WhatWillHappen,IfZeroSpinParticle
PossessesSpinRotationalConstruction,
WithNon-ZeroEigenvaluesOf
SpinAngularMomentum?
ShaoXuRen
InstituteofPhysicalScienceandEngineering
TongjiUniversity,200092,Shanghai,China
Received 8 November 2016; Accepted 2 December 2016; Published 5 December 2016
———————————————————————————————————–
———————————————————————————————————–
Abstract
Thereisnoanyspinrotationalconstructionforzerospinparticle,Casimir
operatorandthethiredcomponentofzerospinparticleare001
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
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1Introduction
Inquantummachenics,themeasurablespinpropertiesofwell-knownallbosons
andfermionsaredemostratedbytwodiagonaloperatorswhichcalledCasimir
operatorandthethirdcomponentofspinparticles.
ThevaluessofspinofanyspinparticleisdiscribedbytheirCasimir
operatorsss1
2
.Thegreaterthes,thegreatertheirCasimiroperators.
Andforthethirdcomponentsofthesespinparticles,themaximumeigenvalueis
justs,theresteigenvaluesoftheirsarealwayslessthanthevalueofs.
Inquantummachenics,zerospinparticle,zsp0hasnorotationalconstruction.
Sothereisactuallynoanyspinrepresentationf orzspinphysicsandMathworld.
CasimirOperatorandthethirdcomponentofzerospinparticlecanbeobviously
andtriviallydepictedasstatements001
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
zerospinparticleremaintobetheformof001I
0

2
,referto(4).
Butthethirdcomponentofzerospinparticleturnsintoaninf initedimensional
matrix,referto(3)ordiagonal(10.0),whichshowsthatbesidesazerovalue
eigenvalue0lyingatthecenter"
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"duetoconventionalspinconceptof001
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
k1,2ofthelosses.
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

jj2
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
ParticleSpinsj1j

(10.4)

3;4,−1
24
2
4
221
4
2
boson54
(10.3)

3;3,−1
15
2
4

3
2

3
2
1
3
2
fermion43
(10.2)

3;2,−1
8
2
4
111
2
2
boson32
(10.1)

3;1,−1
3
2
4

1
2

1
2
1
1
2
fermion21
(10.0)

3;0,−1
0
2
4
001
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/k1,−1

1
k1
{
j,−1
k
0,−1
}(13)
1
k1
{OneBoson kZeroSpinParticles}(13.1)
1
k1
{OneFermion kZeroSpinParticles}(13.2)
Wef i ndthecombinationisanewspinparticlethatsatisfyangularmomentum
rulebelow

j/k1,−1

j/k1,−1
i
j/k1,−1
(14)
Twodiagonalmatricesc)andd)ofspinparticles

j/k1,−1
c)Thethirdcomponents
3,j/k1,−1
withclearfiguresareshowninTable2,
bydirectlysubstitutingtherusultsof(10.j)i nto(15)

3,j/k1,−1

1
k1
{
3,j,−1
k
3,0,−1
}(15)
thenwegetageneralformula(16)
j
2k1

1
k1
{
j
2
k
0
2
}(16)
Theinfluence,ofthenumberkofzerospinparticle(s)onaboson,orona
fermioninformula(16),isdetailedinTable2

Table2TheThirdComponentsofSpinParticleswithdiffenentjandk

3,j/k1,−1

3,j,−1
,k0
3,0,−1
,k0
3;j/2,−1
,k1
3;j/3,−1
,k2
j
2k1
j
2
0
2
j
4
j
6
j
6
6
2
3boson
0
2
zerospin
6
4
threesecond
6
6
1boson
5
5
2
fivesecond
0
2
zerospin
5
4
fivefourth
5
6
fivesixth
4
4
2
2boson
0
2
zerospin
4
4
1boson
4
6
twothird
3
3
2
threesecond
0
2
zerospin
3
4
threefourth
3
6
1
2
fermion
2
2
2
1boson
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
0boson
0
2
zerospin
0
4
0boson
0
4
0boson
–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:
Combinationofa2bosonandazerospinparticle(k1),wouldforma1boson
Combinationofa2bosonandtwozerospinparticles(k2),wouldformathree
thirdspin
2
3
particle
Example2:therowlabelledj2,indicates:
Combinationofa1bosonandazerospinparticle(k1),wouldforma
1
2
fermion
Combinationofa1bosonandtwozerospinparticles(k2),wouldformaone
thirdspin
1
3
particle
Example3:therowlabelledj1,indicates:
Combinationofa
1
2
fermionandazerospinparticle(k1),wouldformaone
fourthspin
1
4
particle
Combinationofa
1
2
fermionandtwozerospinparticles(k2),wouldformaone
sixthspin
1
6
particle
Example4:therowlabelledj0,indicates:
Combinationofa0bosonanda nynumberof0boson(k1,2,...),wouldstill
forma0boson
Note:Thementioneda boveshow,theoriginalboson(k0)ortheorigianl
fermion(k0)seemstobe"dissolvable"(referto(17)and(21)(22),theabsolute
valuesofCasimiroprtatoroftheboson,orthefermionisdeminishing)whenit
combineswithzerospinparticle(s)(instateofk0)toformanewspinparticle
(k1,2,...),theamountofspinofthenewspinparticleisalwayslessthanthe
oneoftheoriginalbosonortheoriginalfermionasbelow
j
2k1

