Spin Forms and Spin Interactions
among Higgs Bosons,
between Higgs Boson and Graviton
ShaoXu Ren
Institute of Physical Science and Engineering
Tongji University, 200092, Shanghai, China
Corresponding email:
Received 16 March 2016; accepted 25 April 2016; published 29 April 2016
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Abstract
This paper offers concrete spin matrix forms of 0spinzero particle, and shows
the existent of the spin interactions among 0spin zero particles.It is obviously
hoping to approach, on the most comprehensive level,to understand what really
Higgs Boson is and what role-play HiggsBoson is acting in particle physics.
As a "particle" of gravitational force, the spin interaction between0spin
zero particle (Higgs Boson) and 2spin particle (Graviton) is given,which maybe
a way that people would find Graviton in future.
Keywords
Higgs Boson; Graviton; Vacuum Bubble Pair; Spin Topological Space, STS;
Casimir operator; right-hand 0spin zeroparticle; left-hand 0spin zero particle
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1. Introduction
Higgs particle is a fundamental particle predicted by the Standard Model, and
confirmed by the Large HadronCollider at CERN.[1]Higgs particle could explain
why thephoton has nomass while Wand ZBosons are very heavy in electron-weak
theory, and endows Fermions such as electron, muon and tau particle and quarks
with their masses.The intrinsic spin angular momentum of Higgs particle is0.
In conventional quantum mechanics,the representation of spin zero particle is a
zero-matrix of one dimensional space,this means,in fact, Higgs particle hasno
matrix representation for itsspin property in the Standard Model.What a pity !
magical Higgs particle could create masses of the particles in universe, however,
failing to write out its own non-trivial spin matrices. Zero is not nothing,zero spin
is not non-twirling.From the pointviewof The Third Kind of Particles,TKP,[2]
the angular momentum property of spin zero particle canbe expressed by infinite
dimensional non-Hermitian matrices whichrelated to Vacuum Bubble Pair,VBP ,

Journal of Modern Physics, 2016, 7, 737-759
Published Online April 2016 in SciRes. http://www.scirp.org/journal/jmp
http://dx.doi.org/10.4236/jmp.2016.78070
How to cite this paper: Ren, S.X. (2016) Spin Forms and Spin Interactions among Higgs Bosons, between Higgs Boson and
Graviton. Journal of Modern Physics, 7, 737-759. http://dx.doi.org/10.4236/jmp.2016.78070

these pairs could be excited into 0spin particle formphase transitions of
Vacuum Spin Particle,VSP, whose Casimir Operator is −1
42I0,less than
zero. [3]( CasimirOperatorofHiggsparticleis02I0,of Graviton is 62I0)
In conventional quantum mechanics, each particle has its own spin space: one
spin prticle, one spin space; two spin particles,two spin spaces; .......; nspin
particles, nspin spaces.These spin spaces are independent each other, and
expressed by VV1⊗V2⊗V3⊗... ⊗Vn.
This paper gives advice: Since the spin angularmomentumconstituents of every
elementary particle are composed of the common series of math elements that based
on the raising operatorsj
1; j,k i2; j,k −and lowering operatorsk −1; j,k –i2; j,k −,
( whichcomposeVBP,TKP ),then anewtypeof spin space,the so-called Spin
Topological Space,STS ,[4] is established. All sorts of spin sparticles are attributed
to this spin space, STS.
In traditional views, there are no any spininteractions among spinzero particles.
However, by means of STS concept, on the contrary, it is shown there are the spin
interactions. 0Spin zero particle not only possesses spin phenomena but also
appears out right-circumrotationand left-circumrotation, such kind of propertiesmay
exist in Higgs Boson world of the Standard Model.
Same reason for,there should be spin particle interactions between Higgs
Boson and graviton, and spin interactions among gravitons,detecting gravitational
force,after the interference effect of gravitational wave is comfirmed.[5]
2. Higgs Boson’s Spin angular Momentum matrices1l,2l,3l
The mathematical structure of 1l,2l,3land 3
2l,1 2l2 2lare
given in matrix series(1),(2),(3) shown below.
They satisfy angular momentum commutation rules
iljl−jlilikl(0.1)
i,j,k1, 2, 3arecirculative
Casimir Operator
2l1
2l2 2l3 2l0I0001I0(0.2)
Using raising operator, lowering operator
l1li2l；－l1l−i2l(0.3)
Then (0.1) turns to
3ll−l3ll(0.4)
3l−l−−l3l−−l(0.5)
l−l−−ll23l(0.6)
What follows are the explicit spinmatrix representations of three generations
(l1,2,3)of 0Boson, (Higgs Boson).

