QuaternionicRepresentationofMaxwell-Dirac IsomorphismasanAlternativeto Barut-DiracEquation
V.Christianto∗
V.ChristiantoViaFlorentinSmarandache, DepartmentofMathematics,UniversityofNewMexicoGallup,NM87301,USA
Received5November2005,Accepted5January2006,Published20September2006
Abstract: ItisknownthatBarut’sequationcouldpredictleptonandhadronmass withremarkableprecision.Recentlysomeauthorshaveextendedthisequation,resulting inBarut-Diracequation.Inthepresentarticlewearguethatitispossibletoderive anewwaveequationasalternativetoBarut-Dirac’sequationfromtheknownexact correspondence(isomorphism)betweenDiracequationandMaxwellelectromagneticequations viabiquaternionicrepresentation.Furthermore,inthepresentnotewesubmittheviewpoint thatitwouldbemoreconceivableifweinterpretthe vierbein ofthisequationintermsof superfluidvelocity,whichinturnbringsustothenotionoftopologicalelectronicliquid.Some implicationsofthispropositionincludequantizationofcelestialsystems.Wealsoarguethatit ispossibletofindsomesignaturesofBose-Einsteincosmology,whichthusfarisnotexplored sufficientlyintheliterature.Furtherexperimentalobservationtoverifyorrefutethisproposition isrecommended.
c ElectronicJournalofTheoreticalPhysics.Allrightsreserved.
Keywords:RelativisticQuantumMechanics,Barut’sequation,Barut-Dirac’sequation PACS(2006):03.65.Pm,03.65.Ca,11.10.-z
1.Introduction
ItisknownthatBarut’sequationcouldpredictleptonandhadronmasswithremarkableprecision[1].AplausibleextensionofBarut’sequationisbyusingBarut-Dirac’s modelviainclusionofelectronself-field.Furthermore,anumberofauthorshasextended ∗ vxianto@yahoo.com
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thisequationusingnon-linearfieldtheory[2a][5][5a].Barut’sequationisasfollows[5a]: iγν ∂ν a∂ 2 µ/m + κ Ψ=0(1) where ∂ν = ∂/∂xν andrepeatedindicesimplyasummation[5a].Theremainingparameterscomefromsubstitutionofvariables: m = κ/α1 and a/m = α2/α1 [5a].Inthe meantimeBarut-Dirac-Vigier’sequationcouldbewrittenas: cα.pE + β(mc 2+ ∈ e 2/r) Ψ= [(∈ αe 2)/(4πmc 2 r 2)]iβαΨ(2)
DespitethisapparentlyremarkableresultofBarut’sequation,nonethelessthereisquestionconcerningthephysicalmeaningofhisequation,inparticularfromtheviewpointof non-linearfieldtheory[2a].Thisquestionseemsveryinteresting,inparticularconsideringtheunsolvedquestionconcerningthephysicalmeaningof wavefunction inQuantum Mechanics[4a].Itisknownthatsomeproponentsof‘realism’interpretationofQuantumMechanicspredictthatthereshouldbeacomplete‘realism’descriptionofphysical modelofelectron,wherenon-localhiddenvariablescouldbeincluded[4][1a].Weconsider thatthisquestionremainsopenfordiscussion,inparticularinthecontextofplausible analogbetweenclassicalelectrodynamicsandnon-localquantuminterferenceeffect,via Aharonov-Cashereffect[8].
Inthepresentarticlewearguethatitispossibletoderiveanewwavequantum relativisticequationasanalternativetoBarut-Dirac-Vigier’sequation.Ourdescription isbasedontheknownexactcorrespondence(isomorphism)betweenDiracequationand Maxwellelectromagneticequationsviabiquaternionicrepresentation.Infact,wewill discussfiveapproachesasalternativetoBarut-Diracequation.Andwewouldarguethat thequestionofwhichoftheseapproachesisthemostconsistentwithexperimentaldata remainsopen.OurpropositionofalternativetoBarut(-Dirac)equationwasbasedon characteristicsofBarutequation:
• itisasecond-orderdifferentialequation(1);
• itshallincludethephysicalmeaningofvierbeininquantummechanicalequation;
• ithasneatlinkagewithotherknownequationsinQuantumMechanicsincluding Diracequation[5a],whileitssolutioncouldbedifferentfromDiracapproach[11];
• ourobservationassertsthatitshallalsoincludeaproperintroductionofLorentz force,andaccelerationfromrelativisticfluiddynamics.
Furthermore,inthepresentnotewesubmittheviewpointthatitwouldbemoreconceivableifweinterpretthe vierbein ofthisequationintermsofsuperfluidvelocity[12][13], whichinturnbringsustothenotionoftopologicalelectronicliquid[27].Itsimplicationstoquantizationofcelestialsystemsleadustoargueinfavorofsignaturesof Bose-Einsteincosmology,whichthusfarhasnotbeenexploredsufficientlyyetinthe literature[49a][49b].
Whatwewouldargueinthepresentnoteisthatonecouldexpecttoextendfurtherthisquaternionrepresentationintotheformofunifiedwaveequation,inparticular usingUlrych’srepresentation[7].Whilesuchanattempttointerpret vierbein ofDirac equationhasbeenmadebydeBroglie(intermsof‘Diracfluid’[41]),itseemsthatan
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exactrepresentationintermsofsuperfluidvelocityhasneverbeenmadebefore.From thisviewpointonecouldarguethatthesuperfluidvierbeininterpretationwillmakethe pictureresemblessuperfluidbivacuummodelofKaivarainen[20][21].Furthermore,this propositionseemstosupportprevioushypotheticalargumentbyProf.J-P.Vigieronthe furtherdevelopmentoftheoreticalQuantumMechanics[6]:
“..arevival,inmodern covariantform,oftheetherconceptofthefoundingfathers ofthetheoryoflight(Maxwell,Lorentz,Einstein, etc.).Thisisacrucialquestion,and itnowappearsthatthevacuumisarealphysicalmedium,whichpresentssurprising properties(superfluid, i.e. negligibleresistancetoinertialmotions)...“
Providedthispropositionofunifiedwaveequationintermsofsuperfluidvelocity vierbein correspondstotheobservedfacts,andthenitcouldbeusedtopredictsome newobservations,inparticularinthecontextofcondensed-matteranalogofastrophysics [16][17][18].Thereforeinthelastsectionwewillextendthispropositiontoarguein favorofsignaturesofBose-Einsteincosmology,includingsomerecentrelevantobservation supportingthisargument.
WhilequaternionicQuantumMechanicshasbeenstudiedbeforebyAdleretc.[14c][28], andalsobiquaternionicQuantumMechanics[2][3],itseemsthatinterpretingtherighthand-sideoftheunifiedwaveequationassuperfluid4-velocityhasnotbeenconsidered before,atleastnotyetinthecontextofcylindricalrelativisticfluidofCarterandSklarzHorwitz.
Inderivingtheseequationswewillnotrelyonexactitudeofthesolutions,because asweshallseetheknownproperties,likefinestructureconstantofhydrogen,canbe derivedfromdifferentapproaches[11][15][19][22a].Instead,wewilluse‘correspondence betweenphysicaltheories’asaguidingprinciple,i.e.wearguethatitispossibletoderive somealternativestoBarutequationviageneralizationofvariouswaveequationsknown inQuantumMechanics.Morelinkagebetweentheseequationsimpliesconsistency.
Furtherexperimentalobservationtoverifyorrefutethispropositionisrecommended.
