latedlethergo

JournalofModernPhysics,2011,2,849-856doi:10.
4236/jmp.
2011.
28101PublishedOnlineAugust2011(http://www.
SciRP.
org/journal/jmp)Copyright2011SciRes.
JMPFormulaeforEnergyandMomentumofRelativisticParticleRegularatZero-MassStateRobertM.
Yamaleev1,21JointInstituteforNuclearResearch,LIT,Dubna,Russia2DepartamentodeFsica,FacultaddeEstudiosSuperiores,UniversidadNacionalAutonomadeMexicoCuautitlánIzcalli,Campo1,MéxicoE-mail:iamaleev@servidor.
unam.
mxReceivedOctober22,2010;revisedMarch2,2011;acceptedApril13,2011AbstractInthispaperwesubstantiateanecessityofintroductionofaconceptthecounterpartofrapidityintotheframeworkofrelativisticphysics.
Itisshown,formulaeforenergyandmomentumdefinedviacounterpartofrapidityareregularnearthezero-massandspeedoflightstates.
Therepresentationfortheenergy-mo-mentumisrealizedasamappingfromthemassless-stateontothemassiveonewhichlookslikeasa"q"-deformation.
Quantizationoftheenergy,momentumandthevelocitynearthelight-speedispresaged.
Ananaloguebetweentherelativisticdynamicsandthestatisticalthermodynamicsofamicro-canonicalensembleisbroughttolight.
Keywords:RelativisticDynamics,ComplexAlgebra,Rapidity,Energy-Momentum,BackgroundEnergy,StatisticalThermodynamics1.
IntroductionThedevelopmentsofthebasictheoriesinthefieldsofsolidstateandelementaryparticlesexhibitacrucialim-portanceofthebehaviorofthephysicalsystemsnearthecriticalpoints.
Theexperimentalresultsandtheirtheo-reticaltreatmentshavediscoveredthenewphysicalphe-nomenanamedasspontaneouslybrokensymmetriesnearthegroundstate.
ThatagroundstateofaquantalsystemneednotpossesstheHamiltonian'ssymmetries,andthereforedegenerate,wasfirstappreciatedandrealizedinnon-relativisticmanybodysystemsandinmanycon-densedmattersituations.
Alsothathadbeenrealized,thestateofsuperconductivitypossesseswithlowerentropy(hence,withhigherorder)thenanormalstate.
FollowingthisunderstandingHeisenberg[1]andNambu[2]madetheseminalsuggestionthatthismayalsobetrueforthevacuumstateofarelativisticquantumfieldtheory.
Fortherelativisticmechanicsthestatewith=0,m=vc,i.
e.
,thestatewithpropermassequaltozeroandthevelocityequaltospeedoflight,isasingularpointofthetheory.
Themassiveparticles,accordingtolawsoftherelativisticmechanics,cannotattainthevelocityequaltospeedoflight.
TheformulaeoftheLorentztransformationsandformulaefortheenergy-momentumoftherelativisticmassiveparticlearesingularnearthevelocityequaltospeedoflight,whereastheyarewellde-finedforthereststate.
Ingeneral,itissupposedthatnearthespeedoflightthedynamicsofamassiveelementaryparticlemoredoesnotobeytheclassicalmechanicsofasingleparticleandonemustworkinthescopeofthequantumfieldtheory.
Infact,accelerationofthechargedparticleinducesradiationofelectromagneticfieldsandleadstocumulativeprocessofcreationofacascadeofelementaryparticles.
Noteworthyacrucialgapbetweenmassiveandmasslessstates:theparticlewithextremelysmallvalueofmasscanstayinthereststate,meanwhileitsneighbour,theparticlewith=0m,hastomovewiththespeedoflight.
Theirregularbehavioroftheconventionalrepresenta-tionsforenergy-momentumnearthezero-masspointisawell-knownproblemoftherelativisticmechanics.
Infact,thecelebratedformulaefortheenergy-momentum0,ppdefinedviavelocityv,202222=,=11mvmcpcpvvcc(1.
1)couldnotbeusedtoobtainanyreasonablelimitatthepoints=vc,=0m,becausetheindeterminacyoftypeR.
M.
YAMALEEVCopyright2011SciRes.
JMP85000,meanwhile,theenergyandmomentumoftheparticlewith=,=0vcmaregivenbyacertainfinitevalue.