Be brief, in Spin Topological Space,STS,[4], the above spin matrices
{ (1.1), (1.2), (1.3);(2.1), (2.2), (2.3);(3.1), (3.2), (3.3) }ofHiggsBosoncanbe
rewritten in the spin forms of (4.1),(4.2),(4.3)
0, 11{11,21,31} (4.1)
−1,12{12,22,32} (4.2)
−2,13{13,23,33} (4.3)
For an example of 0, 11, now, (4.1) is denotedby (5.0):
0, 11{1;0, 11,2;0, 11,3;0, 11} (5.0)
1;0, 111
20
1 −11(5.1)
2;0, 111
2i 0
−1 −21(5.2)
3;0, 111
i{1;0, 112;0, 11−2;0, 111;0, 11}31
(5.3)
3. Spin Interactions between Two HiggsBosons
a) First we deal with two-body systemthat compose of Higgs Boson a,
aandHiggsBosonb,b.aand bare their spin angular momentum
matrix operators.Then show a case of a spin coupling interaction (6.0)
between aand b,the scalar products S2l, or Casimir operators of their
three generations as follows
S2lSlSl(6.0)
Where Slalbl(7.0)
Or S0, 1;−2, −110, 11−2, −11(7.1)
S−1,1;−3,−12−1,12−3, −12(7.2)
S−2,1;−4,−13−2,13−4, −13(7.3)
After careful calculation, for S2(6.0),we have (8.0)
S2lSlSl0I0(8.0)
Or S0, 1;−2, −1
210I0(8.1)
S−1,1;−3,−1
220I0(8.2)
S−2,1;−4,−1
230I0(8.3)
I0diag{...,1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,...}(9.0)
(8.0) and (8.1),(8.2),(8.3) show: there are no any effects of spin coupling
bwtween two 0zero spin particles, aland bl.

b) How can we find no-trivial spin-coupling interactionrather than (8),
by using augular momentum operators of 0zero spinparticle ?
Actually there are two types of 0zerospinparticles,whichisawayto
overcome the obstacle
Assume 0zero spin particles 0, 11and −2, −11to be thought of as
two right-hand spin particles,frmula (7.1) could be written as (10).
ARS0, 1;−2, −110, 11−2, −11(10)
Further the formula (8.1) is expressed as (11)
ARI0ARS0, 1;−2, −1
210I0(11)
On the other side,AL
⊙is markedas the adjoint counter of ARwith
metric cofficient operator,then 0, 1
⊙1and −2, −1 ⊙1are left-hand zero
spin particles accordingly.we have
AL
⊙S0, 1;−2, −1
⊙10, 1
⊙1−2, −1 ⊙1(12)
We are now ready to take further our discussion of scalar product of
right-hand-to-right-hand, ARI0AR(11),to psecudo-scalar product of
left-hand-to-right-hand, AL
⊙AR(13), as follows
ARI0ARAL
⊙AR(13)
Here metric cofficient operator isselected as