2.Biquaternion,Imaginaryalgebra,UnifiedrelativisticWave Equation BeforewediscussbiquaternionicMaxwellequationsfromunifiedwaveequation,first weshouldreviewUlrych’smethod[7]bydefiningimaginarynumberrepresentationas follows[7]: x = x0 + j.x1,j 2 = 1(3) Thisleadstothemultiplicationandaddition(orsubstraction)rulesforanynumber, whichiscomposedofrealpartandimaginarynumber: (x ± y)=(x0 ± y0)+ j.(x1 ± y1), (4) (xy)=(x0y0 + x1y1)+ j.(x0y1 + x1y0) (5)
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Fromthesebasicimaginarynumbers,Ulrych[7]arguesthatitispossibletofindanew relativisticalgebra,whichcouldberegardedasmodifiedformofstandardquaternion representation.
Oncewedefinethisimaginarynumber,itispossibletodefinefurthersomerelations asfollows[14].Given w = x0 + j.x1,thenitsD-conjugateofwcouldbewrittenas: w = x0 j.x1 (6)
Alsoforanygiventwoimaginarynumbers w1,w2 ∈ D,wegetthefollowingrelations[14]: w1 + w2 =¯ w1 +¯ w2 (7) w1 • w2 =¯ w1 • w2 (8) |w|2 =¯ w • w = x 2 0 x 2 1 (9) |w1 • w2|2 = |w1|2 •|w2|2 (10)
Alloftheseprovideusnothingnew.ForextensionoftheseimaginarynumbersinQuantumMechanics,see[33].Nowwewillreviewafewelementarydefinitionsofquaternions andbiquaternions,whichareprovedtobeuseful.
ItisknownthatbiquaternionscoulddescribeMaxwellequationsinitsoriginalform, andsomeoftheuseofbiquaternionswasdiscussedin[2][34]. Quaternionnumber,Qisdefinedby[33][60]: Q = a + b.i + c.j + d.ka,b,c,d ∈ R, (11a) where i2 = j 2 = k2 = ijk = 1(11b)
Alternatively,onecouldextendthisquaternionnumbertoCliffordalgebra[3a][3][6][25][41], becausehigher-dimensionsCliffordalgebraandanalysisgivethepossibilitytogeneralize thefactorisationsintohigherspatialdimensionsandeventospace-timedomains[70a]. InthisregardquaternionsH ∼ C 0,2,whilestandardimaginarynumbersC∼ C 0,1 [70a].
Biquaternionisanextensionofthisquaternionnumber,anditisdescribedhereusing Hodge-bracketoperator,inlieuofknownHodgeoperator(∗∗ = 1)[5a]: {Q}∗ =(a + iA)+(b + iB).i +(c + iC).j +(d + iD).k, (12a) wherethesecondpart(A,B,C,D)isnormallysettozeroinstandardquaternions[33]. Forquaterniondifferentialoperator,wedefinequaternionNablaoperator: ∇q ≡ c 1.∂/∂t +(∂/∂x)i +(∂/∂y)j +(∂/∂z)k = c 1.∂/∂t + i.∇ (12b)
Andforbiquaterniondifferentialoperator,wedefineaquaternionNabla-Hodgebracketoperator: {∇q }∗≡ (c 1.∂/∂t + c 1.i∂/∂t)+ {∇}∗ (12c)
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whereNabla-Hodge-bracketoperatorisdefinedas: {∇}∗≡ (∂/∂x + i∂/∂X).i +(∂/∂y + i∂/∂Y ).j +(∂/∂z + i∂/∂Z).k. (13a) Itisworthnotingherethatequations(4)-(10)arealsoapplicableforbiquaternionnumber. Whileequations(3)-(12a)areknownintheexistingliterature[33][59],andsometimes called‘biparavector’(Baylis),weprefertocallit‘imaginaryalgebra’withemphasison theuseofHodge-bracketoperator.Itisknownthatdeterminantanddifferentiationof quaternionicequationsaredifferentfromstandarddifferentialequations[59],therefore solutionforthisproblemhasonlybeendevelopedinrecentyears.
TheHodge-bracketoperatorproposedhereincouldbecomemoreusefulifweintroduce quaternionnumber(11a)intheparavectorform[70]:
q = 3 k=0 qk.ek when {qk}⊂ C, {ek |k =1, 2, 3 } (13b) ande0 istheunit.Therefore,biquaternionnumbercouldbewritteninthesameform [70]: {q}∗ = q + iq = 3 k=0 qk.ek + i{ 3 k=0 qk.ek} (13c) NowwearereadytodiscussUlrych’smethodtodescribeunifiedwaveequation[7],which arguesthatitispossibletodefineaunifiedwaveequationintheform[7]:
Dφ(x)= m 2 φ.φ(x), (14) whereunified(wave)differentialoperatorDisdefinedas: D = (PqA)µ PqA µ (15) ToderiveMaxwellequationsfromthisunifiedwaveequation,heusesfreephotonfields expression[7]: DA(x)=0, (16) wherepotentialA(x)isgivenby: A(x)= A0(x)+ jA1(x), (17) andwithelectromagneticfields: Ei(x)= ∂ 0Ai(x) ∂iA0(x), (18) Bi(x)=∈ijk ∂j Ak(x) (19) Insertingtheseequations(17)-(19)into(16),onefindsMaxwellelectromagneticequation [7]:
−∇• E(x) ∂ 0C(x) +ij∇• B(x) j(∇xB(x) ∂ 0E(x) −∇C(x)) i(∇xE(x)+ ∂ 0B(x))=0
(20)
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ThegaugetransformationofthevectorpotentialA(x)isgivenby[7]:
A (x)= A(x)+ ∇Λ(x)/e, (21) whereΛ(x)isascalarfield.Asequations(17)-(18)onlyusesimpledefinitionsofimaginarynumbers(3)-(5),thenanextensionfrom(20)and(21)tobiquaternionicformof Maxwellequationsispossible[2][34].
InordertodefinebiquaternionicrepresentationofMaxwellequations,wecouldextend Ulrych’sdefinitionof unifieddifferentialoperator [7]toitsbiquaternioncounterpart,by usingequation(12a),tobecome:
{D}∗≡ ({P }∗−q{A}∗)µ {P }∗−q{A}∗ µ , (22a) orbydefinition P = i ∇and(13a),equation(22a)couldbewrittenas: {D}∗≡ ( {∇}∗−q{A}∗)µ ( {∇}∗−q{A}∗)µ , (22b)
whereeachcomponentisnowdefinedinitsbiquaternionicrepresentation.Thereforethe biquaternionicformofunifiedwaveequationtakestheform: {D}∗ φ(x)= m 2 φ.φ(x), (23) ifweassumethewavefunctionisnotbiquaternionic,and {D}∗{φ(x)}∗ = m 2 φ.{φ(x)}∗ . (24) ifwesupposethatthewavefunctionalsotakesthesamebiquaternionicform. Now,biquaternionicrepresentationoffreephotonfieldscouldbewritteninthesame waywith(16),asfollows: {D}∗ A(x)=0(25)
Wewillnotexploreherecompletesolutionofthisbiquaternionequation,asithasbeen discussedinvariousliteraturesaforementionedabove,including[2][33][34][59].However, immediateimplicationsofthisbiquaternionformofUlrych’sunifiedequationcanbe describedasfollows.
Ulrych’sfermionwaveequationinthepresenceofelectromagneticfieldreads[7]: (PqA)µ PqA µ ψ = m 2.ψ, (26) whichassertsc=1(conventionallyusedtowritewaveequations).Inaccordancewith Ulrych[7]thisequationimpliesthatthedifferentialoperatorofthequantumwaveequation(LHS)iscomposedofthemomentumoperatorPmultipliedbyitsdualoperator, andtakingintoconsiderationelectromagneticfieldeffectqA.Andbyusingdefinitionof momentumoperator: P = i ∇ (27)
Sowegetthree-dimensionalrelativisticwaveequation[7]:
[( i ∇µ qAµ)( i ∇µ qAµ) ψ]= m 2 .c 2.ψ. (28) whichisKlein-Gordonequation.Its1-dimensionalversionhasalsobeenderivedby Nottale[67,p,29].Aplausibleextensionofequation(28)usingbiquaterniondifferential operatordefinedabove(22a)yields:
[( {∇µ}∗−q{Aµ}∗)( {∇µ}∗−q{Aµ}∗) ψ]= m 2 .c 2.ψ, (29) whichcouldbecalledas‘biquaternionic’Klein-Gordonequation.
