS′.WeassumethatthesystemsareorientedparalleltoeachotherandthatS′ismovingrelativetoSwiththevelocityuinthedirectionofthepositivex-axis.Wesupposethattheasymmetryexpressedbytheparameteraisparalleltothevelocityu.Therelativistictransformationsofthecoordinatesandtimearegivenby

x′1=γ(u)[x1−ut],

Theinverserelations

x1=γ(u)[(1+2au)x′1+ut′],

xi=x′i,

i=2,3,

(43)(44)

x′i=xi,

i=2,3,

(41)(42)

t′=γ(u)[(1+2au)t−ux1].

t=γ(u)[t′+ux′1]

areobtainedasthesolutionofEqs.(41)and(42)withrespecttotheunprimedvariables.Theycanbealsoderivedfromtheequationsbytheinterchange󰁃x↔󰁃x′,t↔t′,u↔u′,andbytherelation

u

u′=−

and

referenceframe.Inconnectionwiththetransformationformulae,itisconvenienttointroducethenotations

1

(49)γ=

222(1+󰁃a·󰁃u)−(1+a)u

g=

(1+󰁃a·󰁃u)γ−1

.1+2󰁃a·󰁃u

Accordingtothesubstitution,thereexistthesymmetryproperties

γ(󰁃u′)=(1+2󰁃a·󰁃u)γ(󰁃u),g(󰁃u′)=(1+2󰁃a·󰁃u)2g(󰁃u),

Exploitingtheproperties,theinverserelations

󰁃x=󰁃x′+󰁃u[γ(t′+󰁃a·󰁃x′)+g󰁃u·󰁃x′],t=t′+[γ−(t′+󰁃a·󰁃x′)+g−󰁃u·󰁃x′]

γ±(󰁃u′)=γ∓(󰁃u),g±(󰁃u′)=(1+2󰁃a·󰁃u)g∓(󰁃u).

(54)

(55)(56)

(57)(58)

withrespecttoEqs.(52)and(53)followimmediately.Weexpresstherelativistictransfor-mationsinamorecompactform

x′=D(󰁃u)x,(59)where

D(󰁃u)=

Theinversematrixreads

󰀇

δij+guiuj−γuiaj−γui−g+uj+γ+aj1+γ+

󰀇

󰀈

.

󰀈

(60)

D(󰁃u′)=D−1(󰁃u)=

δij+guiuj+γuiaj+γui

+g−uj+γ−aj1+γ−

.(61)

Thetransformationmatricescanbedecomposedintotheproduct

1󰁃)Ax(󰁃D(󰁃u)=A−a)Λ(βa).x(󰁃

(62)

13

Here

Ax(󰁃a)=

󰀇

√

√

ThematrixΛdependsonthevector

󰁃≡β󰁃uβ󰀥=

√

β2

.(65)

1+󰁃a·󰁃u

.(66)

󰁃↔−β󰁃.TheLetusnoticethattheinterchange󰁃u↔󰁃u′isequivalenttothesymmetryβ

relativistictransformations(59)preservetheinvariant(47).Thisfollowsfromtherelation

D†(󰁃u)ˆaD(󰁃u)=aˆ=A†xηAx,

(67)

whereηstandsforthediagonalmatrixη=diag(-1,-1,-1,+1).

Thetransformationscomplytheprincipleofrelativity.Mathematicallyitisexpressedbytheirgroupproperties.LetD(󰁃u)andD(󰁃v′)betwosuccessiverelativistictransformationsrepresentedbythematrices(60).Thecompositionofthetransformationshastheform

󰁃)D(󰁃Ωx(φv)=D(󰁃v′)D(󰁃u),

provided

󰁃v=

󰁃v′+󰁃u[γ(1+󰁃a·󰁃v′)+g󰁃u·󰁃v′]

(68)

providedtheasymmetryofspace-timeisexpressedbythevector󰁃a.AsconcernsEq.(69),it

canbeobtainedfromtheusualrelativisticcompositionofthefactorsβ

󰁃givenbyEq.(66).Theinverserelation

󰁃v′=

󰁃v−󰁃u[γ(1+󰁃a·󰁃v)−g󰁃u·󰁃v]1+γ−(1+󰁃a·󰁃v′)+g,(76)

−󰁃u·󰁃v′

1+󰁃a·󰁃v′

=γ

(1+󰁃a·󰁃u)(1+󰁃a·󰁃v)−(1+a2)󰁃u·󰁃v

1+2au−uv,

v′

(81)

1

i=v√

i

1+2au−uv,i=2,3.1

Theinverserelationscanbeobtainedbytheinterchange󰁃v↔󰁃v′andu↔u′.UsingEq.(45),

theycanbewrittenasfollows

vv′+u+2auv1=

1′1

1+2au−u2

reliesonthescaleswearedealingwith.Fortheinfiniteresolutionitshouldbeaperfectpointwhosetrajectoryisafractalcurve.Foranarbitrarysmallbutstillfiniteresolutionε−1theperfectpointisapproximatedbyaparticlewhichwecall‘elementary’withrespecttothisresolution.Itisthereforenaturaltoassumethattheconceptsofthemomentum,energy,massandthevelocityofthe‘elementary’particlehavegoodphysicalmeaningalsoatthescaleswherespace-timeisexpectedtobreakdownitsisotropy.

󰁃Wedenotethevaluesofthemomentumandtheenergyoftheelementaryparticleby(P

󰁃′andE′)inthereferencesystemsSorS′,respectively.InconsistencewiththeandE)or(P

principleofrelativityandtheideaspresentedabove,wesearchforrelationsconnectingthesequantities.Inordertodothat,letusfirstdefineassociativevariablesπµ={󰁃π,π0}withthefollowingproperty.ThecomponentsofthevariablesdeterminedrelativetothesystemsSandS′transformintheway

π′=Π(󰁃u)π,(83)where

Π(󰁃u)=

󰀇󰀇

Theinversetransformationπ=Π−1(󰁃u)π′isgivenbythematrix

Π−1(󰁃u)=

δij+guiuj−γaiuj−g+ui+γ+ai−γuj1+γ+

󰀈

.(84)

δij+guiuj+γaiuj+g−ui+γ−ai+γuj1+γ−

󰀈

.(85)

ThereexistsmutualcorrespondencebetweenthetransformationmatricesΠandDgivenby

thematrixtransposition

Π(󰁃u)=D†(󰁃u).(86)Accordingtotherelation,thematrixΠcanbeexpressedintheform

1󰁃)Aπ(󰁃Π(󰁃u)=A−a)Λ(βa)π(󰁃

(87)(88)

where

1A−a)=A†a).π(󰁃x(󰁃

Thegrouppropertiesofthetransformations(83)aredeterminedwithrespecttothecompo-sition

󰁃)Π(󰁃Ωπ(φv)=Π(󰁃v′)Π(󰁃u),(89)providedthevelocities󰁃u,󰁃v′,and󰁃vsatisfytherelation(69).Here

󰁃)=A−1R(φ󰁃)Aπ.Ωπ(φπ

(90)

WeshowthatEqs.(68)and(89)areconsistentwithrelation(86).LetustransposeEq.(68).