Infact,inordertoprovethisassertionwemayusetheexpressionoftheenergyviamomentum222=Ecpmc(1.
2)Forsmallvaluesofthemass,mcp,wecanusethefollowingapproximation22232=1=2mcmcEpcpcpp(1.
3)Hence,when0m,0ppthevelocitytendstospeedoflight,0/=/1vcpp.
Theenergyandmo-mentumatthislimitnumericallyareequaltoeachotherandequalsomefinitevalue.
Thefollowingquestiongivesarise:whatkindofthelawofrelativisticdynamicscanhelpustosolvethisindeterminacyOr,intheotherwords,howwemustmo-dify,orgeneralize,theconventionalframesofthetheoryinordertoelaboratesomepathwayfromthestatewithnon-vanishingpropermasstothestatewithzero-massInordertoanswerthesequestionswehavetodeepentheconceptofrapidity.
Therapidityiswell-knownquan-tityoftherelativistickinematics,thisisanhyperbolicangleusedasaparameteroftheLorentz-boostintheLorentzgroupoftransformations.
Mostphysicistsmain-tainedthattherapidityisamerelyformalquantitybe-causetherapiditydidnotreceiveanyphysicalinterpre-tationintheframeworkoftherelativistickinematics[3].
Nevertheless,ithadbeennotedthattherapidityplaysanimportantroleinestablishingsomelinkbetweenthehyperbolicgeometryandrelativistickinematics.
ThislinkfirstlyhadbeendiscoveredbyV.
Varicak[4],H.
Hergoltz[5]andA.
A.
Rob[6].
Contemporarymodelofrelativistickinematicsbasedonnon-associativealgebrasandontheconceptofgruppoidshadbeenconstructedbyA.
Ungar[7].
Accordingtoourpointofview,therapidityisoneofprincipalnotionsoftherelativisticdynamics.
Aness-entialroleplaysaquantitydualtotherapidity,wede-nominateasthecounterpartofrapidity.
Introductionofthecounterpartofrapidityisrelatedwithdiscoveryofanewfeaturesofdynamicalvariablesoftherelativisticmechanics.
Thecounterpartofrapidityisarapidityre-latedwiththesystemsofreferencescomingfromthelight-speedstatetowardtothereststate.
Thereforetheex-pressionsforenergy,momentumandvelocityexpressedviathecounterpartofrapidityareregularatthestatewith=0,=mvc.
Thenewrepresentationforenergyandmomentumcanbeconsideredasamappingfromthemasslessstateontothestatewithmass.
Animportantfeatureofthecounterpartofrapidityisthatthehyperbolicangleisproportionaltothepropermassofaparticle.
Thepartofthehyperbolicanglein-dependentofthemasscanbeinterpretedviatheconceptofbackgroundenergy.
Therelativisticenergy-momentumwithintheframeworkofthenewrepresentationformallycoincideswithformulaeofstatisticalthermodynamicsofansingleoscillatorwhichallowsonetogiveaninter-pretationofthebackgroundenergybyintroducingthequantitiesofthestateliketemperatureandentropy.
ThepaperbesidesoftheIntroductionandConclusionsispresentedbythefollowingsections.
Section2presentselementsoftherelativisticdynamicsofchargedparticle.
InSection3,anevolutiongeneratedbymass-shellequationisexplored.
Transmissionbetweentranslationandhyperbolicrotationisestablished.
InSec-tion4,therepresentationforthemomentaismodifiedbyintroducingafundamentalconstantofmass.
Themodifiedformulaisinterpretedasamappingfrommasslessstateontothestatewithnontrivialmass.
Ahypothesisonquan-tizationofthevelocitynearlightvelocityissuggested.
InSection5,ananaloguebetweenformulaeofrelativisticmechanicsandformulaeofstatisticalthermodynamicsisdemonstrated.
2.
ElementsofRelativisticDynamicsofChargedParticleInthissectionwewillremindonlyselectedelementsoftherelativisticdynamicsofchargedparticlenecessaryinsubsequentsections.
ConsideramotionoftherelativisticparticlewithchargeeintheexternalelectromagneticfieldsEandB.
TherelativisticequationsofmotionwithrespecttothepropertimearegivenbytheLorentz-forceequations[8]:00dd=),=ddpeeepmcmmcpEpBEp(2.
1)0dd=,=ddpctmmrp(2.