010
100
001
(14)
Attention:
right-hand 0zero spin particles0, 11and −2, −11obey angular
momentumcommutation rules of right-handed coordinate system；
left-hand 0zero spin particles 0, 1
⊙1and −2,−1 ⊙1obey angular
momentumcommutation rules of left-handed coordinate system.
For moreconcise, the symbols (15) are given in the futurediscussions
ARj,k;r,s≡j,k1r,s1,3i,j≡3; i,j1(15)

c) Let us have a look at an example of psecudo-scalar product of
left-hand-to-right-hand of spin zero particles, AL
⊙AR.Aftercareful
calculation,we get two groups of 0spin interactions, Group-A and
Group-B.
By way of illustration, we shall refer to the feature B(1) of Group-B：
Formulas (16.1) (16.2)and formulas (17.1), (17.2)are the third
compoments of initial state iand final state fof psecudo-scalar spin
interaction of the first generation spin particles(l1).
Initial state
030, 1diag {...6, 5, 4, 3, 2,1,0,-1, -2, -3, -4,...}(16.1)
03−2, −1diag {...4, 3, 2, 1,0, -1, -2, -3, -4, -5, -6...}(16.2)
Final state
030, −1diag {...5, 4, 3, 2, 1, 0,-1, -2, -3, -4, -5,...}(17.1)
13−2, 1diag {...5, 4, 3, 2, 1,0, -1, -2, -3, -4, -5,...}(17.2)
And the conservation of the third compoment of spin angular momentums
between initial state iand final state fis obtained as (18.0)
30, 1; −2, −130, −1; −2, 1(18.0)
ininal sum of s.a.mfinal sum of s.a.m
Where
30, 1; −2, −130, 13−2,−1(18.1)
30, −1; −2, 130, −13−2,1(18.2)
And the conservation of psecudo-scalar spin interactionabout ininl state i
and final state fis obtained as(19.0)
LB,i(1) LB,f(1) LB(19.0)
Where
LB,i(1) BL,1
⊙0,1; −2, −1BR,1 0,1; −2, −1(19.1)
LB,f(1) BL,1
⊙0,−1; −2, 1BR,1 0,−1; −2, 1(19.2)
LB−2 diag {...102,8
2,6 2,4 2,2 2, 02,2 2,4 2,6 2,8 2,10 2, ,...}(20)
What mentioned above isso-called fission of 0zero spin particles, refer
to Fig4.
Conservation (18.0) and conservation (19.0) imply that if initial state i
and final state fexchange places, so-called fusion of other spinparticles is
given, refer to Fig3.
Fig1, Fig2 of Group-A are obtained by the analogy to those of Fig3,
Fig4 of Group-B above.

A(3) 0−2, 1↖LA,f(3) 1
32LA↗−3,0 0A(3)
A(2) 0−1, 1↖LA,f(2) 1
22LA↗−2,0 0A(2)
A(1) 00,1↖LA,f(1) 1
12LA↗−1,0 0A(1)
1 0
fusions A
A(3) −/6 −2,0 ↗LA,i(3) 1
32LA↖−3, 1/6 A(3)
A(2) −/4 −1,0 ↗LA,i(2) 1
22LA↖−2, 1/4 A(2)
A(1) −/2 0, 0↗LA,i(1) 1
12LA↖−1, 1/2 A(1)
Fig1zero spin particles are formedby the fusions of other spin particles
A(3) −/6 −2,0 ↖LA,f(3) 1
32LA↗−3, 1/6 A(3)
A(2) −/4 −1,0 ↖LA,f(2) 1
22LA↗−2, 1/4 A(2)
A(1) −/20, 0↖LA,f(1) 1
12LA↗−1, 1/2A(1)
fissions A
1 0
A(3) 0−2, 1↗LA,i(3) 1
32LA↖−3,0 0A(3)
A(2) 0−1, 1↗LA,i(2) 1
22LA↖−2,0 0A(2)
A(1) 00,1↗LA,i(1) 1
12LA↖−1,0 0A(1)
Fig2other spin particles are released by the fissions of zerospinparticles
LA,f(3) ≡AL,3
⊙−2,0 ; −3, 1AR,3 −2, −1; −3, 1(21.1)
LA,i(3) ≡AL,3
⊙−2, 1; −2, 1AR,3 0, 1; −3,0 (21.2)
LA,f(2) ≡AL,2
⊙−1,0 ; −2, 1AR,2 −1, −1; −2, 1(22.1)
LA,i(2) ≡AL,2
⊙−1, 1; −1, 1AR,2 −1, 1; −2,0 (22.2)
LA,f(1) ≡AL,1
⊙0, 0;−1, 1AR,1 0, 0;−1, 1(23.1)
LA,i(1) ≡AL,1
⊙0,1; −1,0 AR,1 0,1; −1,0 (23.2)
LA−2 diag {...112,9
2,7 2,5 2,3 2, 12,1 2,3 2,5 2,7 2,9 2, ,...}(24)