Exploitingthecorrespondence(86)andusingEqs.(72),(88),and(90),wecanwrite

󰁃)=Π(󰁃󰁃Π(󰁃v)Ωπ(−φv)Ω†u)Π(󰁃v′).x(φ)=Π(󰁃

(91)

WeapplythetranspositionoperationonEq.(70)too.AsthematricesΛareinvariantunder

󰁃uandβ󰁃v′inthemutualreversetheoperation,weobtainthecompositionoftheparametersβ

16

order.Fromthesymmetryreasons,thecompositionmustbeofthesameformasEq.(70).Wehavetherefore

R(−󰁃φ)Λ(β󰁃w)=Λ(β󰁃v)R(−φ󰁃)=Λ(β󰁃u)Λ(β󰁃v′).(92)Thevectorβ󰁃wcorrespondstothevelocityw󰁃accordingtoEq.(66).Thevelocityisgivenby

theformula(69)inwhichthevelocities󰁃uand󰁃v′aremutuallyinterchanged.MultiplyingEq.

(92)bytheA−π1

fromtheleftandbytheAπfromtheright,weget

Ωπ(−󰁃φ)Π(w󰁃)=Π(󰁃v)Ωπ(−󰁃φ

).(93)

TogetherwithEq.(91)onehas

Ωπ(−󰁃φ)Π(w󰁃)=Π(󰁃u)Π(󰁃v′).

(94)Afterperformingtheinterchange󰁃u↔󰁃v′,weobtainEq.(89).Itwasthusshownthatthecom-positionoftwosuccessivetransformationsofthevariablesπfollowsfromthecompositionofthecorrespondingtransformationsofthecoordinatesandtime,providedtheirtransformationmatricesareconnectedbytherelation(86).

Unlikethetransformationsofthecoordinatesandtime,theinvariantcombination

π20−󰁃π2+2π0󰁃a·󰁃π

(95)

constructedfromthevariablesπdoesnotcorrespondtothemetrics(48).Inordertoremove

thisdefectwehavetodeterminethe4-momentumoffreeparticlebymeansofnewvariables.Thetransformationsofsuchvariablesshouldpreservethesamemetricinvariantasthetrans-formationsoftheirkinematicalcounterparts,thecoordinatesandtime.Weshowthatthere

existstwosetsofthevariablespµs={P󰁃s,E},s=L,Rdefinedbytherelation

π=As(󰁃a)ps,

(96)

with

As(󰁃a)=

󰀇

δij±εijkak0

01

󰀈

,(97)

whichcomplytherequirement.HereεijkistheLevi-Civitasymbol.Theplus(inthenext

everyupper)signandtheminus(inthenexteverylower)signcorrespondstos=Lands=R,respectively.Wewillregardthevariablespµspace-timecharacterizedbytheasymmetrysasthe4-momentumofanelementaryparticlein󰁃a.Weattributethefirstsetofthevariables(s=L)totheparticlewhichwecallleft-handed.Thesecondset(s=R)correspondsparticlerevealingright-handedtypeofmotion.TherelationbetweenthemomentaP

󰁃tothe

sandtheaboveconsideredvariable󰁃πreads

󰁃π=P󰁃s±P󰁃s×󰁃a,

P

󰁃π±󰁃a×󰁃π+(󰁃a·󰁃π)󰁃as=󰁃Thetransformationsofthevariablespreservetheinvariant

2󰁃2+2ξ0󰁃󰁃s.ξ0−ξa·ξs

(100)

󰁃s/dξ0.Theyarerelatedtothevelocities󰁃󰁃s=dξLetusintroducetheparametersUuasfollows

󰁃s=󰁃Uu∓󰁃u×󰁃a,

󰁃u=

󰁃s∓󰁃󰁃s+(󰁃󰁃s)󰁃Ua×Ua·Ua

,1+a2

G±=

g±

,(105)󰁃1+2󰁃a·U

whichtogetherwiththesymmetryproperties(55)and(56)determinetheinversematrix

󰁃)=∆(U󰁃′)=∆−1(U

󰀇

δij+GUiUj+G+aiUjGUai+G−Ui

+γUj1+γ−

2

󰀈

.(106)

Thetransformationmatrixescanbewrittenintheway

󰁃󰁃s)=A−1(󰁃∆(Ua),psa)Λ(β)Aps(󰁃

where

Aps(󰁃a)=Aπ(󰁃a)As(󰁃a)=

11+a2

󰀇

(107)

δij±εijkak√−ai

0

1+γ−+

󰁃′GU2󰁃a·V󰁃·V󰁃′+G−U

.(110)

Formula(109)isconsequenceofEqs.(70)and(107).ThematrixΩphasthestructure

󰁃)≡Ωps(φ󰁃)=A−1R(φ󰁃)Aps=A−1Ωπ(φ󰁃)As.Ωp(φpss

18

(111)

TheinverserelationtoEq.(110)reads

󰁃′=V

󰁃−U󰁃γ+G−󰁃󰁃−GU󰁃·V󰁃Va·V

󰀄

󰀆

󰁃2+m2󰁃(1+a2)Pa·P.0−󰁃

(117)

Wewillnotconsiderherethesolutionwithminussignbeforethesquarerootcorresponding

󰁃.Ithasasingletoanti-particles.Theenergy(117)ispositiveforarbitraryvaluesof󰁃aandP

minimumforthemomentumandenergy

󰁃0=M0󰁃Pa,

19

󰁃0)=M0.E(P

(118)

