WhatWillHappen,IfZeroSpinParticle PossessesSpinRotationalConstruction, WithNon-ZeroEigenvaluesOf SpinAngularMomentum? ShaoXuRen InstituteofPhysicalScienceandEngineering TongjiUniversity,200092,Shanghai,China Correspondingemail:shaoxu-ren@hotmail.com Received 8 November 2016; Accepted 2 December 2016; Published 5 December 2016 ———————————————————————————————————– ———————————————————————————————————– Abstract Thereisnoanyspinrotationalconstructionforzerospinparticle,Casimir operatorandthethiredcomponentofzerospinparticleare001 2 and0 respectively.Further,therearenospininteractionsbetweenzerospinparticleand otherspinparticles. Thispapershows:inSpinTopologicalSpace,STS[1],thethirdcomponentof zerospinparticlepossessesnon-zeroeigenvaluesbesidesoriginalzerovalue,this leadstoamiraculousspininteractionphenomenonbetweenzerospinparticleand otherspinparticles.InSTS,zerospinparticlecould"dissolveotherspinparticles", degradethevaluesoftheirCasimiroperator,anddecaythesespinparticlesinto otherformsofspinparticle. Keywords zerospinparticle;non-Hermitianmatrix;non-zeroeigenvalues;Casimiroperator; thethirdcomponent;SpinTopologicalSpace,STS;bindingenergyofspinparticles ———————————————————————————————————– ———————————————————————————————————– 1Introduction Inquantummachenics,themeasurablespinpropertiesofwell-knownallbosons andfermionsaredemostratedbytwodiagonaloperatorswhichcalledCasimir operatorandthethirdcomponentofspinparticles. ThevaluessofspinofanyspinparticleisdiscribedbytheirCasimir operatorsss1 2 .Thegreaterthes,thegreatertheirCasimiroperators. Andforthethirdcomponentsofthesespinparticles,themaximumeigenvalueis justs,theresteigenvaluesoftheirsarealwayslessthanthevalueofs. Inquantummachenics,zerospinparticle,zsp0hasnorotationalconstruction. Sothereisactuallynoanyspinrepresentationf orzspinphysicsandMathworld. CasimirOperatorandthethirdcomponentofzerospinparticlecanbeobviously andtriviallydepictedasstatements001 2 and0,whichdonotcontradict angularmomeutumtheory. Journal of Modern Physics, 2016, 7, 2257-2265 http://www.scirp.org/journal/jmp ISSN Online: 2153-120X ISSN Print: 2153-1196 DOI: 10.4236/jmp.2016.716194 December 5, 2016
Paragraph2shows:inSpinTopologicalSpace,STS,thecaseaboutthetwo operatorsmentionedabovehaveaslightlydiff e rentbehavior:Casimiroperatorof zerospinparticleremaintobetheformof001I 0 2 ,referto(4). Butthethirdcomponentofzerospinparticleturnsintoaninf initedimensional matrix,referto(3)ordiagonal(10.0),whichshowsthatbesidesazerovalue eigenvalue0lyingatthecenter" 0"of(3)ordiagonal(10.0),zerospinparticle couldpossesssnon-zeroeigenvalues,whichevenbegreaterorlessthan0! Paragraph3describesthebasicsofSTS.InSTS,thespinspaceofeachspin particleisnolongerdependenteachotherasweusuallyfimilliarwithbefore.Now, well-knownallbosonsandfermionsareabtributedtoonespinspace,STS. Furtherwecanuseagroupofunifiedsubscriptsjandasubscript−1to describespinclassficationaboutbosonsandfermions.Raisingoprtators j (8), loweringoperators −1 − (9)andthethirdcomponents(10)givedetailedaccountof thefunctionofsubscriptsjinspinclassfication. Asaspecialexample,theestablishmentofnon-trivialspinrepresentations(1), (2),(3),(4)ofzerospinisjustduetotwoinfinitedimensionalnon-Hermitian matrices j (8.0)and −1 − (9). Paragraph2andParagraph3prepareconceptualtoolstodiscussParagraph4. BymeansofadditionofspinangularmomentaintheframeofSTS,pragraph4 