j
2
k1,2,3,...(17)

d)Casimiroprtator,thesumofsquare
j/k1,−1
2
of
j/k1,−1
Bymeansof(10)and(18),(19),

3;j/k1,−1

1
2
{
j/k1


−1
−
–
−1
−

j/k1

}(18)

1;j/k1,−1
2

2;j/k1,−1
2

1
2
{
j/k1


−1
−

−1
−

j/k1

}(19)
Thetotalsquare

j/k1,−1
2
isgiven

j/k1,−1
2

1;j/k1,−1
2

2;j/k1,−1
2

3;j/k1,−1
2

j{j2k1}
2
4k1
2
(20)
Theconcreteresultsof(20)withk1,2aregiveninTable3
Table3CasimirOperatorsofSpinParticleswithdiffenentjandk

j/k1,−1
2

j,−1
2
,k0
0,−1
2
,k0
j/2,−1
2
,k1
j/3,−1
2
,k2
j{j2k1}
4k1
2
j{j2}
4
0
4
j{j4}
16
j{j6}
36
j
6
48
2
4
331
0
2
4
001
15
2
4

3
2

3
2
1
8
2
4
111
5
35
2
4

5
2

5
2
1
0
2
4
001
45
2
16

5
4

5
4
1
55
2
36

5
6

5
6
1
4
24
2
4
221
0
2
4
001
8
2
4
111
10
2
9

2
3

2
3
1
3
15
2
4

3
2

3
2
1
0
2
4
001
21
2
16

3
4

3
4
1
3
2
4

1
2

1
2
1
2
8
2
4
111
0
2
4
001
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
001
5
2
16

1
4

1
4
1
7
2
36

1
6

1
6
1
0
0
2
4
001
0
2
4
001
0
2
4
001
0
2
4
001
–1
–1
2
4

–1
2

–1
2
1
0
2
4
001
–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/k1,−1
2

k→
lim
j{j2k1}
4k1
2
→
k→
lim
1
k
→001
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

jj2
4
,(11).Andafter
thecombinations(13),thecontributionsofspinparticle

3,j/k1,−1
are

A
2

j/k1,−1
2

j{j2k1}
4k1
2
,(20).
Δ
(j,k)below,thedifferencebetween
A
2
and
B
2
,iscalledloseofCasimir
oprtator
Δ
(j,k)
j/k1,−1
2
−
j,−1
2
–
kj
4k1
2
{
j2k2−2
1
}(22)
Δ
(j,1) –
j
16
{
3j2−2
1
}(22.1)
Δ
(j,2) –
2j
36
{
4j2−2
1
}(22.2)

Table4Δ(j,k)LossesofCasimiroprtatorwithk1,2(unit
2
)

j/k1,−1
2

j/2,−1
2
,k1
j/3,−1
2
,k2
j,−1
2
,k0Δj,1Δj,2
j{j2k1}
4k1
2
j{j4}
16
j{j6}
36
j{j2}
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/k1,−1
2

j{j2k1}
2
4k1
2
istheattributeofthefigureofspinparticlelablledwithdifferentjand
k.Theattributemayberewrittenintheformoftherotationalenergyofspin
particleasbelow.
E
r
j,k

j/k1,−1
2
2I
j,k
(23)
And
ΔE
r
j,k
1
2I
j,k
{
j/k1,−1
2
−
j,−1
2
}
Δj,k
2I
j,k
(24)
(24)iscalledasthebindingenergy,theenergyreleasedwhentheconstituent
spinparticles,

j,−1
andk
0,−1
cometogethertoformspinparticle
j/k1,−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