ThemassM0(theminimalenergy)dependsontheasymmetryparameter󰁃abytherelation

M0=

m0

.1+a2

(119)

Beyondtheminimum,asthemomentumincreases,theenergytendstoinfinity.Itconsistsoftwoterms.Thefirsttermisthefreeenergy

E=

󰀁

1+a2(121)

󰁃0)andtheparameter󰁃isexpressedintermsoftheminimalenergyEmin=E(Pa.Weconjecture

thatsimilarconsiderationsconcernalsootherintrinsiccharacteristicsoftheparticles,suchasspinandcharge.Onecanconsiderthephysicalquantitiesasrelatedtothegeometricalstructuresofparticletrajectoriesinthefractalspace-time.Weanticipatethatspinofaparticlemaybeconnectedtospecialerraticcharacteroftheleft-handedorright-handedfractal-liketrajectoryatsmallscales.Inthedomain,wherethefractalattributesofthemotionexpire,thevalueof󰁃adiminishesandthefractaldynamicswillconvertintotherelativisticdynamicsinsmoothspace.

Wemakesomecommentsontheenergymomentumconservation.Letusconsideraclosedsystemwiththemassm0which󰀁splitsintotwoparts.Thedecayisgovernedbytheenergy

∗∗∗2

momentumconservation,m0=󰁃q2+m2q1=−󰁃q2,asdescribedinthe2and󰁃

systemrestframe.Thesimilarisvalidinspace-timewithbrokenisotropy.Denotingtheenergymomentumfour-vectorsofthedecayproductsbyp1andp2,onecanwrite

222

m20=(p1+p2)=m1+m2+2p1p2

󰀆󰀆󰀄󰀄

222222󰁃󰁃󰁃󰁃󰁃󰁃a·P2−(󰁃a×P2)=E1−P1+2E1󰁃a·P1−(󰁃a×P1)+E2−P2+2E2󰁃

22󰁃󰁃󰁃󰁃󰁃󰁃=(E1+E2)−(P1+P2)+2(E1+E2)󰁃a·(P1+P2)−󰁃a×(P1+P2).

2

󰁃1·P󰁃2+E1󰁃󰁃2+E2󰁃󰁃1−(󰁃󰁃1)(󰁃󰁃2)+2E1E2−Pa·Pa·Pa×Pa×P

󰀄

󰀄󰀆

Weseethatifthefourvectorsp1andp2arecharacterizedbytheinvariant(47),theirsum

p1+p2possessesthispropertytoo.Thisimpliestheconservationofthetotalenergyand

20

󰀆

(122)

󰁃=P󰁃1+P󰁃2,whichresultsintheconservationofthefreeenergymomentum,E=E1+E2andP

󰀁

22󰁃1P+M1+

󰀁

√

√

calculation,theformulaeareconsistentwiththeproportionalitybetweenthemomentumP󰁃theinvariant(47)and(115).Thecoefficientof

sandthevelocity󰁃visdenotedbythesymbol

Mandrepresentstheinertialmassoftheparticle.Theinertialmassdependsonthevelocityintheway

M(󰁃v)=M0γ(󰁃v).(130)TheM0istherestmassofthe‘elementary’particlegivenbyEq.(119).Therestmass

correspondstotheminimalenergy(118).

LetusnowderivetheinverseexpressionwithrespecttoEq.(128).Besidestheinvariant(47)and(115),onecanconstructtheinvariantrelation

(A†xηAps)µνxµpν=tE−x󰁃

·P󰁃s+2󰁃a·󰁃xE∓󰁃a·(󰁃x×P󰁃s)=τM0.(131)

Itrepresentstheequationoftheelementaryparticletrajectoryexpressedintermsofitsmo-mentum,energy,anditsmassM0.Thesolutionoftheequationis󰁃x=󰁃vt,where

󰁃v=

P󰁃s±P󰁃s×󰁃a−󰁃aE

1+a2

.(134)

Weseethatforthezerovalueofthemomentumthereexiststhenon-zerovalueofthevelocity

󰁃v0=−

󰁃a

√

thefactorγ.AsshownintheAppendixB,itcorrespondstotheminimallengthcontractionsandtotheminimaldilatationsoftime.

Ingeneral,forarbitraryenergyofaparticle,wehaveshownthefollowingresult.Itconsistsoftheclaimthatinspace-timewithbrokenisotropythemomentumoftheparticleisnotpar-alleltoitsvelocity.Approximatingthefractalspace-timebyafamilyofthespacesRεwithdifferentiablegeometry,thevelocityfluctuateswithrespecttothemomentumindependenceonthestochasticnatureoftheanisotropyparameter󰁃a.Accordingtothefluctuations,the‘point-like’particlemovesarounditsmomentumpassinganunpredictableandchaotictrajec-torycharacteristicforfractals.Independenceonthefluctuatinganisotropy󰁃a,thevelocityoftheparticlecanbearbitrarylarge.Thisisconnectedwithapossibilityofpropagationofphysicalsignalswithvelocitiesexceedingthespeedoflightintheisotropicspace-time.Thepropertyis,however,compensatedbytheextremeirregularandrandomshapeofthetrajec-toriesalongwhichthesignalismediated.Westressherethatthestatementsarerelativeanddependonthescaleoftheobserveraswell.When‘measuring’thefractalpropertiesoftheparticlemotion,theobserverexpressesthemintermsofitsownfractalcharacteristicsbeingafractalitself.Thisistypicalfortheparameter󰁃awhichisafunctionofthescalestructuresofboththeobservedparticleandtheobserver(seesectionV.).Accordingtoouropinion,theparametercouldhaverelevancetomoredeepercontextofthemetricpotentialswhichhaverelationtotheintimatestructureofspace-time.Itmaybeconnectedwitha‘fieldofthespace-timeasymmetry’reflectingthestructureatsmallscales.Existenceofthe‘field’wouldresultintoadisparitybetweentheenergy-momentumandthecoordinatesandtime.Herethedisparityisdemonstratedbythefollowingcommutationrelation

†

A†psηAx−AxηAps=

󰀇

±ǫijk2ak−2ai

2aj0

󰀈

.(137)

Thecommutatorisnon-zeroprovidedthenon-zerovalueofthefield.Inthepresentpaper

weapproximatethefieldofthespace-timeasymmetryintermsoftheanisotropyvector󰁃aandconsideritasarandomandchaoticquantity.AsshowninsectionV.,theanisotropyhasrelevancetotheanomalousexcessofthetopologicaldimensions.Theinvestigationsinthisdirectionrequire,however,moredetailedandfundamentalstudy.