B(3) 0−2, 1↖LB,f(3) 1
32LB↗−4, 10B(3)
B(2) 0−1, 1↖LB,f(2) 1
22LB↗−3, −10B(2)
B(1) 00,1↖LB,f(1) 1
12LB↗−2, −10B(1)
1−1
fusions B
B(3) −/3 −2,−1↗LB,i(3) 1
32LB↖−4, 1/3B(3)
B(2) −/2 −1,−1↗LB,i(2) 1
22LB↖−3, 1/2 B(2)
B(1) 00,−1↗LB,i(1) 1
12LB↖−2, 11B(1)
Fig3zero spin particles are formedby the fusions of other spin particles
B(3) −/3 −2, −1↖LB,f(3) 1
32LB↗−4, 1/3 B(3)
B(2) −/2 −1, −1↖LB,f(2) 1
22LB↗−3, 1/2 B(2)
B(1) 00,−1↖LB,f(1) 1
12LB↗−2, 11∗B(1)
fissions B
1−1
B(3) 0−2, 1↗LB,i(3) 1
32LB↖−4, −10B(3)
B(2) 0−1, 1↗LB,i(2) 1
22LB↖−3, −10B(2)
B(1) 00,1↗LB,i(1) 1
12LB↖−2, −10B(1)
Fig4other spin particles are released by the fissions of zerospinparticles
LB,f(3) ≡BL,3
⊙−2, −1; −4, 1BR,3 −2, −1; −4, 1(25.1)
LB,i(3) ≡BL,3
⊙−2, 1; −4, −1BR,3 0, 1; −4, −1(25.2)
LB,f(2) ≡BL,2
⊙−1, −1; −3, 1BR,2 −1, −1; −3, 1(26.1)
LB,i(2) ≡BL,2
⊙−1, 1; −3, −1BR,2 −1, 1; −3, −1(26.2)
LB,f(1) ≡BL,1
⊙0,−1; −2, 1BR,1 0,−1; −2, 1(27.1)
LB,i(1) ≡BL,1
⊙0,1; −2, −1BR,1 0,1; −2, −1(27.2)
LB−2 diag {...102,8
2,6 2,4 2,2 2, 02,2 2,4 2,6 2,8 2,10 2, ,...}(28)