2)Theseequationsimplythefirstintegralofmotion22220=ppMc(2.
3)Theconstantofmotionhasobtaineditsinterpretationasasquareofpropermassoftherelativisticparticle,sothat,22=Mm.
Inthecaseofstationarypotentialfield,i.
e.
when=(),eVrEtheequationsimplytheotherconstantofmotion,theenergyoftherelativisticparticle0=()EcpVr(2.
4)Inordertogiveamainideaweshallrestrictourselvesbyconsideringonlylengthsofthemomenta.
ForthatR.
M.
YAMALEEVCopyright2011SciRes.
JMP851purposeletusconsidertheprojectionofEquation(2.
1)ontodirectionofmotion.
Inthiswaywecometothefollowingsetofequations00ddddddppeppEmc(2.
5)where=,=EppnEn(2.
6)Equation(2.
5)canbeintegratedwithrespecttopara-meter.
Wefind0=coshsinh,=sinhcoshpABpAB(2.
7)Let0=forthereststatewhere=0p.
Then(2.
7)isredefinedasfollows000=cosh,=sinhpmcpmc(2.
8)Velocitywithrespecttocoordinatetimeisdefinedby00==tanhvpcp(2.
9)ComparingthisformulawiththeformulaeusedintherelativistickinematicswecometoconclusionthattheparametercoincideswithhyperbolicangledenominatedinLorentz-kinematicsastherapidity.
InEquation(2.
5)therapidityispresentedasacomplementarydy-namicalvariable.
TheevolutiongovernedbyLorentz-forceEquation(2.
1)changesthequantities0,ppinasuchwaythatre-maininvariantthemass-shellEquation(2.
3).
Therapidityduringofthisevolutionundergoestotranslations.
Notice,howeverthisisnotuniqueformofvariationof0,ppremaininginvariantthemass-shellEquation(2.
3).
Nowletusexploreanotherformofevolutionsoftheenergy-momentumwhichalsoremaininvariantthemass-shellequation.
Followingreferences[9,10],letusintroducetwoquantities21>>0qqby202121121122pqqmcqqpqq(2.
10)Inversely,1020=,=.
qpmcqpmc(2.
11)Thequantities12,qqformthesetofeigenvaluesofthequadraticpolynomial22222002=0,0XcpXcppp(2.
12)withrespecttowhichrelationships(2.
10)playtheroleofVieta'sformulae.
Undertranslation0=YXpthequadraticpolynomialtakestheformofmass-shellequa-tion:222220==.
YppmcObservation2.
1Themass-shellequationremainsinvariantunderadditivechanges(simultaneoustranslations)ofthepairofquantities12,qq,1122=(0),=(0).
qqqq(2.
13)Thesetranslationsresultanadditivechangeofthekineticpartoftheenergy00=(0)pp(2.
14)Inthesamespiritasasolutionofthequadraticequation2=1x,theimaginaryi,generatesalgebraofcomplexnumbers,thesolutionofquadraticEquation(2.
12)generatesanalgebraofgeneralcomplexnumbers[12,13].
Inreferences[14,15],somegeometricalandalgebraicalpropertiesoftheevolutiongovernedbybyquadraticEquation(2.
12)hadbeenexplored.
Since12,qqarerootsofthequadraticEquation(2.
10),thefollowingEulerformulaeholdtrue0010exp1,2.
kkxgppxgppk(2.
15)Formthefollowingratio21exp2=qDmcqD(2.
16)where0010;,=;,gppDgpp(2.
17)Translations=,=1,2kkqqkremaininvariantm,hence21exp2=qDmcqD(2.
18)Let,=D.
Then,201exp2=qmcq(2.
19)Theformula(2.
19)isaKey-formulaofthehyperboliccalculus.
Thisformulaestablishessomeinterrelationbe-tweensimultaneoustranslationsofdenominatorandnu-meratorofthefractionandthehyperbolicrotation.
3.
CounterpartofRapidityandRepresentationsforEnergy-MomentumRegularNearZero-MassStateFromKey-formula(2.
19)bytakingintoaccount(2.
11)weget000=cothpmcmc(3.
16)Anysetofmomenta20{,}ppwith220>pprealR.
M.
YAMALEEVCopyright2011SciRes.
JMP852positivefinitenumbersmayserveasageneratoroftheevolutionwheretheinitialpointisdefinedby202201exp2==pmcqmcpmcq(3.