Theideastackledinthissectionconcerngeodesicreferencesystemsintheimmediatesur-roundingsofagivenpointPinthe4-space.Thesurroundingsdependontheresolutionwearedealingwith.Onecanintroducesuchsystemsintheproximityofeverypointofthegeodesicallines.Accordingtotheideasaboutfractalpropertiesofspace-timeatsmallscales,wecharac-terizethegeodesicsystemsofinertiabythemetrictensorsaˆ.Themetricsreflectssignificantpropertyofthefractalstructureofspace-timewhichisbreakingitsisotropy.Thestructureisrevealedindependenceontheleveloftheresolution.Foragivenresolution,itispossibletotransformawaytheanisotropyofthespace-timelocally,exploitingnewpseudo-Cartesian

󰁃K0}.Wecanintroducethevariablesinthewaycoordinatesrµ={󰁃r,r0}andkµ={K,

r=Axx,

Theexplicitformoftheequationsreads√󰁃r=

√

k=Apsps.

(138)

Unlikethexandpthepseudo-Cartesianvariablesrandkarefunctionsoftheanisotropy󰁃a.Usingthevariables,onecanwritethecorrespondingrelativisticinvariantintheform

2r0(󰁃a)−󰁃r2(a)=τ2,

22󰁃2(󰁃K0−Ka)=M0(a).

(141)

Thespace-timeanisotropyisthusremovablelocallybutcannotberemovedcompletely,i.e.

simultaneouslyforeverypointofthe4-space.Hence,weconsidertheanisotropyatsmallscalestobetheintrinsicpropertyofspace-timeitself.Itsadequatedescriptionassumesapproacheswithinafractalgeometry.

5Interactionsofasymmetricfractalsystems

Theabilityoffractalstostructurespace-timewasdiscussedinRef.[17].Suchapproachgivesuspossibilitytoattributegeometricalnotionstothestructuralparameterscharacter-izingfractaltrajectoriesoffreeparticles.Weconsideroneoftheparameterstobethescaledependentcoefficient󰁃areflectingbreakingofthespace-timeisotropy.Thequantityisassumedtohavestochasticandirregularnaturerepresentingthefractalpropertiesofthestructuresatsmalldistances.Thenaturalquestionariseswhetheronecanorganizearegioninwhichthestructurescouldbesomehoworiented.Weanswerthequestionpositivelyandarguethatsuchregioncouldbecreatedintheinteractionsofhadronsandnuclei.Thisconcernshighenergieswheretheobjectsrevealfractalcompositionintermsofthepartoncontentinvolved.Thefractalityresultsfromnonexistenceoflowercutoffatwhichthestructureswouldstop.Weconjecturethattheinteractionsofthefractalobjectsaffectthecharacterofspace-timeatsmallscales.Onecanimaginethatthechaoticcharacterofthespace-timeanisotropycanbeorientedandspace-time‘polarized’bytheinteractionsoffractalspossessingmutuallydifferentanomalousdimensions.Inotherwords,weconjecturethattheinteractionsoftheasymmetricfractalsystemsresultinpolarizationofthe(QCD)vacuum.Thevacuumfluctuationsbecomeorientedformingaregionofthespace-timeasymmetry.Wedenotetheasymmetrycorrespond-ingtotheregionbythevector󰁃a¯.Withouttheorganization,theparameterrepresentscaledependentrandomquantity󰁃a.Aswewillshow,the󰁃a¯canbeconnectedwiththeanomalous(fractal)dimensionsoftheinteractingfractals.

Letusconsiderthecollisionoftheasymmetricfractalobjects.Theneedtosatisfytheprinciplesofthescale-motionrelativityimpliesreplacementofthescaleindependentphysicallawsbythescaledependentequations.Thisconcernstheenergyandmomentumwhichinthepresenceofaspace-timeanisotropyareconvertedtothevariablessatisfyingtheformula(117).Weapplytheformulatotherelationsconnectingthevariablesoftherecoilparticlewiththecorrespondingmomentumfractionsintheconstituentinteraction.Weinferonfractalcharacterregardingthemotionoftheparticleformtherequirementswhichleadtotherela-tions.Theyresultfromthephenomenologicalanalysisofthezscalingvariableandconcerntheminimalresolutionε−1withwhichonecansingleouttheconstituentinteractionunder-lyingtheproductionoftheinclusiveparticlem1.Theassumptionisreflectedbytheformofthemomentumfractionsχ1andχ2whichfollowsfromtheconditionforthemaximumofthecoefficient(3).Accordingtotherequirement,therecoilparticlehastheenergyE′expressedintheway

󰀁2E′2

+µ2(142)=χ1+χ2=ω22−(ω1−ω2).s

24

Forthesakeofsimplicity,allmassesmiandMiareneglected.WeidentifytheenergyE′withtheexpression(117).Thisgives

󰀁

ω21

+µ21+

󰀁

√

E,χ⊥=

2P⊥Pmax

′s

=⊥

(1+a¯2)(χ2z+χ2⊥)

=

2+µ22,(145)a¯χ󰀁

ω2

z=ω1Theobtainedsystemfortheunknownvariablesχ−ω2.

(146)

zandχ⊥dependsontheparametera¯.The

variationrangeofthevariablesisgivenbytheconditionχ1+χ2and(143),itcanberewrittenasfollows

≤1.AccordingtoEqs.(142)(χz−a¯)2+(1+¯a2)χ2≤1+¯

a2⊥.(147)

Theχzandχ⊥areboundedinsidetheellipsoidgivenbytheasymmetrya¯.Ifweapproach

thephase-spacelimitofthereaction(1),thevariablestendtotheirboundaryvalues

χz→χ˜z=

Pzmax

E′,(148)

max

andsatisfytheequationoftheellipsoid.Similarappliestheelementaryinteraction.Theparticle’smomentumP

󰁃foranyotherparticleproducedin

andenergyE′areconnectedbythedispersionrelation(117).Inthezeromassapproximation,therelationcanbeexpressedintheway

󰀂PzE′

󰀅2=1+a¯2

.(149)AsfollowsfromEqs.(142)and(143),itisidenticaltotheequation

󰀇

χ2

z

χ=1+a¯2,

(150)