ThereforeweconcludethatthereisneatcorrespondencebetweenUlrych’sfermion waveequationandKleinGordonequation,inparticularviabiquaternionicrepresentation.ItisalsoworthnotingthatitcouldbeshownthatSchrodingerequationcouldbe derivedfromKlein-Gordonequation[11],andKlein-GordonequationalsoneatlycorrespondstoDuffin-Kemmer-Petiauequation.Furthermoreitcouldbeprovedthatmodified (quaternion)Klein-GordonequationcouldberelatedtoDiracequation[7].Allofthese linkagesseemtosupportargumentbyGurseyandHesteneswhofindplentyofinterestingfeaturesusingquaternionicDiracequation.Inthisregard,Meessenhasproposeda methodtodescribeelementaryparticlefromKlein-Gordonequation[30].
Byassigningimaginarynumberstoeachcomponent[7,p.26],equation(26)couldbe rewrittenasfollows(bywritingc=1):
arewrittenexplicitly.Nowitispossibletorewriteequation (30)incompletetensorformalism[7],ifPaulimatricesandelectromagneticfieldsare expressedwithantisymmetrictensor,soweget:
Notethatequation(31)isformalidenticalto quadraticform ofDiracequation[7],which supportsargumentsuggestingthatmodified(quaternion)Klein-Gordonequationcould berelatedtoDiracequation.Interestingly,equation(31)isalsoknownintheliterature asFeynman-Gell-Mann’sequation,anditsimplicationswillbediscussedinsubsequent section[5].Interestingly,ifweneglectcontributionoftheelectromagneticfield(qande) component,andusingonly1-dimensionalofthepartialdifferentiation,onegetsawave equationfromFeynmanrules[56,p.6]: ∂µ∂µ + m 2 Ψ=0, (33) whichhasbeenusedtodescribequantum-electrodynamicswithoutrenormalization[56].
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(PqA)µ (PqA)µ eEiijσi eBi σi + m 2 ψ =0, (30)
wherePaulimatrices σi
(PqA)µ PqA µ eσµν F µν + m 2 ψ =0, (31) where Fµν =(∂µAν ∂ν Aµ). (32)
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Furtherextensionofequation(28)couldbemadebyexpressingitintermsof4velocity: [( i ∇µ qAµ)( i ∇µ qAµ) ψ]= pµpµ.ψ. (34) Inthecontextofrelativisticfluid[10][11],onecouldarguethatthis4-velocitycorresponds tosuperfluidvierbein[13][16][17].ThereforewecoulduseCarter-Langlois’equation[12]: µρ.µρ = c 2 .µ 2 , (35)
byreplacingmwiththeeffectivemassvariable µ.Thisequationhasthemeaningof cylindricallysymmetricsuperfluidwithknownmetric[12]: gρσ .dxρ.dxσ = c 2.dt2 + dz2 + r 2.dφ2 + dr2 (36)
Furtherextensionofequation(35)ispossible,asdiscussedbyFischer[13],wherethe effectivemassvariabletermalsoappearsintheLHSofvelocityequation,bydefining momentumofthecontinuumas: pα = µ.uα (37) Thereforeequation(35)nowbecomes: µ 2 .uα.uα = c 2 .µ 2 , (38) wheretheeffectivemassvariablenowacquiresthemeaningofchemicalpotential[13]: µ = ∂ ∈ /∂ρ, (39) and ρ.pα/µ = K/ 2 pα = jα, (40) K = 2 (ρ/µ) . (41a)
ThequantityKisdefinedasthestiffnesscoefficientagainstvariationsoftheorderparameterphase.Alternatively,frommacroscopicdynamicsofBose-Einsteincondensate containingvortexlattice,onecouldwritethechemicalpotentialintheform[57]: µ = µ0 1 (Ω0/ω⊥)2 2/5 (41b)
wherethequantityΩcorrespondstotheangularfrequencyofthesampleandisassumed tobeuniform, ω istheoscillatorfrequency,andchemicalpotentialintheabsenceof rotationisgivenby[57]: µ0 =( ωho/2)(Na/0.0667aho)2/5 (41c)
andNrepresentsthenumberofatomsandaisthecorrespondingoscillatorlength[57]: aho = /Mωho (41d)
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Nowthesoundspeed cs couldberelatedtotheequationsabove,forabarotropicfluid [13],as:
cs = d (ln µ) /d (ln ρ)= K/ 2 d2 ∈ /dρ2 (42)
Usingthisdefinition,thenequation(42)couldberewrittenasfollows: pα = K 1 2 jα =(jα/cs).d2 ∈ /dρ2 , (43)
Introducingthisresult(43)intoequation(34),weget: [( i ∇µ qAµ)( i ∇µ qAµ) ψ]= (jα/cs).d2 ∈ /dρ2 2 .ψ (44)
whichisanalternativeexpressionofrelativisticwavefunctionintermsofsuperfluidsound speed, cs.Notethatthisequationcouldappearonlyifweinterpret4-velocityinterms of superfluidvierbein [11][12].ThereforethisequationisKlein-Gordonequation,where vierbeinisdefinedintermsofsuperfluidvelocity.Alternatively,inconditionwithout electromagneticcharge,thenwecanrewriteequation(44)intheknownformofstandard Klein-Gordonequation[36]: [DµDµψ]= (jα/cs).d2 ∈ /dρ2 2 .ψ. (45)
Therefore,thisalternativerepresentationofKlein-Gordonequation(45)hasthephysical meaningofrelativisticwaveequationforsuperfluidphonon[37][38].
Aplausibleextensionof(44)isalsopossibleusingourdefinitionofbiquaternionic differentialoperator(22a):
{D}∗ ψ = (jα/cs).d2 ∈ /dρ2 2 ψ (46)
whichisanalternativeexpressionfromUlrych’s[7]unifiedrelativisticwaveequation, wherethevierbeinisdefinedintermsofsuperfluidsoundspeed,cs.Thisisthemain resultofthissection.Asalternative,equation(46)couldbewrittenincompactform: [{D}∗ +Γ]Ψ=0, (47) wheretheoperatorΓisdefinedaccordingtothequadraticofequation(43): Γ= (jα/cs).d2 ∈ /dρ2 2 (48a)
Forthesolutionofequation(44)-(47),onecouldreferforinstancetoalternativedescriptionofquarksandleptonsviaSU(4)symmetry[28][58].Aswenoteabove,equation (31)isalsoknownintheliteratureasFeynman-Gell-Mann’sequation,andithasbeen arguedthatithasneatlinkagewithBarutequation[5].Thisassertioncouldmademore conceivablebynotingthatequation(31)isquadraticformofDiracequation.Inthis regard,recentlyKruglovhasconsideredaplausiblegeneralizationofBarutequationvia third-orderdifferentialextensionofDiracequation[60]:
(γµ∂µ + m1)(γν ∂ν + m2)(γα∂α + m3) ψ =0 (48b)
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Itisalsointerestingtonotethatinhispreviouswork,Kruglov[60a]hasarguedinfavor ofDirac-Kahlerequation: (dδ + m) ψ =0, (48c) wheretheoperator(dδ)istheanalogofDiracoperator γµ∂µ.Itseemsplausible, therefore,inthecontextofKruglov’srecentattempttogeneralizeBarutequation[60]to arguethatfurthergeneralizationtobiquaternionicformispossiblebyrewritingequation (47)inthethird-orderequation,byusingourdefinition(12c):
(48d) Therefore,wecouldconsiderthisequationasthefirstalternativeto(generalized)Barut equation.Notethatweusehereequation(12c)insteadof(22a),inaccordancewith Kruglov[60]definition: ∂ν = ∂/∂xν =(∂/∂xm,∂/∂(it))(48e) Insubsequentsections,wewillconsideranumberofotherplausiblealternativestoBarutDirac’sequation,inparticularfromtheviewpointofsuperfluidvierbein.