consindersthespincoupling(13)ofasingleboson,orasinglefermionwithk zerospinparticlesandobtainsthegeneralformulaforthecoupling.For understandingthephysicalpictureof"WhatWillHappen",detailedaccountofk 1,2aregiveninTable2andTable3. Theresultsareincredible,theboson,orthefermionseemslikethesolute.And zerospinparticleseemslikeasolventwhichhasmiraculouspowertodiluteand reducethevalueofCasimiroperatorofbosonorfermionintheprocessofthe spincoupling(13). Whenthenumberofzerospinparticleincrease,thevaluesofCasimiroperator ofbosons,fermionsbecomelessandless,theyaredegradedbyzsp,anddecayinto otherformsofspinparticle.Singleboson,singlefermiongraduallydissolveinthe solventcomprisedofzerospinparticles,when"thedensityofthesolvent",orthe numberkofzspapproachstoinfinite. Inquantummachenics,everyspinparticle,besideszerospinparticle,is "Something",ascomesdowntothespinphenomena.Bycontrast,zerospinparticle isjust"Nothing"duetoconventionalspinconceptof001 2 and0. Table2andTable3showifmatrices(1),(2),(3),(4),thef iguresofzerospin particleinSTS,areintroducedtotakepartinspininteractions,what"whatwill happen..."is"SomethingPlusNothing,equalstoSomethingforless" Theamountof"less",whichrefertothediff e renceoftwoCasimiroperators possessedbyspinparticlesbeforeandaftertheircombination(13)respectively,is calledlossesofCasimiroperatorΔ (j,k).Table4givesthedetailsforthecases k1,2ofthelosses. Whenphysicsdimension 2 ofCasimiroperatori sconnectedtotherotational energyofspinparticle,thenlossesofCasimiroperatorwouldleadtotheresearch onso-calledbindingenergyofspinparticles,onwhichaglimpseofcommentis givenintheendofthispaper.
3BosonsandFermionsinSpinTopologicalSpace Threeoperatorsin(6)below,satisf yspinangularmomentumcommutationrelus(7) { j , −1 − , 3;j,−1 }(6) j −1 − – −1 − j 2 3;j,−1 (7.1) 3;j,−1 j – j 3;j,−1 j (7.2) 3;j,−1 −1 − – −1 − 3;j,−1 − −1 − (7.3) Thesethreeoperatorsareraisingoperators j ,loweringoperators −1 − andthe thirdcomponentoperaters 3;j,−1 ofdifferentspinparticleswhichlabelledby differentvaluesofj−1,0,1,2,3,4,... Writeouttheexplicitexpressionsofraisingoperators j (8)andlowering operator −1 − (9),whichappearin(6),(7) 4 diag{,9,8,7,6,5,4,3,2,1,0,-1,,} 1 (8.4) 3 diag{,8,7,6,5,4,3,2,1,0,-1,-2,,} 1 (8.3) 2 diag{,7,6,5,4,3,2,1,0,-1,-2,-3,,} 1 (8.2) 1 diag{,6,5,4,3,2,1,0,-1,-2,-3,-4,,} 1 (8.1) 0 diag{,5,4,3,2,1,0,-1,-2,-3,-4,-5,,} 1 (8.0) −1 diag{,4,3,2,1,0,-1,-2,-3,-4,-5,-6,,} 1 (8.-1) −1 − diag{,-4,-3,-2,-1,0,1,2,3,4,5,6,,} −1 (9) Subscripts,"1"in(8)diag{,,} 1 and"–1"in(9),diag{,,} −1 ,representthe firstminortop-rightdiangonalandthefirstminordown-leftdiangonalresepectively. (8.0)and(9)constructzerospinparticlerepresentions(1),(2),(3),whichmentioned inparagraph2 Twodiagonalmatricesa)andb)ofspinparticles j,−1 a)Thethirdcomponents 3;j,−1 (10)areobtainedbyusingaboveexpressions (8),(9)and(7.1)asbelow 3;j,−1 1 2 { j −1 − – −1 − j }(10) 3;4,−1 diag{,7,6,5,4,3,2,1,0,-1,-2,-3,,} 0 (10.4) 3;3,−1 1 2 diag{,13,11,9,7,5,3,1,-1,-3,-5,-7,,} 0 (10.3) 3;2,−1 diag{,6,5,4,3,2,1,0,-1,-2,-3,-4,,} 0 (10.2) 3;1,−1 1 2 diag{,11,9,7,5,3,1,-1,-3,-5,-7,-9,,} 0 (10.1) 3;0,−1 diag{,5,4,3,2,1,0,-1,-2,-3,-4,-5,,} 0 (10.0) 3;−1,−1 1 2 diag{,9,7,5,3,1,-1,-3,-5,-7,-9,-11,,} 0 (10.-1) subscript"0"in(5)representsthemajordiangonal,sometime,"0"isomitted ifnoconfusion.