1+χ2

󰀈where

Pzs

P⊥

s

χ1+χ,

2

χ1+χ.(151)

2

Thevaluesofχz/(χ1+χ2)arelimitedwithintheinterval

a¯−≤

χz

with

a¯±=a¯±

√

1+a¯2.(154)

Substitutingtheexpression(154)intotherelation(146),onearrivesattheequationforthe

asymmetrya¯.Itssolutionwhichcompliesthephysicalrequirementsonthekinematicsofthesubprocessreads

α−1a¯=λc,(155)

α

where󰀃,λc≤1.(156)

(1−λ1)(1−λ2)

UsingEqs.(8),(9),(146),(150)and(155),onecanexpressthevariablesχzandχ⊥inasimpleform

√

χz=µ1−µ2,χ⊥=2

λc=

αsin2(θ/2)−

1

cos2(θ/2),α

χ⊥→χ˜⊥=sinθ.

(159)

Alltheexpressionsaregivenintermsofthecoefficientαwhichistheratiooftheanomalous

fractaldimensionsofthecollidingobjects.Thecollisionsoftheasymmetricfractalsystemsarecharacterizedbythedifferentfractaldimensionsandthuswithα=1.Intheconsideredscenario,itresultsincreationofthedomaininwhichtheisotropyofspace-timeisviolated.Thespace-timeanisotropyintheinteractionregionisgivenbytheformula(155).Ifα=1,thereisnopolarizationofspace-timeinducedbytheinteraction.Thiscorrespondstothecollisionsofthefractalspossessingequalfractaldimensions.Similarsituationconcernstheinteractionoftheobjectswhichrevealnofractal-likesubstructure.Theasymmetrya¯becomesnon-zeroforα=1.Itchangesitssignifλ1↔λ2andα↔α−1,i.g.iftheinteractingfractalsaremutuallyinterchanged.Theparametera¯istheproductoftheinducedasymmetry

α−1

(160)a¯0=

αandthefactorλc.Theinducedasymmetryofspace-timeresultsfromtheinteractionofthefractalscharacterizedbymutuallydifferentanomalous(fractal)dimensions.Thevalueoftheasymmetrywasidentified[7]withthespacecomponentofthefourvelocity

V

=a¯0.

1−V2

26

(161)

ThevelocityVhasitsoriginintheasymmetryoftheinteractionandvanishesinthecollisionsofobjectswhichpossessequalfractalstructures(α=1).Itcanbeexpressedbytheform

V=

α−1

provided

1+V1V2α=α1α2.

,(163)(164)

Ifweexploittheexperimentallyestablished[7]relationδA=Aδ,thelastequationcanberewrittenasfollows

A3A2

A2

sNN≥20GeVwherethezscalingbecomes

valid.Inordertodealwithsufficientasymmetry,wehavetoconsidertheprocessesinwhichthefactorλcislargeenough.Thisconcernstheinteractionswithlargetransversemomentaoftheobservedsecondaries.Wehaveestimatedtheexpectedasymmetryinthecaseoftheexperimentallymeasuredinclusivereactions[24]at400GeVprotonincomingenergy.Forthe

√

pAinteractionsα=Aandλc≃E⊥/(A).Theasymmetrywasevaluatedaccordingtotheformula(155)inthemostoptimisticcaseofE⊥=7GeV.Wehaveobtainedthevaluesa¯∼0.09÷0.13forvarioustargetnuclei.Therelativelyhighestimatesareratherheuristicandshouldnotbetakenliterally.Oneaprioridoesnotknowwhetherthefullasymptoticregimewithrespecttothefractalpropertiesoftheinteractionisachievedattheconsideredcenter-of-massenergy.Thismayoccurathigherenergieswhere,forthegiventransversemomentum,theprojectionfactorλcbecomesmuchsmaller.Theexperimentalsearchfortheeffectshouldthusrelayonthedetectionoftheparticleswithstillhighermomenta.Itisconnectedwithdifficultiesinmeasurementsofsmallcrosssectionsatwhichtheparticlesareobservable.Thisconcernsalsothestatisticalanalysisofanexperimentfromwhichonecouldinferontheexistenceofthepossibilitytoinduceapolarizationofspace-time.

27

6Summary

Thequestionsaddressedinthepaperconcerngeneralpropertiesoftheparticleproductionathighenergies.Thepropertiesareconnectedwiththenotionssuchaslocality,self-similarityandfractalityinthecollisionsofhadronsandnuclei.Theyaremanifestedmostlyintherelativisticregimeoflocalpartoninteractionswhichunderlietheproductionoftheobservedsecondaries.Inthisregime,thedescriptionoftheinclusivecrosssectionsrevealsscalingbehaviorindependenceonthesinglevariablez.

Wehavediscussedsomeaspectsoftherelationbetweenthefractalityoftheinteractingobjectsandthefractalpropertiesofspace-time.Itisrelevantforsmallscaleswherethepartoncompositionoftheobjectsissupposedtorevealafractal-likesubstructure.Theas-sumptionhasfundamentalconsequencewhichisbreakingofthereflectioninvarianceattheinfinitesimallevel.Specialattentionisdedicatedtotheelaborationoftheformalismconcern-ingtherelativityinspaceswithbrokenisotropy.Ourtreatmentcorrespondstoachangeintheenergyformulaintherelativisticcase.Wehaveobtainedexplicitrelationsbetweentheenergy/momentumandthevelocityinspace-timecharacterizedbytheasymmetry󰁃a.Inviewoftheseresults,increaseofstochasticityoftheparameterwithdecreasingscaleswouldresultinunpredictablefractal-likemotionofparticleswithrespecttotheirmomenta.ThisimplieschangeoftherestmassM0independenceonthevalueof󰁃aaswellaspossibilityofmotionwiththevelocitiesexceedingthespeedoflightinisotropicspace-time.Wehavedeterminedthecoefficientcharacterizingtheanisotropyofspace-timeintheinteractionsoftheasymmetricfractalsystems.Itisexpressedintermsoftheanomalousdimensionsofthefractalobjects(hadronsandnuclei)collidingathighenergies.Therelationisilluminatedwithrespecttothechoiceofthescalingvariablez.Thevariablezrepresentsafractalmeasureproportionaltotheformationlengthofaproducedparticle.Thescalinghypothesisstatesthatthedifferen-tialcrosssectionfortheproductionoftheparticledependsathighenergiesonitsformationlengthuniversallyandinanenergyindependentway.TheevolutionoftheformationprocessisexpressedbythescalingfunctionH(z).Theproposedscenarioisstressedbytheresultsofouranalysisconcerningexperimentaldataathighenergies.Namely,basedontheconfronta-tionofthezscalingschemewiththeexperimentaldata,wehaveshownthattheanomalousfractaldimensionsfortheinclusiveproductionofpions(δ∼0.8)andforjets(δ∼1)nearlycorrespondtotherelationD=1+δ=2.TherelationcharacterizesfractaldimensionofFeynmantrajectoriesandisadirectconsequence[17]oftheHeisenberguncertaintyrelations.Presentedapproachtothezscalingshowsthattheobservedregularitycanhaverelevancetofundamentalprinciplesofphysicsatsmallscales.Thegeneralassumptionsandideasdiscussedhereunderlineneedofsearchingnewapproachestophysicsatultra-relativisticenergies.Thisconcernsbetterunderstandingofthemicro-physicaldomaintestedbylargeacceleratorsofhadronsandnuclei.