3.Alternative#2:Barut-Dirac-Feynman-Gell-MannEquation
Itisargued[5,p.4]thatBarutequationisthesumofDiracequationandFeynmanGell-Mann’sequation(31).Butfromtheaforementionedargument,itshouldbeclear thattheFeynman-Gell-Mann’sequationisnothingmorethanUlrych’sfermionwave equation,whichisindeedaquadraticofDiracequation.Therefore,itseemsthatthere shouldbeotherroutetoderiveBarut-Diractypeequation.Inthisregard,wesubmit theviewpointthattheintroductionofelectronself-fieldwouldleadtoanalternativeof Barutequation.
First,letusrewriteequation(31)withassigningtherealcinlieuofc=1:
tion,inparticularifwethenintroduceequation(43)intotheLHS.
Inthisregard,wecanintroduceIbison’sdescriptionofelectronself-energyfromZPE [38]:
[{∇q µ}∗ +pµ][{∇q ν }∗ +pν ][{∇q α}∗ +pα]Ψ=0
(PqA)µ PqA µ eσµν F µν +
c 2 ψ
[( i ∇µ qAµ)( i ∇µ qAµ) eσµν F µν + pµpµ]Ψ=0,
or [( i ∇µ qAµ)( i ∇µ qAµ)+ pµpµ]Ψ=(eσµν F µν )Ψ
m 2
=0, (49) Byusingequation(34),thenFeynman-Gell-Mann’sequationbecomes:
(50)
, (51) whichcanbecalledFeynman-Gell-Mann’sequationwithsuperfluidvierbeininterpreta-
eσµν F µν = m0aµ m0τ0 daλ/dτ + a λ aλuµ/c2
τ0 = e 2/6πε0m0c 3
(52) where
(53)
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Thefirsttermintherighthandsideofequation(52)couldbewrittenintheLorentz form[42][24a,p.12]: m0aµ = m[dv/dt]= e[E + vxB](54) where: E = −∇φ, (55) B = ∇xA. (56) Therefore,bydefininganewparameter[24a,p.12]: ∀ = e[E + vxB]µ m0(e 2/6πε0m0c 3) daλ/dτ + a λ aλuµ/c2 , (57) onecouldrewriteequation(51)intermofequation(43): ( i ∇µ qAµ)( i ∇µ qAµ)+ (jα/cs).d2 ∈ /dρ2 2 Ψ= ∀Ψ, (58) whichcouldberegardedasasecondalternativeexpressionofBarutequation.Therefore weproposetocallitBarut-Dirac-Feynman-Gell-Mannequation.Implicationsofthis equationshouldbeverifiedviaexperiments,inparticularwithcondensed-matterphysics.
4.Alternative#3:SecondOrderDifferentialFormofSchr¨odingerTypeEquation
ItisknownthatBarutequationisatypicalsecond-orderdifferentialequation,which isthereforenon-linear.ThereforeagoodalternativetoBarutequationcouldbederived fromsimilarapproachwithSchr¨odinger’soriginalequation,butthistimeitshouldbe differentiatedtwice.
Inthisregard,itseemsworthnotingherethatitismorepropertouseNoether’sexpressionoftotalenergyinlieuofstandardderivationofSchr¨odinger’sequation(E = p2/2m). AccordingtoNoether’stheorem[39],thetotalenergyofthesystemcorrespondingtothe timetranslationinvarianceisgivenby: E = mc 2 +(cw/2) ∞ 0 γ 2 4πr 2.dr = kµc2 (59) wherekis dimensionless function.Itcouldbeshown,thatforlow-energystatethetotal energycouldbefarlessthan E = mc2.InterestinglyBakhoum[22]hasalsoarguedin favorofusing E = mv2 forexpressionoftotalenergy,whichexpressioncouldbetraced backtoLeibniz.Thereforeitseemspossibletoarguethatexpression E = mv2 ismore generalizedthanthestandardexpressionofspecialrelativity,inparticularbecausethe totalenergynowdependsonactualvelocity[39].
Fromthisnewexpression,itisplausibletorederivequantumrelativisticwaveequation insecond-orderdifferentialexpression,anditturnsoutthenewequationshouldalso includeaLorentz-forceterminthesamewayofequation(57).Thisfeatureisseemingly interesting,becausetheseequationsarederivedfromdifferentapproachfrom(57).
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WestartwithBakhoum’sassertionthatitismoreappropriatetouse E = mv2 , insteadofmoreconvenientform E = mc2.Thisassertionwouldimply[22]:
H 2 = p 2 .c 2 m 2 o.c 2 .v 2 . (60)
Therefore,forphononspeed(cs)inthelimit p → 0,wewrite[37]: E(p) ≡ cs. |p| . (61)
AbitremarkconcerningBakhoum’sexpression,itdoesnotmeantoimplyortointerpret E = mv2 asanassertionthatitimplieszeroenergyforarestmass.Actuallytheproblem comesfrom’mixed’interpretationofwhatwemeanwith’velocity’.InoriginalEinstein’s paper(1905)itisdefinedas’kineticvelocity’,whichcanbemeasuredwhenstandard ’steelrod’hasvelocityapproximatesthespeedoflight.Butinquantummechanics, weareaccustomedtomakeuseitdeliberatelytoexpress’photonspeed’=c.According toBakhoum,togetaconsistentinterpretationbetweenspecialrelativityandquantum mechanics,weshouldtreatthisdefinitionofvelocityaccordingtoitscontext,inparticular toitslinkagewithelectromagneticfield.Therefore,inspecialrelativity1905paper,it shouldbebettertointerpretitas’speedoffreeelectron’,whichapproximatesc.For muon,Spohn[42]hasobtainedv=0.9997cwhichisveryneartoc,butnotexactly=c. Forhydrogenatomwith1electron,theelectronoccupiesthefirstexcitation(quantum numbern=1),whichimpliesthattheirspeedalsoapproximatec,whichthenitisquite safetoassume E ∼ mc2.Butforatomswithlargeamountofelectronsoccupyinglarge quantumnumbers,asBakhoumshowedthatelectronspeedcouldbefarlessthanc, thereforeitwillbemoreexacttouse E = mv2,whereherevshouldbedefinedas ’averageelectronspeed’.Furthermore,inthecontextofrelativisticfluid,wecoulduse Eα = µ.uαuα fromequation(37).
Inthefirstapproximationofrelativisticwaveequation,wecouldderiveKlein-Gordontyperelativisticequationfromequation(60),asfollows.Byintroducinganewparameter: ζ = i(v/c), (62) thenwecanrewriteequation(60)intheknownprocedureofKlein-Gordonequation: E 2 = p 2 .c 2 + ζ 2 m 2 o.c 4 , (63) where E = mv2.[22]Byusingknownsubstitution: E = i.∂/∂t,p = ∇/i, (64) anddividingby( c)2,wegetKlein-Gordon-typerelativisticequation: c 2∂Ψ/∂t + ∇2Ψ= ko 2 Ψ, (65) where ko = ζmoc/ (66)
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OnecouldderiveDirac-typeequationusingsimilarmethod.Buttheuseofnewparameter (62)seemstobeindirect,albeititsimplifiesthesolutionbecauseherewecanusethe samesolutionfromKlein-Gordonequation[30].