17)and0=coth,=sinhmcpmcmcpmc(3.
18)Fromtheseequationsthefollowingusefulrelationshipisderived0exp=pmcmcp(3.
19)Withrespecttotheevolutionofenergy-mo-mentumisgivenbyfollowingequations:200dd=,=ddppppp(3.
20)Theevolutionofkineticenergyisdescribedbynon-linearequation22200d=0dppmc(3.
21)Thus,wepossessnowwithtwodifferentrepre-sen-tationsfortheenergyandmomentumviahyperbolictrigonometry.
Oneisafunctionoftherapiditygivenby0()=cosh(),=sinh().
Ipmcpmc(3.
22)Andtheotheronedescribesanevolutionwithrespecttocounterpartofrapidity=mc:0()=coth,=sinhmcIIpmcmcpmc(3.
23)Theformerisregularatthereststate,theseformulaeweshalldenominatedasthelow-speedrepresentation.
Thelatterisregularatthestateofspeedoflight,theseformulae,correspondingly,weshallnamedashighspeedrepresentation.
Fromthelow-speedrepresentationwecometotheformulaforenergynearthereststateforslowmotion,thisis,so-called,non-relativisticlimit.
Inordertoobtainthenon-relativisticlimitweuseexpansionof(3.
22)forsmallvaluesof1:201=1,=.
2pmcpmcRemovingfromtheseequations,weget20=2ppmcmc(3.
24)or,2()0==2nonrelpEcpmcm(3.
25)where()nonrelEisexpressionforkineticenergyofNew-tonianmechanics.
Thehigh-speedrepresentationallowstoobtainanalo-gousexpressionforenergynearthelight-speedstate.
Forsmallvaluesof1wecometothefollowingex-pansion0111=coth=,=.
2pmcmcpmcRemovingfromtheseequations,weget220=.
2mcppp(3.
26)Fromcomparisonofformulae(3.
25)and(3.
26)weconcludethatthemassandmomentumattheselimitsaremutuallyreplaced.
Formulae(3.
23)nearthelight-speedstatearegivenby22011=coth=,=.
2mcpmcmcpItisseen,atthelimit=0mwehave0011=0==0=.
pmpmc(3.
27)Hereweintroducedthenewquantity0whichisequaltoenergy-momentumoftherelativisticsystematthestate=0,=mvc.
Inthereststatetheenergyisequaltotheproperinertialmass(inenergyunits)and,inthesamemanner,inthestateofthelight-speedtheenergyisequalto0.
Thus,therelativisticdynamicsoftherelativisticparticlebesidetheinertialmassmcon-tainsaparameterdualtothepropermass(wesuggesttodenominatethisvalueaslight-mass).
Theparameter0determinesthevalueofthekineticenergyofthemotion.
Thisquantitycorrespondstotheenergyoftheparticleinitsmasslessstate.
Nowletusunderlinesomereciprocitybetweentwohyperbolicanglesand.
Forthatpurposeintro-ducecomplementarytovvelocityvobeyingthefo-llowingequation222=vvc(3.
28)Noticethatvisrelatedwithhyperbolicparameter()inamannerquitesimilarasvisexpressedviarapidity():222222220==1=tanhpvcvccp(3.
29)InterrelationbetweenandisexpressedbytheR.
M.
YAMALEEVCopyright2011SciRes.
JMP853followingformulaeofreciprocityexp=coth,exp=coth22(3.
30)Fromtheseformulaeitfollowsthat=0and=when=0v,and=and=0when=vc.
Insomesensethepairs(,)vand(,)varereciprocaltoeachother.
4.
q-DeformationandQuantizationoftheEnergy-MomentumLetusintroducesomeparameterinunitofmassandlabelthisparameterby.
Defineadimensionlessvariableby=c(4.
1)Re-writeformulaefortheenergy-momentum(3.
18)inthesevariables0=,=cothsinhmcmppmcm(4.
2)Hereweshouldnoticethattheformulaforthemo-mentumadmitsthefollowingintegralrepresentation221/20021/200sinh==expdmcmcxxcpmc(4.
3)In[16]ithasbeenshownthat1geometricallycanbeinterpretedasacurvatureofahyperbolicspace.
Inthisspacethelengthofthecirclewithradiusmisde-finedbyformulae[17]:=1:=2sinhmL(4.