Acknowledgment

ThisworkhasbeenpartiallysupportedbytheGrantoftheCzechAcademyofSciencesNo.1048703(128703).

28

AppendixA

Wewouldliketopresentsomepropertieswhichfollowfromthedeterminationofthevari-ablesusedinourscheme.Theelementaryinteractionofconstituentsischaracterizedbythemomentumfractionsx1andx2.Therelationbetweenthefractionsisgivenbytheminimumrecoilmasshypothesisintheconstituentinteraction.ThevariablesaredeterminedinawaytomaximizethevalueofΩ,whichgivestheminimalresolutionε−1.Eachinteractingconstituentconsistsfromaleadingpartcarryingthemomentumfractionλiandofaparton‘coat’whichisafractalcloudoftinypartonswiththemomentumfractionχi.Whatpenetratesthecloudisusuallydeterminedbythevirtualityofaprobeandisconnectedwiththeresolution.Thesituationis,however,differentascomparedtothedeepinelasticprocesseswherethe‘elemen-tary’interactionisfixedbythekinematicalcharacteristicsoftheleptonscattering.Inthecollisionsofthecompositeobjectssuchashadronsandnuclei,onecan,inprinciple,recognizetheinteractionsofconstituentswhichunderlietheproductionprocesses,aswell.Thelevelofrecognitionisgivenbytheresolutionε−1,withwhichonecansingleoutthecorrespondingsub-processes.Itconcernsbothhardandsoftcollisionscharacterizedbythedifferentmomentumtransfer.Thisinturndeterminesthevirtualityofaprobecarryingthemomentumtransferredandpenetratingintothefractalsubstructureoftheveryconstituents.Thesquaresofthe4-2

momentatransferred−Q21and−Q2fromthefirstandthesecondinteractingconstituentareasfollows

22

Q2Q2(166)1=(x1P1−q),2=(x2P2−q).

Inthezeromassapproximation,thequantitiesarecorrelatedwiththesquareofthesubprocess

energysxviatherelation

2

sx+Q2(167)1+Q2=0.Thetransferredmomentaareusuallyconsideredasvirtualitiesoftheprobesthatpenetratethe

internalstructureoftheinteractingobjects.Iftheunderlyinginteractionofconstituentsdoesnotpossesthecontactcharacter,thevirtualitiesarecarriedbythequantaofthecalibrationfields.Thenthefieldsmediatetheinteractionbetweentheconstituents.Thetransferred

2

momenta−Q21and−Q2areconnectedwithresolution.Theyareequalfor

χ1

.(168)λ2

Theconditiondeterminestheboundarybetweenthephasespacehemispheres[7]belongingto

2

theinteractingobjects1and2.Wehave−Q21>−Q2inthehemispherecorrespondingtotheobject2.Thisregionispreferabletostudyoftheprocessesinwhichtheconstituentsfromtheobject1penetratedeeperintothestructureoftheobject2andtestitinmoredetail.

22

For−Q21<−Q2itisviceversa.Withtheincreasingvaluesof−Qi,theinteractionoftheconstituentstakeplaceonstillsmallerdistances.Thisinturnincreasesthespatialresolutionnecessaryforinvestigationoffractalityatsmallscales.

Nextwewillshowthatourdeterminationofthemomentumfractionsaccountsfortheback-to-backtopologyintheconstituent’scenter-of-masssystemSc.First,letusconsiderthemomentaoftheinclusiveparticleanditsrecoilinthetotalcenter-of-masssystemS.Forthenucleon-nucleoncollisionstheparameterα=1andthe‘coats’oftheinteractingconstituentscarrythemomentumfractions

χ1→µ¯1≡λ

󰀃

1−λ2

,29

χ2→µ¯2≡λ

󰀃

1−λ1

.(169)

Theconstituentsareindistinguishableforx1=x2.Inthisregion,eachofthempossessesthecloudoftinypartonswiththesamemomentagivenbyµ¯1=µ¯2.Thisisnotlongervalidforx1=x2.Letusassumethatx1>x2.ItfollowsfromEqs.(6)and(8)thatλ1>λ2andµ¯1<µ¯2.Thisimpliesthesituationwhentherecoilobjectmovesinthedirectionnotpreciseoppositetotheinclusiveparticlem1inthesystemS.Forthesakeofsimplicity,wedemonstratethisstatementintheapproximationwhenallmassesareneglected.Weusethenotations

E−qzP2q

√→,(170)λ1=

P1P2s

¯−qP2q¯E¯z

√µ¯1=→,(171)

P1P2s

χ1=

P2q′

√

′

E′−qz→,P1P2s

(172)

¯andqintroducingtheenergyandmomentumfortherecoilparticlebythesymbolsE¯(orE′

andq′forα=1)inthecenter-of-masssystemS.Theanglescontainedbythemomenta󰁃q,󰁃q¯,and󰁃q′withthecollisionaxisorientedinthedirectionofmotionofthecollidingobject1aregivenbytheexpressions

tan(θ/2)=

󰀃

λ1

,

¯2)=tan(θ/

󰀃

µ¯1

,

tan(θ′/2)=

󰀃

χ1

,(173)