Alternatively,onecouldderiveanewquantumrelativisticequation,bynotingthat expressionoftotalenergy E = mv2 isalreadyrelativisticequation.Wewillderivehere twoapproachestogetrelativisticwaveequationfromthisexpressionoftotalenergy. Thefirstapproach,isusingUlrych’s[7]methodasfollows: E = mv 2 = p.v (67)
Takingsquareofthisexpression,weget: E 2 = p 2 .v 2 (68) or p 2 = E 2/v2 (69)
NowweuseUlrych’ssubstitution[7]: (PqA)µ PqA µ = p 2 , (70)
andintroducingstandardsubstitutioninQuantumMechanics(64),onegets: (PqA)µ PqA µ Ψ= v 2 (i.∂/∂t)2Ψ, (71) or ( i ∇µ qAµ)( i ∇µ qAµ) (i/v.∂/∂t)2 Ψ=0. (72a) whichcanbecalledasNoether-Ulrych-Feynman-Gell-Mann’s(NUFG)equation.Thisis thethirdalternativetoBarut-Diracequation.
Alternatively,byusingstandarddefinitionp=m.v,wecanrewriteequation(71)in formofequation(43): (PqA)µ PqA µ Ψ= m 2 (jα/cs).d2 ∈ /dρ2 2 (i.∂/∂t)2Ψ (72b)
InordertoverifythatwecanusethesamemethodwithSchr¨odingerequationtoderive nonlinearwaveequation,letusconsiderOleinik’snonlinearwaveequation.Itisargued thattheproperequationofmotionisnottheDiracorSchr¨odingerequation,butanequationwithanewself-energyterm[24].Thiswouldmeanthatthereisapairwavefunction toincludeelectroninteractionwithitssurroundingmedium.Therefore,thestandard Schr¨odingerequationbecomesnonlinearequationsofmotion[24]:
i∂/∂t + ∇2/2mU (x) Ψ(x) Ψ(x) =0(73) whereweuse =1forconvenience.
Fromthisequation,onecangettherelativisticversioncorrespondingtoDiracequation[24].Interestingly,Froelich[66]hasconsideredequationofmotionforthefew-body
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systemsassociatedwiththehydrogen-antihydrogenpairsusingradialSchr¨odinger-type equation.Therefore,itseemsinterestingtoconsiderequation(73)alsointhecontextof hydrogen-antihydrogenmolecule.
Andbecauseequation(73)isderivedfromthestandarddefinitionoftotalenergy E = p2/2m,thenourmethodtouseequation(60)seemstobealogicalextension ofOleinik’smethod.Togetnonlinearversionsimilartoequation(73),thenwecould rewriteequation(72a)as: ( i ∇µ qAµ)( i ∇µ qAµ) (i/v.∂/∂t)2 Ψ(x) Ψ(x) =0 (74)
What’smoreinterestinghere,isthatOleinik[24a,p.12]hasshownthatequation(73) couldleadtoanexpressionofNewtonian-Lorentzforcesimilartoequation(54): m0aµ = m[d2r/dt2]= e[E + v × B](75)
ThisverifiesouraforementionedpropositionthatagoodalternativetoBarut’sequation shouldincludeaLorentz-forceterminwaveequation.Inotherwords,fromequation (73)wefindneatlinkagebetweenSchr¨odingerequation,nonlinearwave,andLorentzforce.Wewillusethislinkageinthefollowingsection.Itturnsoutthatwecanfind apropergeneralizationofBarut’sequationviaintroductionofNewtonian-acceleration fromvelocityoftherelativisticfluidinsimilarformofLorentzforce.
5.Alternative#4:Lorentz-force&NewtonianAcceleration Method
Forthefourthmethod,wewillintroduceLeibnizrule[40]intoequation(67)via differentiationwithrespecttotime,whichyields:
dE/dt = d[p.v]/dt = v.[dp/dt]+ p.[dv/dt](76)
ThenextstepistakingderivationoftheknownsubstitutioninQM:
dE/dt = i.∂ 2/∂t2 , (77) dp/dt = d( i ∇)/dt = i ˙ ∇ Now,substitutingbackequation(77)and(64)intoequation(76),weget: (i.∂ 2/∂t2)Ψ=(v.[ i ˙ ∇] [dv/dt].i ∇)Ψ (78)
Atthispoint,wenotethatthesecondtermintherighthandsideofequation(78)could bewrittenintheLorentzforceform[42],andfollowingequation(54): [dv/dt]= e/m.(E + vxB)(79) where: E = −∇φ, (80)
B = ∇xA. (81)
Therefore,wecanrewriteequation(78)intheform:
(i.∂2/∂t2)Ψ=(v.[ i ∇] e/m.[E + vxB].i ∇)Ψ, (82)
whichisanewwaverelativisticquantumequationasalternativetoBarutequation. Toourpresentknowledge,thisalternativewaveequation(82)hasneverbeenderived elsewhere.
Asanalternativetoequation(79),wecanrewriteLorentzformintermofNewtonianacceleration.Inthisregard,itisworthnotingthatthedefinitionofaccelerationof relativisticfluidisnotwidelyacceptedyet[10].Thereforewewillusehereresultfrom relativisticfieldequationsfromPoissonprocess[46],fromwhichwegetanexpressionof acceleration[46]:
dv/dt = /2m.(∂ 2u/∂x2) v.∂u/∂x + u.∂v/∂xm 1.∂V/∂x = ∃ (83)
Therefore,bysubstitutingthisequationinto(78),weget:
(i.∂ 2/∂t2)Ψ=(v.[ i ˙ ∇] −∃.i ∇)Ψ, (84) whichcanbeconsideredasabetteralternativetoequation(82).
6.Alternative#5:Schr¨odinger-Ginzburg-LandauEquationand QuantizationofCelestialSystems
Intheprecedingsection(#4),wehavefoundtheneatlinkagebetweenSchr¨odinger equation,nonlinearwave,andLorentz-force,whichindicatesapossibilitytobeconsidered asalternativetoBarutequation.Now,asthefifthalternativemethod,itwillbeshown thatwecanexpecttogeneralizeSchr¨odingerequationtodescribequantizationofcelestial sytems.Whilethisnotionofmacro-quantizationisnotwidelyacceptedyet,aswewill seethelogarithmicnatureofSchr¨odingerequationissufficienttoensureitsapplicability tolargersystems.Asalternative,wewillalsodiscussanoutlineforderivingSchr¨odinger equationfromsimplificationofGinzburg-Landauequation.ItisknownthatGinzburgLandauequationexhibitsfractalcharacter.
First,letusrewriteSchr¨odingerequation(73)initscommonform: i∂/∂t + ∇2/2mU (x) Ψ=0(85) whereweuse =1forconvenience,or (i∂/∂t)Ψ= H.Ψ(86)
Now,itisworthnotingherethatEnglman&Yahalom[4a]arguethatthisequation exhibitslogarithmiccharacter: lnΨ(x,t)=ln(|Ψ(x,t)|)+ i. arg(Ψ(x,t))(87)
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Schr¨odingeralreadyknewthisexpressionin1926,whichthenheusedittopropose hisequationcalled‘eigentlicheWellengleichung’ [4a].Thereforeequation(85)canbe rewrittenasfollows: 2m(∂ ln |Ψ| /∂t)+2∇ ln |Ψ| ∇ arg[Ψ]+ ∇ ∇ arg[Ψ]=0(88)
Interestingly,Nottale’sscale-relativisticmethod[43][44]wasalsobasedongeneralization ofSchr¨odingerequationtodescribequantizationofcelestialsystems.Itisknownthat Nottale-Schumacher’smethod[45]couldpredictnewexoplanetsingoodagreementwith observeddata.Nottale’sscale-relativisticmethodisessentiallybasedontheuseoffirstorderscale-differentiationmethoddefinedasfollows[43][44]:
∂V/∂(ln δt)= β(V )= a + bV + (89)
Nowitseemsclearthatthelogarithmicderivation,whichisessentialinscale-relativity approach,alsohasbeendescribedproperlyinSchr¨odinger’soriginalequation[4a].In otherword,itslogarithmicformensuresapplicabilityofSchr¨odingerequationtodescribe macroquantizationofcelestialsystems.