b)Casimiroprtator,thesumofsquare j,−1 2 of j,−1 Thetotalsquareof j,−1 isdefinedas j,−1 2 j,−1 j,−1 1;j,−1 2 2;j,−1 2 3;j,−1 2 s 2 −1 2 2 2 jj2 4 (11) here 1;j,−1 2 2;j,−1 2 1 2 { j −1 − −1 − j }(12) Theconcreteresultsofa)(10)andb)(11)aregiveninTable1 Table1BosonsandFermionsinSTS 3;j,−1 j,−1 2 j 2 ParticleSpinsj1j (10.4) 3;4,−1 24 2 4 221 4 2 boson54 (10.3) 3;3,−1 15 2 4 3 2 3 2 1 3 2 fermion43 (10.2) 3;2,−1 8 2 4 111 2 2 boson32 (10.1) 3;1,−1 3 2 4 1 2 1 2 1 1 2 fermion21 (10.0) 3;0,−1 0 2 4 001 0 2 boson10 (10.-1) 3;−1,−1 – 1 2 4 – 1 2 – 1 2 1− 1 2 negativefermion0–1 4WhatWillHappen... ACombination(13)ofaboson,orafermionspinparticlewithkzerospin particlesisintroducedasbelow j/k1,−1 1 k1 { j,−1 k 0,−1 }(13) 1 k1 {OneBoson kZeroSpinParticles}(13.1) 1 k1 {OneFermion kZeroSpinParticles}(13.2) Wef i ndthecombinationisanewspinparticlethatsatisfyangularmomentum rulebelow j/k1,−1 j/k1,−1 i j/k1,−1 (14) Twodiagonalmatricesc)andd)ofspinparticles j/k1,−1 c)Thethirdcomponents 3,j/k1,−1 withclearfiguresareshowninTable2, bydirectlysubstitutingtherusultsof(10.j)i nto(15) 3,j/k1,−1 1 k1 { 3,j,−1 k 3,0,−1 }(15) thenwegetageneralformula(16) j 2k1 1 k1 { j 2 k 0 2 }(16) Theinfluence,ofthenumberkofzerospinparticle(s)onaboson,orona fermioninformula(16),isdetailedinTable2
Table2TheThirdComponentsofSpinParticleswithdiffenentjandk 3,j/k1,−1 3,j,−1 ,k0 3,0,−1 ,k0 3;j/2,−1 ,k1 3;j/3,−1 ,k2 j 2k1 j 2 0 2 j 4 j 6 j 6 6 2 3boson 0 2 zerospin 6 4 threesecond 6 6 1boson 5 5 2 fivesecond 0 2 zerospin 5 4 fivefourth 5 6 fivesixth 4 4 2 2boson 0 2 zerospin 4 4 1boson 4 6 twothird 3 3 2 threesecond 0 2 zerospin 3 4 threefourth 3 6 1 2 fermion 2 2 2 1boson 0 2 zerospin 2 4 1 2 fermion 2 6 onethird 1 1 2 1 2 fermion 0 2 zerospin 1 4 onefourth 1 6 onesixth 0 0 4 0boson 0 2 zerospin 0 4 0boson 0 4 0boson –1 −1 2 Nfermion 0 2 zerospin −1 4 Nonefourth −1 6 Nonesixth Note:(16)andTable2show:thenewspinparticle 3;j/2,−1 or 3;j/3,−1 maybe eitheranewbosonoranewfermion,orneitherabosonnoraf ermionatall. Example1oftherowlabelledj 4,indicates: Combinationofa2bosonandazerospinparticle(k1),wouldforma1boson Combinationofa2bosonandtwozerospinparticles(k2),wouldformathree thirdspin 2 3 particle Example2:therowlabelledj2,indicates: Combinationofa1bosonandazerospinparticle(k1),wouldforma 1 2 fermion Combinationofa1bosonandtwozerospinparticles(k2),wouldformaone