4a)Correspondingly,thelengthofthecirclewithradiusmisequal=2sinhmL(4.
4b)Takingintoaccountthiscorrespondenceletusper-formthefollowingmodificationsintheformulaeforenergy-momentum20sinh=sinh=sinhmmcmccmpp(4.
5)2000=coth=sinhcothppmcmmmcc(4.
6)Noteworthy,theconstantnowisnotanarbitraryconstant,butithastobeunderstoodasafundamentalconstantofthetheory.
Letusremembertheformulaofq-deformationofaquantityN:1():=NNqqqNqq(4.
7)Fromthispointofviewthelastexpressionin(4.
5)isq-deformationofwithparameterofdeformation=exp(/)qm.
Innotationsof(4.
7),Equation(4.
5)canbewrittenasfollowsexpqcmqp(4.
8)Notice,=1=1qforanyq.
Therefore0,=1=.
pmcOntheotherhand,if=0mand=1qthenfrom(4.
8)itfollows0=0==cpmc(4.
9)Henceatthepoint=1momentumoftheparticlewithmassmisequaltothemomentumofthemasslessparticle00,=1==0,=1==pmpmcc(4.
10)Notice,however,thevelocityoftheparticleatthispointisnotequaltothelightvelocity.
Infact,thesefor-mulaeimplyexistenceofapointontheaxisofmomentawherethemomentaofthemassiveandmasslessparticlesareequal.
Atthispointtheenergyandthevelocityaregivenby20=1=cosh,=coshmccpcvm(4.
11)Nowrememberonintegralrepresentation(4.
3).
Nowthisfractionwasreplacedby0sinh=sinhmcmcpp(4.
12)Itisinterestingtoobservethatforthenewfraction(4.
12)weshallobtainasuminsteadofanintegralifweassumethatisanintegralnumber.
LetJbeahalf-integernumberwith=0,1/2,1,3/2,2,J,and=21=1,2,3,J.
Thenthefollowingequationholdstrue.
R.
M.
YAMALEEVCopyright2011SciRes.
JMP854=sinh21==exp.
sinhJnJmJcmnmpThisformulapromptsustointroduceahypothesisonquantizationof.
Experimentallythequantizationcanbeobservednearthelightvelocitywherethevelocityofthemassiveparticlebringsnearerthelightvelocityspas-modicallyaccordingtolaw=.
cosh21cvmJ5.
AnaloguewithStatisticalThermodynamicsInthissectionwearegoingtoobserveaninterestinganalogybetweentheformulaeobtainedintheprevioussectionsfortheenergy-momentumandthewell-knownformulaeofthestatisticalthermodynamics.
Thisana-logueexhibitsacorrespondencebetweenthepresentre-presentationofenergy-momentumandtheaverageenergyofthesinglequantummechanicaloscillator.
Inthiswaywewillcometotheinterpretationofasanabsolutetemperatureofaheatreservoir.
Letusstartfromformulaforthemomentumoftheform1=22sinhpmcmc(5.
1)Noticethatthisexpressioncanberepresentedasasumofgeometricalseries.
Infact,=0exp1==exp2sinh1exp2nnmcEcmcmc(5.
2)heretheexpression21=22nEmcncanbeconsideredasaspectrumofthequantumoscillatorwheretheroleofzeropointenergy2istakenoverbytheenergyattherest2mcoftheparticle.
Furthermore,itispuzzledthattheformulaforthelengthofmomentumperunitofmassisquitesimilartotheformulaforsingle-particlepartitionfunctionZ.
Thepartitionfunctionforthesingleoscillatoris[18]:exp2,,1=1expZTV(5.
3)where1=,TwithabsolutetemperatureTgiveninenergyunit.
Con-tinuingtheobservationletusnoticethattheexpressionfortheenergy0cpoftheform2011=22exp21cpmcmc(5.
4)isquitesimilartheexpressionforthemeanenergyofthesingleoscillator112exp1U(5.
5)Theseobservationsleadustothefollowingcorres-pondencebetweendynamicvariablesoftherelativisticmechanicsandthestatisticalthermo-dynamics:201112,,mccT(5.
6)Inreference[15]wehavejustifiedthefollowingre-lationshipbetweenenergyandmomentum0d=ln.
dcpp(5.
7)Thisequationquiteanalogoustothewell-knownequa-tionofthermodynamicsconnectinginternalenergyUwiththepartitionfunctionZ:d=lndUZ(5.