Toemphasizethisassertionofthepossibilitytodescribequantizationofcelestial systems,letusreturnforawhiletotheprecedingsectionwhereweuseFischer’description [13]ofrelativisticmomentumof4-velocity(37)-(38).InterestinglyFischer[13]arguesthat thecirculationleadingtoequation(37)-(38)isintherelativisticdensesuperfluid,defined astheintegralofthemomentum: γs = pµdxµ =2π.Nv , (90)
andisquantizedintomultiplesofPlanck’squantumofaction.ThisequationisthecovariantBohr-Sommerfeldquantizationof γs.AndthenFischer[13]concludesthatthe Maxwellequationsofordinaryelectromagnetismcanbecastintotheformofconservation equationsofrelativisticperfectfluidhydrodynamics [10],ingoodagreementwithVigier’s guessasmentionedabove.Furthermore,thetopologicalcharacterofequation(90)correspondstothenotionoftopologicalelectronicliquid,wherecompressibleelectronicliquid representssuperfluidity[27].
Itisworthnotinghere,becauseherevorticesaredefinedaselementaryobjectsinthe formofstabletopologicalexcitations[13],thenequation(90)couldbeinterpretedas signaturesofBohr-Sommerfeldquantizationfromtopologicalquantizedvortices.Fischer [13]alsoremarksthatequation(90)isquiteinterestingforthestudyofsuperfluidrotation inthecontextofgravitation.Interestingly,applicationofBohr-Sommerfeldquantization tocelestialsystemsisknowninliterature[47][48],whichhereinthecontextofFischer’s argumentsitseemsplausibletosuggestthatquantizationofcelestialsystemsactually correspondstosuperfluid-quantizedvorticesatlarge-scale[27].Inouropinion,thisresult supportsknownexperimentssuggestingneatcorrespondencebetweencondensedmatter physicsandvariouscosmologyphenomena[16]-[19].
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Tomaketheconclusionthatquantizationofcelestialsystemsactuallycorrespondsto superfluid-quantizedvorticesatlarge-scaleabitconceivable,letusconsideranillustration ofquantizationofcelestialorbitinsolarsystem.
Inordertoobtainplanetaryorbitpredictionfromthishypothesiswecouldbeginwith theBohr-Sommerfeld’sconjectureofquantizationofangularmomentum.Thisconjecture mayoriginatefromthefactthataccordingtoBCStheory,superconductivitycanexhibit macroquantumphenomena[16][65].Inprinciple,thishypothesisstartswithobservation thatinquantumfluidsystemslikesuperfluidity,itisknownthatsuchvortexesaresubject toquantizationconditionofintegermultiplesof2π,or vs.dl =2π.n/m4.Asweknow, forthewavefunctiontobewelldefinedandunique,themomentamustsatisfyBohrSommerfeld’squantizationcondition: Γ
p.dx =2π.n (91)
foranyclosedclassicalorbitΓ.Forthefreeparticleofunitmassontheunitspherethe left-handsideis[49]: T 0
v 2.dτ = ω 2.T =2π.ω (92) whereT=2π/ω istheperiodoftheorbit.Hencethequantizationruleamountstoquantizationoftherotationfrequency(theangularmomentum):ω = n .Thenwecanwrite theforcebalancerelationofNewton’sequationofmotion[49]: GMm/r2 = mv 2/r (93)
UsingBohr-Sommerfeld’shypothesisofquantizationofangularmomentum,anewconstantgwasintroduced: mvr = ng/2π (94)
JustlikeintheelementaryBohrtheory(beforeSchr¨odinger),thispairofequationsyields aknownsimplesolutionfortheorbitradiusforanyquantumnumberoftheform[49]: r = n 2 .g 2/(4π 2.GM.m2)(95) whichcanberewrittenintheknownform[43][44]: r = n 2.GM/v2 o (96) wherer,n,G,M,vo representsorbitradii,quantumnumber(n=1,2,3,...),Newton gravitationconstant,andmassofthenucleusoforbit,andspecificvelocity,respectively. Inthisequation(96),wedenote: vo =(2π/g).GMm (97)
Thevalueofmisanadjustableparameter(similartog).[43][44]
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Usingthisequation(96),wecouldpredictquantizationofcelestialorbitsinthesolar system,whereforJovianplanetsweuseleast-squaremethodanddefineMintermsof reducedmass µ =(M1.M2)/(M1 + M2).FromthisviewpointtheresultisshowninTable 1below[49]:
Table1.Comparisonofpredictionandobservedorbitdistanceof planetsinSolarsystem(in0.1AUunit)[49]
Object No. Bode Nottale CSV Observed ∆(%) 1 0.4 0.428 2 1.7 1.71
Mercury 3 4 3.9 3.85 3.87 0.52
Venus 4 7 6.8 6.84 7.32 6.50
Earth 5 10 10.7 10.70 10.00 -6.95
Mars 6 16 15.4 15.4 15.24 -1.05
Hungarias 7 21.0 20.96 20.99 0.14
Asteroid 8 27.4 27.38 27.0 1.40
Camilla 9 34.7 34.6 31.5 -10.00
Object No. Bode Nottale CSV Observed ∆(%)
Jupiter 2 52 45.52 52.03 12.51
Saturn 3 100 102.4 95.39 -7.38
Uranus 4 196 182.1 191.9 5.11 Neptune 5 284.5 301 5.48
Pluto 6 388 409.7 395 -3.72
2003EL61 7 557.7 520 -7.24
Sedna 8 722 728.4 760 4.16
2003UB31 9 921.8 970 4.96
Unobserved 10 1138.1
Unobserved 11 1377.1
Forcomparisonpurpose,wealsoincludesomerecentobservationbyM.Brown et al.fromCaltech[50][51][52][53].ItisknownthatBrown etal.havereportednotless thanfournewplanetoidsintheoutersideofPlutoorbit,including2003EL61(at52AU), 2005FY9(at52AU),2003VB12(at76AU,dubbedas Sedna.) AndrecentlyBrownandhis teamreportedanewplanetoidfinding,called2003UB31(97AU).Thisisnottoinclude Quaoar (42AU),whichhasorbitdistancemoreorlessnearPluto(39.5AU),thereforethis objectisexcludedfromourdiscussion.Itisinterestingtoremarkherethatallofthose
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new‘planetoids’arewithin8%boundfromourpredictionofcelestialquantizationbased ontheaboveBohr-Sommerfeldquantizationhypothesis(Table1).Whilethisprediction isnotsoprecisecomparedtotheobserveddata,onecouldarguethatthe8%boundlimit alsocorrespondstotheremainingplanets,includinginnerplanets.Thereforethis8% uncertaintycouldbeattributedtomacroquantumuncertaintyandotherlocalfactors.
Whileourpreviouspredictiononlylimitsnewplanetfindinguntiln=9ofJovian planets(outersolarsystem),itseemsthatthereareenoughreasonstosupposethat moreplanetoidsaretobefoundinthenearfuture.Thereforeitisrecommendedto extendfurtherthesamequantizationmethodtolargernvalues.Forpredictionpurpose, weincludeinTable1newexpectedorbitsbasedonthesamequantizationprocedurewe outlinedbefore.ForJovianplanetscorrespondingtoquantumnumbern=10andn=11, ourmethodsuggeststhatitislikelytofindneworbitsaround113.81AUand137.71AU, respectively.Itisrecommendedtherefore,tofindnewplanetoidsaroundthesepredicted orbits.