thirdspin 1 3 particle Example3:therowlabelledj1,indicates: Combinationofa 1 2 fermionandazerospinparticle(k1),wouldformaone fourthspin 1 4 particle Combinationofa 1 2 fermionandtwozerospinparticles(k2),wouldformaone sixthspin 1 6 particle Example4:therowlabelledj0,indicates: Combinationofa0bosonanda nynumberof0boson(k1,2,...),wouldstill forma0boson Note:Thementioneda boveshow,theoriginalboson(k0)ortheorigianl fermion(k0)seemstobe"dissolvable"(referto(17)and(21)(22),theabsolute valuesofCasimiroprtatoroftheboson,orthefermionisdeminishing)whenit combineswithzerospinparticle(s)(instateofk0)toformanewspinparticle (k1,2,...),theamountofspinofthenewspinparticleisalwayslessthanthe oneoftheoriginalbosonortheoriginalfermionasbelow j 2k1 j 2 k1,2,3,...(17)
d)Casimiroprtator,thesumofsquare j/k1,−1 2 of j/k1,−1 Bymeansof(10)and(18),(19), 3;j/k1,−1 1 2 { j/k1 −1 − – −1 − j/k1 }(18) 1;j/k1,−1 2 2;j/k1,−1 2 1 2 { j/k1 −1 − −1 − j/k1 }(19) Thetotalsquare j/k1,−1 2 isgiven j/k1,−1 2 1;j/k1,−1 2 2;j/k1,−1 2 3;j/k1,−1 2 j{j2k1} 2 4k1 2 (20) Theconcreteresultsof(20)withk1,2aregiveninTable3 Table3CasimirOperatorsofSpinParticleswithdiffenentjandk j/k1,−1 2 j,−1 2 ,k0 0,−1 2 ,k0 j/2,−1 2 ,k1 j/3,−1 2 ,k2 j{j2k1} 4k1 2 j{j2} 4 0 4 j{j4} 16 j{j6} 36 j 6 48 2 4 331 0 2 4 001 15 2 4 3 2 3 2 1 8 2 4 111 5 35 2 4 5 2 5 2 1 0 2 4 001 45 2 16 5 4 5 4 1 55 2 36 5 6 5 6 1 4 24 2 4 221 0 2 4 001 8 2 4 111 10 2 9 2 3 2 3 1 3 15 2 4 3 2 3 2 1 0 2 4 001 21 2 16 3 4 3 4 1 3 2 4 1 2 1 2 1 2 8 2 4 111 0 2 4 001 3 2 4 1 2 1 2 1 4 2 9 1 3 1 3 1 1 3 2 4 1 2 1 2 1 0 2 4 001 5 2 16 1 4 1 4 1 7 2 36 1 6 1 6 1 0 0 2 4 001 0 2 4 001 0 2 4 001 0 2 4 001 –1 –1 2 4 –1 2 –1 2 1 0 2 4 001 –3 2 16 –1 4 –1 4 1 –5 2 36 –1 6 –1 6 1 Ask,thenumberofzerospinparticlesincresing,thenewspinparticle gradually"dissolveinto"azerospinparticle k→ lim j/k1,−1 2 k→ lim j{j2k1} 4k1 2 → k→ lim 1 k →001 0,−1 2 (21) e)LossesofCasimiroperator Beforethecombinations(13),thecontributionsofasinglespinparticle j,−1 andkzerospinparticlesk 0,−1 are B 2 j,−1 2 k 0,−1 2 j,−1 2 jj2 4 ,(11).Andafter thecombinations(13),thecontributionsofspinparticle 3,j/k1,−1 are A 2 j/k1,−1 2 j{j2k1} 4k1 2 ,(20). Δ (j,k)below,thedifferencebetween A 2 and B 2 ,iscalledloseofCasimir oprtator Δ (j,k) j/k1,−1 2 − j,−1 2 – kj 4k1 2 { j2k2−2 1 }(22) Δ (j,1) – j 16 { 3j2−2 1 }(22.1) Δ (j,2) – 2j 36 { 4j2−2 1 }(22.2)