8)Theevolutionwithrespecttonowobtainsitsinterpretationviastatisticalthermodynamics.
In(5.
4),(5.
5)theterms2,aswellasquantity2mc,areexactlythecontributionofthezero-pointenergytothetotalenergy.
Furthermore,nowweareabletointroducethenotionsofthefreeenergyandtheentropy.
Thefreeenergyis002=ln2sinh=lnmcFmcP(5.
9)Theentropyis(inunit=1k,k-constantofBoltzmann)=cothln2sinSmcmcmc(5.
10)ConsiderNdistinguishableoscillators[19].
Itisveryinstructivetocalculatefromthepartitionfunction2e()=1eNmcmcZN(5.
11)R.
M.
YAMALEEVCopyright2011SciRes.
JMP855thecorrespondingdensityofstatesoftheN-oscillatorsystem.
Withtheaidofthebinomialexpansion=01!
1=!
(1)!
1NlNlxlNxwewrite=01)!
1=exp2!
1!
2lNlZNmcNllNComparingthiswiththegeneralformulaforZwithdiscreteenumerableenergies=expZNdEgEEcwefind21)!
=2,=2!
1!
llNNlEmclglN(5.
12)TheenergieslEarejustthezero-pointenergiesoftheNoscillatorspluslquantaofenergy22mc.
Thereareexactlylgwaystodistributetheseindis-tinguishableenergyquantaamongtheNdistinguishableoscillators.
Thedistributionofindistinguishablequantainsteadofenumeratedparticlesisthestartingpointofquantumstatistics.
ConsideracollectionofNidenticalquantumoscillators,supposedtheirdistinguishable.
Thetotalnumberofdistinguishablestates,correspondingtotheenergyE,andalltheprobabilitiesturnouttobeequal,asexpected.
Itissimply=lg.
Thuswecancheckwhethertheentropy=lnScoincideswithexpression(5.
10).
Using,1lNandStirling'sformulaweget1!
=ln,=,1!
!
=lnlnln.
MlSNlSlNlNllNNToobtain(,)SENthe2=/2/2lEmcNmustbeinserted:2222=ln2222lnln2222ENENSmcmcENENNNmcmc(5.
13)IfwewanttocomparethiswithEquation(5.
10),wemustexpresstheenergyintermsofthe"temperature",2222221=|=ln,2exp21=exp21NSENmcEmcENmcmcEorNmcmc(5.
15)whichisidenticaltoEquation(5.
4).
Itisinterestingtocomparethelimitingcasesforsta-tisticalthermodynamicsandfortherelativisticme-chanics.
Inthecaseofhightemperatures,0,TkTtheclassicallimitisrecovered,becausethecharacteristicparametermeasurestheratiooftheenergylevelsoftheoscillatorcomparedtothemeanthermalenergykTwhichisavailable.
Inthemechanicsthelimit=0correspondstothestateofmasslessparticle,ortothestateoflight.
Atthelimitoflowtemperature,onehasthelargestdeviationsfromtheclassicalcase,where=.
2NUInthemechanicsthelimitcorrespondstothereststate=0pwiththerestenergy20=cpmc.
6.
ConclusionsandCommentsInthepresentpaperwegaveagroundfornewrepre-sentationforenergy,momentumandvelocityviahy-perbolictrigonometrythe(hyperbolic)angleofwhichisproportionaltothepropermass.
Thehyperbolicangledualtotherapiditywasdenominatedascounterpartofrapidity.
Wehaveworkedprimarilyinthescopesofthedynamics,notkinematics,andrestrictedourselvesonlywithonedimensionalcase.
Itiswellknownthatinthecovariantformulationtherapidityispresentedbyanti-symmetrictensorinfour-dimensionalMinkowskispace.
Inthatcontextthequestiongivesarise:whatkindtensorialobjectwillpresentthecounterpartofrapiditywithintheframeworkofcovariantformulationThisquestionhasacertainanswer:inthecovariantformulationthecounter-partofrapidityispresentedbythefour-vector[20].
7.
References[1]W.
Heisenberg,"ZurTheoriedeElementarteilchen,"ZeitNaturforsch,Vol.
14a,1959,pp.
441-451.
[2]Y.
Nambu,"DynamicalModelofElementaryParticlesBasedonanAnalogywithSuperconductivity,"PhysicalReview,Vol.