Asaninterestingalternativemethodsupportingthispropositionofquantizationfrom superfluid-quantizedvortices(90),itisworthnotingherethatKiehnhasarguedinfavor ofre-interpretingthesquareofthewavefunctionofSchr¨odingerequationasthevorticity distribution(includingtopologicalvorticitydefects)inthefluid[61].Fromthisviewpoint, Kiehnsuggeststhatthereis exactmapping fromSchr¨odingerequationtoNavier-Stokes equation,usingthenotionofquantumvorticity[61].Interestingly,deAndrade&Sivaram [62]alsosuggestthatthereexistsformalanalogybetweenSchr¨odingerequationandthe Navier-Stokesviscousdissipationequation:
∂V/∂t = ν.∇2V (98) where ν isthekinematicviscosity.Theirargumentwasbasedonpropagationtorsion modelforquantizedvortices[62].WhileKiehn’sargumentwasintendedforordinaryfluid, nonethelesstheneatlinkagebetweenNavier-Stokesequationandsuperfluidturbulence isknowninliterature[63][64][21].
Therefore,itseemsinterestingtoconsideraplausiblegeneralizationofSchr¨odinger equationinparticularinthecontextofviscousdissipationmethod.First,wecouldwrite Schr¨odingerequationforachargedparticleinteractingwithanexternalelectromagnetic field[61]intheformofequation(28)and(85): [( i ∇µ qAµ)( i ∇µ qAµ)Ψ]=[ i2m.∂/∂t +2mU (x)]Ψ (99)
Inthepresenceofelectromagneticpotential[69],onecouldincludeanothertermintothe LHSofequation(99): [( i ∇µ qAµ)( i ∇µ qAµ)+ eAo]Ψ=2m [ i∂/∂t + U (x)]Ψ. (100)
ThisequationhasthephysicalmeaningofSchr¨odingerequationforachargedparticleinteractingwithanexternalelectromagneticfield,whichtakesintoconsiderationAharonov effect[69].Topologicalphaseshiftbecomesitsimmediateimplication,asalreadyconsideredbyKiehn[61].
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Therefore,inthecontextofquaternionicrepresentationofSchr¨odingerequation[70], onecouldwriteequation(100)intermsofequation[22a]: [{D}∗ +eAo]Ψ=2m [ i∂/∂t + U (x)]Ψ (101)
Inthecontextoftopologicalphaseshift[69],itwouldbeinterestingthereforetofindthe scalarpartofequation(101)inexperiments[8].
Asdescribedabove,onecouldalsoderiveequation(96)fromscale-relativisticSchr¨odinger equation[43][44].Itshouldbenotedhere,however,thatNottale’smethod[43][44]differsappreciablyfromtheviscousdissipativeNavier-StokesapproachofKiehn,because NottaleonlyconsidershisequationintheEuler-Newtonlimit[67][68].Nonetheless,as weshallsee,itispossibletofindageneralizationofSchr¨odingerequationfromNottale’s approachinsimilarformwithequation(101).Inordertodoso,firstwecouldrewrite Nottale’sgeneralizedSchr¨odingerequationviadiffusionmethod[67][71]:
i2mγ (iγ + a(t)/2)(∂ψ/∂x)2 ψ 2 + ∂ ln ψ/∂t +iγa(t) (∂ 2ψ/∂x2) /ψ =Φ+ a(x) (102) where ψ,a(x),Φ, γ eachrepresentsclassicalwavefunction,anarbitraryconstant,scalar potential,andaconstant,respectively.Ifthefunctionf(t)issuchthat
a(t)= i2γ,α(x)=0, (103) γ = /2m (104) thenonerecoverstheoriginalSchr¨odingerequation(85).
Furthergeneralizationispossibleifwerewriteequation(102)inquaternionform similartoequation(101):
i2mγ [ (iγ + a(t)/2)({∇}∗)2ψ 2 + ∂ ln ψ/∂t] +iγ.a(t) ({∇ }∗) /ψ =Φ+ a(x) (105)
Alternatively,withrespecttooursuperfluiddynamicsinterpretation[13],onecouldalso getSchr¨odingerequationfromsimplificationofGinzburg-Landauequation.Thismethod willbediscussedsubsequently.ItisknownthatGinzburg-Landauequationcanbeused toexplainvariousaspectsofsuperfluiddynamics[16][17][18].
AccordingtoGross,Pitaevskii,Ginzburg,wavefunctionofNbosonsofareducedmass m*canbedescribedas[55]: ( 2/2m∗) ∇2ψ + κ |ψ|2 ψ = i.∂ψ/∂t (106)
Forsomeconditions(wherethetemperaturedependenceofthedensityofCooperpairs, ns,isjustthesquareoforderparameter.Or |ψ|2 ≈ ns = A(Tc T )),thenitispossible
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toreplacethepotentialenergyterminequation(106)withHulthenpotential.This substitutionyields:
( 2/2m∗).∇2ψ + VHulthen.ψ = i.∂ψ/∂t (107) where VHulthen(r)= κ |ψ|2 ≈−Ze2.δ.e δr/(1 e δr)(108)
Thisequation(108)hasapairofexactsolutions.Itcouldbeshownthatforsmall valuesof δ,theHulthenpotential(108)approximatestheeffectiveCoulombpotential,in particularforlargeradius[14b]:
V eff Coulomb = e 2/r + ( +1) 2/(2mr 2)(109)
Thereforeequation(109)couldberewrittenas: 2∇2ψ/2m ∗ + e 2/r + ( +1) 2/(2mr 2) .ψ = i.∂ψ/∂t (110) Forlargeradii,secondterminthesquarebracketofLHSofequation(110)reducesto zero[54], ( +1). 2/(2mr 2) → 0(111) sowecanwriteequation(110)as: ( 2∇2ψ/2m ∗ +U ).ψ = i.∂ψ/∂t (112) whereCoulombpotentialcanbewrittenas: U = e 2/r (113)
Thisequation(112)isnothingbutSchr¨odingerequation(85).ThereforewehaverederivedSchr¨odingerequationfromsimplificationofGinzburg-Landauequation,inthe limitofsmallscreeningparameter.CalculationshowsthatintroducingthisHulthen effect(108)intoequation(107)willyielddifferentresultonlyattheorderof10 39 m comparedtopredictionusingequation(110),whichisofcoursenegligible.Therefore,we concludethatformostcelestialquantizationproblemstheresultofTDGL-Hulthen(110) is essentially thesamewiththeresultderivedfromequation(85).Now,toderiveequation (96)fromSchr¨odingerequation,thereaderisadvisedtoseeNottale’sscale-relativistic method[43][44].
WhatwewouldemphasizehereisthatthisderivationofSchr¨odingerequationfrom Ginzburg-Landauequationisingoodagreementwithourpreviousconjecturethatequation(90)impliesmacroquantizationcorrespondingtosuperfluid-quantizedvortices.This conclusionisthemainresultofthissection.Itisalsoworthnotingherethatthereis recentattempttointroduceGinzburg-Landauequationinthecontextofmicrotubule dynamics[72],whichimplieswideapplicabilityofthisequation.
Inthefollowingsection,wewouldextendthisargumentbynotingthatmacroquantizationofcelestialsystemsimpliesthe topological characterofsuperfluid-quantizedvortices,andcosmicmicrowavebackgroundradiationisalsoanindicationofsuchtopological superfluidvortices.
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7.FurtherNote:SignaturesofBose-EinsteinCosmology
ItisknownthatCMBRtemperature(2.73K)isconventionallyassumedtocomefrom thehotearlyUniverse,whichthencoolsadiabaticallytothepresentepoch.Nonetheless thisdescriptionisnotwithoutproblems,suchashowtoconsiderthesmalltemperature fluctuationsofCMBRastheseedsthatgiverisetolarge-scalestructuresuchasgalaxy formation[73].FurthermoreitisknownthatCMBRfollowsPlanckradiationlawwith highprecision,soonecouldarguewhetheritalsoindicatesthatlarge-scalestructures obeyquantum-mechanicalprinciples.Thereforewewillconsiderheresomealternative hypothesis,whichsupporttheideaoflow-energyquantummechanicscorrespondingto superfluidvorticesdescribedintheprecedingsection.