¯<π(θ+θ¯>π).Thiscanbeprovedrespectively.Therelationsx1>x2(x1asfollows.Lete.g.x1>x2.Itisequivalenttoλ1>λ2and

1>

󰀃

λ1

󰀃

1−λ1

.(174)

¯2),andconsequentlyExploitingEqs.(169)and(170),wecanwrite1>tan(θ/2)tan(θ/

¯<π.Theinverseinequalitiescanbeprovedequivalently.Wehavethusshownthatinθ+θ

the2→2processesthereisperfectback-to-backcorrelationbetweentheinclusiveparticleanitsrecoilinthereferencesystemSonlyforx1=x2.Thisisvalidalsoforthereactionswheretheparameterα=1.Changeoftheparametercorrespondstoachangeofthescaleofthereferencesystemandresultsinchangingoftheresolution.Inthecenter-of-masssystemS,theconstituentsubprocessrevealsback-to-backtopologyinthespecialcase

cosθ=

1−α

x1x2

Thisallowsustowrite

λ1=

ccP2q

√

x1

cc

P1P2

=

E−

s

c

cqz

󰀃

x1

.(177)

′′

Theanglesθcandθccontainedbythemomenta󰁃qcand󰁃qcwiththecollisionaxiscanbeexpressedinthesystemScintheway

tan(θc/2)=

Thisimplies

󰀃

λ1x2

,

′

tan(θc/2)=

󰀃

χ1x2

.(178)

′

tan(θc/2)tan(θc/2)=1

(179)

′

andconsequentlyθc+θc=π.Really,thesubstitutionofexpressions(178)intoEq.(179)gives󰀁

λ1χ1(λ2+χ2).(180)

Itremainstoexploittherelationλ1λ2=χ1χ2andonegetstheidentity.Wehavethusshown

thatourdeterminationofthemomentumfractionsisconsistentwithback-to-backtopologyofthecollisionsinthecenter-of-masssystemsoftheinteractingconstituents.

AppendixB

InthisAppendixwediscusssomeaspectsconcerningtherelativistictransformationsoftheenergyandthecoordinatesinspace-timewithbrokenisotropy.Reasonabledefinitionofthevariablesassumesthefulfillmentofcertainrequirementsresultingfromthepropercompositionofthevelocities.Itregardstheprincipleofcausalityandtheconstraintonthepositivityoftheenergy.Attheendweaddsomecommentsonthecharacterofthelengthscontractionsandthedilatationsoftime.

Accordingtothespecialtheoryofrelativity,thevaluesoftheparticle’svelocitiesareboundedwithinthesphereu≤1inanyinertialframe.Thisisgivenbythefactorγwhichfor󰁃a=0andforthesuperluminousvelocitiesbecomesimaginary.Thesituationchangesifweadmitthebreakingofthespace-timeisotropyexpressedbythenon-zerovalueof󰁃a.Thevelocityspheredeformstoanellipsoidwiththefocusinthebeginningofthevelocity√space.Centeroftheellipsoidisshiftedintothepoint󰁃u=󰁃aanditslargeraxisbecomes

1+a2,a+=a+

√

composedvelocity(38)isboundedbytheconditiona−≤v≤a+aswell.Ifthelimitingvelocitya−ora+iscomposedwithavelocityu,

a+u+2aa−=

a−−u

1+a,

(183)

+u

onegetsagaina−ora+,respectively.Asfollowsfromtherelations

a=−

a+−

1+2aa,

(184)

+

thelimitingvelocitiesa−anda+aremutuallyinversewithrespecttoEq.(45).Fora>0,theinstantvelocityoftheparticleisboundedfromabovebythevalueofa+whichislargerthanunity.Thisgivespossibilityofthemotionwiththevelocitiesexceedingthespeedoflightcinisotropicspace-time.

Wewillshowthatpropagationofanenergeticsignalwithsuchvelocitiesfulfillstheprincipleofcausality.Accordingtotherequirement,theconsequence-thedetectionofasignalcannotprecedeitsemissioninwhateversystemofreference.Letusassumethatthesignalwasemittedinthepoint(x1,t1)anddetectedat(x2,t2)withrespecttotheframeS,dt=t2velocityofthesignalpropagationisv=dx/dt,dx=x2−t1>0.ThefromthesystemS′

−x1.Letuslookatthetwoevents

movingrelativelytotheinitialonewiththespeedu.Accordingtothetransformation(42)wehave

dt′=γ(u)[(1+2au)dt−udx]=γ(u)dt(1+2au−uv).

(185)

Thefactorontherighthandsideisnon-negativeforanyvelocitiesfromtheintervala−u,v≤a+.Consequentlydt′=t′2signalpropagatesbetweenthepoints−t′1(≥󰁃x1,0.t1)Theandsame(󰁃x2,tis≤2)validwithinthethevelocitygeneral󰁃vcase=d󰁃xwhen/dt.theWeassumethatthesystemS′ismovingwithrespecttothereferenceframeSwiththevelocity󰁃u.AsfollowsfromEqs.(59)and(60),thetimeintervaldt′ofthesignalpropagationrelativetothesystemS′isgivenby

dt′=(1+γ+)dt+γ+󰁃a·d󰁃x−g+󰁃u·d󰁃x=[1+γ+(1+󰁃a·󰁃v)−g+󰁃u·󰁃v]dt.

(186)

WeseefromEq.(79)thattheexpressioninthebracketsisnon-negative.Thisimpliesdt′inagreementwiththecausalityprinciple,whichisnotviolatedinspace-timewithbroken≥0

isotropy.

Thenextstepistoprovethepositivityofthetransmittedenergy.CombiningEqs.(125)and(126),onehas

󰁃π=

(1+a2)2

1+󰁃a·󰁃v

󰀄

(1+󰁃a·󰁃u)(1+󰁃a·󰁃v)−(1+a)󰁃u·󰁃v󰀆

.

(188)

WhenexploitingEqs.(77)and(79),therelationcanberewrittenintotheform

E′

=E

(1+󰁃a·󰁃v′)γ(󰁃v′)

Asfollowsfromtheinequality

validforanyvelocity󰁃vboundedbytheellipsoid(80),thefactorsontherightsideoftherelation(189)arenon-negative.TheenergyofthesignalisthuspositiveineachsystemofreferenceS′whichismovingrelativelytothesystemSwiththevelocity󰁃u.Theabovementionedpropertiesenablethepropagationofphysicalsignalsincludingtransportationoftheenergywiththevelocitiesexceedingthevalueofc-thespeedoflightinisotropicspace-time.Thiscanoccuratsmallscaleswithintheregionswithbrokenspace-timeisotropy.