117,1960,pp.
648-659.
doi:10.
1103/PhysRev.
117.
648[3]J.
-M.
Levy-Leblond,"AGedankenexperimentinScienceHistory,"AmericanJournalofPhysics,Vol.
48,1980,pp.
345-354.
[4]V.
Varicak,"ApplicationofLobachevskianGeometryintheTheoryofRelativity,"PhysikalischeZeitschrift,Vol.
11,1910,pp.
93-96.
[5]H.
Hergoltz,"GeometricalAspectsofRelativityTheory,"Ann.
d.
Phys,Vol.
10,1910,pp.
31-45.
R.
M.
YAMALEEVCopyright2011SciRes.
JMP856[6]A.
A.
Robb,"OpticalGeometryofMotion,aNewViewoftheTheoryofRelativity,"Cambridge,1911.
[7]A.
A.
Ungar,"HyperbolicTrigonometryandItsApplica-tionsinPoincaréBallModelofHyperbolicGeometry,"Computers&MathematicswithApplications,Vol.
41,No.
1-2,2001,pp.
135-147.
doi:10.
1016/S0898-1221(01)85012-4[8]A.
O.
Barut,"ElectrodynamicsandClassicalTheoryofFieldsandParticles,"DoverPublications,Inc.
,NewYork,1980.
[9]R.
M.
Yamaleev,"ExtendedRelativisticDynamicsofChargedSpinningParticleinQuaternionicFormulation,"AdvancesinAppliedCliffordAlgebras,Vol.
13,No.
2,2003,pp.
183-218.
doi:10.
1007/s00006-003-0015-8[10]R.
M.
Yamaleev,"RelativisticEquationsofMotionwithinNambu'sFormalismofDynamics,"AnnalsofPhysics,Vol.
285,No.
2,2000,pp.
141-160.
doi:10.
1006/aphy.
2000.
6075[11]A.
R.
Rodriguez-Dominguez,"EcuacionesdefuerzadeLorentzComoEcuacionesdeHeisenberg,"RevistaMexi-canadeFisica,Vol.
53,No.
4,2007,pp.
270-280.
[12]I.
M.
Yaglom,"ComplexNumbersinGeometry,"Aca-demicPress,NewYork,1968.
[13]P.
FjelstadandS.
Gal,"TwoDimensionalGeometries,Topologies,TrigonometriesandPhysicsGeneratedbyComplex-TypeNumbers,"AdvancesinAppliedCliffordAlgebras,Vol.
11,No.
1,2001,pp.
81-90.
doi:10.
1007/BF03042040[14]R.
M.
Yamaleev,"MulticomplexAlgebrasonPolynomi-alsandGeneralizedHamiltonDynamics,"JournalofMa-thematicalAnalysisandApplications,Vol.
322,No.
2,2006,pp.
815-824.
doi:10.
1016/j.
jmaa.
2005.
09.
073[15]R.
M.
Yamaleev,"GeometricalandPhysicalInterpreta-tionofEvolutionGovernedbyGeneralComplexAlge-bra,"JournalofMathematicalAnalysisandApplication,Vol.
340,No.
2,2008,pp.
1046-1057.
doi:10.
1016/j.
jmaa.
2007.
09.
018[16]R.
M.
Yamaleev,"ComplexAlgebrasonN-orderPoly-nomialsandGeneralizationsofTrigonometry,OscillatorModelandHamiltonDynamics,"AdvancesinAppliedCliffordAlgebras,Vol.
15,No.
1,2005,pp.
123-125.
[17]V.
F.
Kagan,"Gostexizdat,"FoundationsofGeometry,PartI,"Moscow,1949,pp.
331-336.
[18]W.
Greiner,L.
NeiseandH.
Stcker,"ThermodynamicsandStatisticalMechanics,"Springer-Verlag,NewYork,1995,pp.
208-214,ISBN0-387-94299-8.
[19]H.
S.
Robertson,"StatisticalThermodynamics,"PTRPren-ticeHall,EnglewoodCliffs,1993,pp.
110-112,ISBN0-13-845603-8.
[20]R.
M.
Yamaleev,"NewRepresentationforEnergy-Mo-mentumandItsApplicationstoRelativisticDynamics,"NuclearPhysics,Vol.
74,No.
7,2011,pp.
1-8,(inRus-sian).