Inrecentyears,therearealternativeargumentssuggestingthattheUniverseindeed resemblesthedynamicsofNnumberofPlanckianoscillators.Usingsimilarassumption, forinstanceAntoniadis etal.[74]arguethatCMBRtemperaturecouldbederivedusing conformalinvariance symmetry,insteadofusingHarrison-Zel’dovichspectrum.Other hasderivedCMBRtemperaturefromWeylframework[74a].Furthermore,iftheCMBR temperature2.73Kcouldbeinterpretedaslow-energypartofthePlanckdistribution law,thenitseemstoindicatethattheUniverseresemblesBose-Einsteincondensate[75]. Pervushin etal.alsoarguedthatCMBRtemperaturecouldbederivedfromconformal cosmologywithrelativeunits[76].TheseargumentsseemtosupportWinterberg’shypothesisthatsuperfluidphonon-rotonaethercouldexplaintheoriginofcosmicmicrowave backgroundradiation[18][19].
Ofcourse,itdoesnotmeanthatCMBRdatafitsperfectlywithPlanckdistribution law.IthasbeenarguedthatCMBRdatamorecorrespondstoq-deformedPlanckradiationdistribution[77].However,thisargumentrequiresfurtheranalysis.Whatinterests ushereisthattherearereasonstobelievethataquantumuniversebasedonPlanck scaleisnotmerelyapurehypotheticalnotion,inparticularifweconsiderknownanalogy betweensuperfluidityandvariouscosmologyphenomena[16][17].
Anotherargumentcomesfromfractalityargument.IthasbeendiscoveredbyFeynmanthatthetypicalquantummechanicalpathsarenon-differentiableandfractal[67]. Inthisregard,ithasbeenarguedthattheUniverseisembeddedinCantorianfractal spacetimehavingnon-integerHausdorffdimension[78],andfromthisviewpointitcould beinferredthatthecorrelatedfluctuationsofthefractalspacetimeisanalogoustothe Bose-Einsteincondensatephenomenon.Interestingly,thereisalsohypothesissuggesting thatHausdorffdimensioncouldberelatedtotemperatureofidealBosegas[79].
Fromtheseaforementionedarguments,itseemsplausibletosupposethatthatCMBR temperature2.73KcouldbeinterpretedasasignatureofBose-Einsteincondensatecosmology.Inparticular,onecouldconsider[22b]that“thisrelationshipcomesdirectlyfrom Boltzmann’slawN=B.k.T,whereNisthebackgroundnoisepower;Tisthebackground temperatureindegreesKelvin;andBisthebandwidthofthebackgroundradiation.It followsthattheratio(N/kB)forthecosmicbackgroundradiationisapproximatelyequal to”e”,becauseweusuallyconverttheequationtodecibelsbytakingnaturallogarithm.
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Therelationshipisasolidoneinfact.”Fromthisviewpoint,itseemsquiteconceivableto explainwhyCMBRtemperature2.73Kisnearenoughtoknownnumbere=2.71828..., whichseemstosuggestthatthelogarithmicformofSchr¨odingerequation(‘eigentliche Wellengleichung’)[4a] mayhave adeeplinkagewiththisnumbere=2.71828...
Nonetheless,werecognizethatthispropositionrequiresfurtheranalysisbeforewe couldregarditasconclusive.Butwecandescribeheresomeargumentstosupportthe newinterpretationsupportingthisBose-Einsteincosmologyargument:
• FromFischer’sargument[13]weknowthatBohr-Sommerfeldquantizationfrom superfluidvorticecouldexhibitatallscales,includingcelestialquantization.This propositioncomesdirectlyfromhisassertionof thetopologicalcharacter ofsuperfluid vortices,becausesuperfluidistopologicalelectronicliquid[27].
• Extendingfurthertheaforementionedhypothesisoftopologicalsuperfluidvortices, thenitseemsinterestingtocompareitwithtopologicalanalysisofCOBE-DMR data.G.Rocha etal.[80]argueusingwaveletapproachwithMexicanHatpotential thatitispossibletointerpretthedataasclueforafinitetorusUniverse,albeitnot conclusiveenough.
• Interestingly,thisconjecturecouldberelatedtoBulgadaev’sargument[81]suggestingthattopologicalquantumnumbercouldberelatedtotorusstructureasstable soliton [81a].Ineffect,thisseemstoimplythatthebasicstructureofphysical phenomenathroughoutallscalescouldtaketheformoftopologicaltorus.
Inotherwords,thetopologicalcharacterofsuperfluidvorticesimpliesthatitispossibleto generalizesuperfluidvorticestolargescales.AndthetopologicalcharacterofCMBRdata seemstosupportourpropositionthattheuniverseindeedexhibitstopologicalstructures. ItfollowsthenthatCMBRtemperatureistopological[80]inthesensethatthesuperfluid natureofbackgroundtemperature[18][19]couldbeexplainedfromtopologicalsuperfluid vortices.
Interestingly,similarargumenthasbeenpointedoutbyanumberofauthorsby mentioningnon-GaussianpartofCMBRspectrum.However,furtherdiscussiononthis issuerequiresanothernote.
8.ConcludingRemarks
ItisknownthatBarutequationcouldpredictleptonmass(andalsohadronmass) withremarkableprecision.Therefore,inthepresentarticle,weattempttofindplausible linkagebetweenDirac-Maxwell’sisomorphismandBarut-Dirac-Vigierequation.From thispropositionwecouldfindaunifiedwaveequationintermsofsuperfluidvelocity (vierbein),whichthencouldbeusedasbasistoderivesomealternativedescriptionsof Barutequation.Furtherexperimentisrequiredtoverifywhichequationisthemost reliable.
Inthepresentnotewesubmittheviewpointthatitwouldbemoreconceivableifwe interpretthe vierbein oftheunifiedwaveequationintermsofsuperfluidvelocity,which inturnbringsustothenotionoftopologicalelectronicliquid.Nonetheless,theproposed
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imaginaryalgebradiscussedhereinisonlyatitselementaryform,anditrequiresfurther analysisinparticularinthecontextof[5a][7][14][28].Itislikelythatthissubjectwill becomethesubjectofsubsequentpaper.
Furthermore,thenotionoftopologicalelectronicliquidcouldleadtotopologicalsuperfluidvortices,whichmayexplaintheoriginofmacroquantizationofcelestialsystems andperhapsalsotopologicalcharacterofCosmicMicrowaveBackgroundRadiations. Nonetheless,suchapropositionrequiresfurtheranalysisbeforeitcanbeconsideredas conclusive.
Providedtheaforementionedpropositionsofusingsuperfluidvelocity(vierbein)to describeunifiedwaveequationcorrespondtotheobservedfacts,andtheninprincipleit seemstosupportargumentsinfavorofpossibilitytoobservecondensed-matterhadronic reaction.
Acknowledgement
TheauthorwouldthanktoProfs.C.Castro,M.Pitk¨anen,R.M.Kiehn,EzzatG. Bakhoum,A.Kaivarainen,P.LaViolette,F.Smarandache,andE.Scholz,forinsightful discussionsandremarks.SpecialthanksgotoProf.C.CastroforsuggestingfindinglinkagebetweenquaternionicKlein-GordonequationandDuffin-Kemmer-Petiauequation, toProf.EzzatG.BakhoumforhisremarkonBoltzmanndistributionanditslinkageto CMBRtemperature.
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