Basedonthetransformations(59),wecandrawconclusionsregardingthecourseoftimeandthechangeoflengthsoftheelementarysectionsexpressedrelativetothesystemsSandS′.LetusconsideraclockatrestwithrespecttothesystemS′.Timerecordedbytheclockisreferredasthepropertime.AccordingtoEq.(133),theincreaseofthepropertimedτandthecorrespondingincreaseoftimedtinthesystemSarerelatedasfollows

dt(󰁃v)=

dτ

(1+󰁃a·󰁃v)2

−

(1+a2)v2

.

0≤1+aa−≤1+󰁃a·󰁃v

(190)

(191)

Inviewoftheasymmetryrepresentedbythefactor󰁃a,thecourseoftheclocktimecanbeeven

fasterthaninitsrestframe,whenobservedbyamovingobserver.Theminimaltimeintervalbetweentwoevents

dτ

dt(󰁃va)=≤dτ,󰁃va=󰁃a,(192)21+a

isrecordedfromthesystemSinwhichtheclocksaremovingwiththevelocity󰁃va.Theclocksareslowingdown(theirtimeintervalsincrease)iftheirvelocityapproachesthelimitgivenbyEq.(80).

Thechangeofthelengthofasectionwiththevelocityislittlemorecomplicated,thoughtitstransversedimensionswithrespecttothemotionarenotsubjectedtoanychange.Indeed,ifthevelocityisorientede.g.inthedirectionofthex-axis,theyandzcomponentsofthecoordinatesareinvariantwithrespecttothetransformation(59).Asconcernsthelongitudinalcontractions,wediscussherethesimplifiedsituationinwhichthesectionhasnotransversedimensionrelativetoitsmotion.ItisnaturaltodefinethelengthdlofthesectionwithrespecttoSasthedifferencebetweenthesimultaneouscoordinatevaluesofitsend-points.IfthesectionisatrestintheS′system,itsrestlengthisgivenbydl0=x′2−x′1.Accordingtothespecificsituationconsidered,thebothvaluesareconnectedwiththeexpression

dl(󰁃v)=dl0

󰀁

1+a2≥dl0.The

contractionsoftheelementarysectionwithrespecttoitsmaximalvaluedl(󰁃va)increaseifthevelocityofthesectionapproachestheboundarygivenbyEq.(80).

References

[1]ProceedingsoftheXInternationalConferenceonUltra-RelativisticNucleus-NucleusCollisions,

Borlange,Sweden,1993[Nucl.Phys.A566,1c(1994)].

33

[2]ProceedingsoftheXIIInternationalConferenceonUltra-RelativisticNucleus-NucleusColli-sions,Heidelberg,Germany,1996[Nucl.Phys.A610,1c(1996)].[3]ProceedingsoftheXIVInternationalConferenceonUltra-RelativisticNucleus-NucleusColli-sions,Torino,Italy,1999[Nucl.Phys.A661,1c(1999)].[4]A.M.Baldin,ParticlesandNuclei8(1977)429.[5]V.A.Matveev,

V.A.Matveev,S.Brodsky,G.S.Brodsky,G.

R.M.Muradyan,A.N.Tavkhelidze,Lett.Nuov.Cim.5(1972)907;R.M.Muradyan,A.N.Tavkhelidze,ParticlesandNuclei2(1971)7;Farrar,Phys.Rev.Lett.31(1973)1153;Farrar,Phys.Rev.D11(1975)1309.

ˇ[6]I.Zborovsk´y,Yu.A.Panebratsev,M.V.TokarevandG.P.Skoro,Phys.Rev.D54,5548(1996).ˇ[7]I.Zborovsk´y,M.V.Tokarev,Yu.A.Panebratsev,andG.P.Skoro,Phys.Rev.C59,2227(1999).[8]F.W.FullerandJ.A.Wheeler,Phys.Rev.128,919(1962).[9]S.Hawking,Nucl.Phys.B144,349(1978).

[10]J.Ellis,N.E.Mavromantos,andD.V.Nanopoulos,hep-th/0012216.

[11]B.Mandelbrot,TheFractalGeometryofNature(Freeman,SanFrancisco,1982)[12]V.S.Stavinsky,Part.Nuclei10,949(1979).

[13]G.A.Leksin,PreprintITEP-147,Moscow1976,12p.;

G.A.Leksin,ProceedingsoftheXVIIIInternationalConferenceonHighEnergyPhysics,Tbilisi,1976,JINRD1,2-10400,Dubna,1977,A6-3.[14]M.V.TokarevandT.G.Dedovich,Int.J.Mod.Phys.A15,3495(2000);

[15]M.V.Tokarev,JINRReportNo.E2-97-56,Dubna,1997;M.V.Tokarev,O.V.Rogaˇcevski,and

T.G.Dedovich,JINRReportNo.E2-99-313,Dubna,1999.[16]I.Zborovsk´y,inProceedingsoftheXIIthInternationalSeminaronHighEnergyPhysicsProb-lems,Dubna,Russia,1994,editedbyA.M.BaldinandV.V.Burov(JINRReportNo.E2-94-504,

Dubna,1994),p.290.[17]L.Nottale,FractalSpace-TimeandMicrophysics(WorldScientific,Singapore,1992).[18]R.P.Feynman,Rev.Mod.Phys.20,367(1948).

[19]H.vonKoch,Archivf¨urMathematik,AstronomiochPhysik1,681(1904).[20]G.N.Ord,J.Phys.A:Math.Gen.16,1869(1983).

[21]M.V.Lalan,BulletindelaSoci´et´eMath´ematiquedeFrance,65,83(1937).[22]J.C.Pissondes,J.Phys.A:Math.Gen.32,2871(1999).[23]L.W.Thomas,Phil.Mag.3,1(1927).

[24]D.Antreasyanetal.,Phys.Rev.D19,764(1979).

34

Figure1:SuccesiveapproximationsofthevonKochfractalcurve.Itstopologicaldimensionis1,whileitsfractaldimensionisln4/ln3.

35