Table4Δ(j,k)LossesofCasimiroprtatorwithk1,2(unit 2 ) j/k1,−1 2 j/2,−1 2 ,k1 j/3,−1 2 ,k2 j,−1 2 ,k0Δj,1Δj,2 j{j2k1} 4k1 2 j{j4} 16 j{j6} 36 j{j2} 4 (22.1)(22.2) j 6 15 4 212− 132 16 −10 5 45 16 55 36 35 4 − 95 16 − 260 36 42 10 9 6−4− 176 36 3 21 16 3 4 15 4 − 39 16 − 108 36 2 3 4 4 9 2− 20 16 − 56 36 1 5 16 7 36 3 4 − 7 16 − 20 36 000000 –1 –3 16 –5 36 –1 4 1 16 4 36 LossesofCasimiroprtator(16)and(17)mean:SomethingPlusNothing, EqualToSomethingForLess f)Thebindingenergyofspinparticles InSTS,spinparticlesaresymbolledbyjandk.Formular(20) j/k1,−1 2 j{j2k1} 2 4k1 2 istheattributeofthefigureofspinparticlelablledwithdifferentjand k.Theattributemayberewrittenintheformoftherotationalenergyofspin particleasbelow. E r j,k j/k1,−1 2 2I j,k (23) And ΔE r j,k 1 2I j,k { j/k1,−1 2 − j,−1 2 } Δj,k 2I j,k (24) (24)iscalledasthebindingenergy,theenergyreleasedwhentheconstituent spinparticles, j,−1 andk 0,−1 cometogethertoformspinparticle j/k1,−1 . 5Conclusions Sofarzerospinparticleistheonlyspinparticlenotpossessingnon-trivialspin angularmomentumrepresentation,becausezerospinparticlepossessesnospin rotationalconstruction,andplaysthe"nothingrole"ofspininteractionsworld. Thispaper,researchingthespinangularmomentumcouplingbetweenzerospin particleandotherspinparticle,maybeanapproachtojudgewhetherzerospin particlepossessesspinrotationalconstruction. Thispapershows:inSpinTopologicalSpace,STS,zerospinparticlewasno longunabletodoanything,inspininteractions.Theideaofcombinationof"the nothing"ofzerospinparticlewith"thesomething"ofotherspinparticleprovides heuristicmaththoughttounderstandmanyinterestingphysicsphenomena[2],[3].
References [1]ShaoXuRen(2014)JournalofModernPhysics,5,800-869 http:/dx.doi.org/10.4236/jmp.2014.59090 ShaoXuRen(2014)JournalofModernPhysics,5,1848-1879 http:/dx.doi.org/10.4236/jmp.2014.517187 ShaoXuRen(2016)JournalofModernPhysics,7,737-759 http:/dx.doi.org/10.4236/jmp.2016.78070 ShaoXuRen(2016)JournalofModernPhysics,7,1364-1374 http:/dx.doi.org/10.4236/jmp.2016.711123 ShaoXuRen(2015)InteractionoftheOriginsofSpinAngularMomentum ISBN978-988-14902-0-9(20162ndedition), [2]Abbott,B.P.,e tal.(2016)PhysicalReviewLetters,116,ArticleID:061102. http://dx.doi.org/10.1103/physrevlett.116.061102 [3]ScienceAdvances29Apr2016:Vol.2,no.4,e1501748 DOI:10.1126/sciadv.1501748
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