4) Spin Interactions between Higgs Bosons and Gravitons
d) To return to the case of scalar product of right-hand-to-right-hand
particles, ARI AR,because we couldget no-trivial spin-coupling interactions
between 0zero spin particle and 2spin particle. Here two groups of ,
Group-C and Group-D. are given. Analogously, illustration by example of the
feature D(1) of Group-D as follows：
Formulas (29.1) (29.2)and formulas (30.1), (30.2)are the third
compoments of initial state iand final state fof scalar spin interaction of
the first generation spin particles (l1)
Initial state
030, 1diag {...6, 5, 4, 3, 2,1,0,-1, -2, -3, -4,...}(29.1)
23−3, 2diag {...5, 4, 3, 2, 1,0, -1, -2, -3, -4, -5,...}(29.2)
Final state
1/2 30, 21
2diag {...13, 11, 9, 7, 5,3,1, -1, -3, -5,-7,...}
(30.1)
3/2 3−3, 11
2diag {...9,7,5, 3, 1,-1, -3, -5, -7, -9, -11...}
(30.2)
And the conservation of the third compoment of spin angular momentums
between initial state iand final state fis obtained as (31.0)
30, 1; −3, 230, 2; −3, 1(31.0)
ininal sum of s.a.mfinal sum of s.a.m
Where
30, 1; −3, 230, 13−3, 2(31.1)
30, 2; −3, 130, 23−3,1(31.2)
And the conservation of scalar spin interaction about ininl state iand
final statefis obtained as (32.0)
LD,i(1) LD,f(1) LD(32.0)
Where
LD,i(1) DR,10,1; −3, 2I DR,1 0,1; −3, 2(32.1)
LD,f(1) DR,10,2; −3, 1I DR,1 0,2; −3, 1(32.2)
LD8I0(33)
What mentionedabove is so-called fission of Higgs Boson and Graviton,
refer to Fig8.
Conservation (31.0) and conservation (32.0) imply that if initial state i
and final state fexchange places, so-called fusion of other spinparticles is
given, refer to Fig7.
Fig5, Fig6 of Group-C are obtained by the analogy to those of Fig7,
Fig8 of Group-D above.

C(3) 0−3,0 ↖LC,f(3) LC0↗−9, 62C(3)
C(2) 0−2,0 ↖LC,f(2) LC↗−6, 42C(2)
C(1) 0−1,0 ↖LC,f(1) LC↗−3, 22C(1)
0 0
fusions C
C(3) 1−3, 6↗LC,i(3) LC↖−9,0 1C(3)
C(2) 1−2, 4↗LC,i(2) LC↖−6,0 1C(2)
C(1) 1−1,,2↗LC,i(1) LC↖−3,0 1C(1)
Fig5HiggsBoson and Gravitonare formedby fusionsofother spin particles
C(3) 1−3, 6↖LC,f(3) LC↗−9,0 1C(3)
C(2) 1−2, 4↖LC,f(2) LC↗−6,0 1C(2)
C(1) 1−1,,2↖LC,f(1) LC↗−3,0 1C(1)
fissions C
0 0
C(3) 0−3,0 ↗LC,i(3) LC↖−9, 62C(3)
C(2) 0−2,0 ↗LC,i(2) LC↖−6, 42C(2)
C(1) 0−1,0 ↗LC,i(1) LC↖−3, 22C(1)
Fig6other spin particlesare releasedby fissions of HiggsBoson and Graviton
LC,f(3) ≡CR,3
⊙−3, 6; −9,0 I0CR,3 −3, 6; −9,0 (34.1)
LC,i(3) ≡CR,3
⊙−3,0 ; −9, 6I0CR,3 −3,0 ; −9, 6(34.2)
LC,f(2) ≡CR,2
⊙−2, 4; −6,0 I0CR,2 −2, 4; −6,0 (35.1)
LC,i(2) ≡CR,2
⊙−2,0 ; −6, 4I0CR,2 −2,0 ; −6, 4(35.2)
LC,f(1) ≡CR,1
⊙−1, 2; −3,0 I0CR,1 −1, 2; −3,0 (36.1)
LC,i(1) ≡CR,1
⊙−1,0 ; −3, 2I0CR,1 −1,0 ; −3, 2(36.2)
LC8I0(37)

D(3) 0−2, 1↖LD,f(3)LD↗−9, 62D(3)
D(2) 0−1, 1↖LD,f(2) LD↗−6, 42D(2)
D(1) 00,1↖LD,f(1) LD↗−3, 22D(1)
1 0
fusions D
D(3) 5/6 −2, 6↗LD,i(3) LD↖−9, 17/6 D(3)
D(2) 3/4 −1, 4↗LD,i(2) LD↖−6, 15/4 D(2)
D(1) 1/2 0,2↗LD,i(1) LD↖−3, 13/2 D(1)
Fig7Higgs Boson and Graviton are formedby fusions of other spin particles
D(3) 5/6 −2, 6↖LD,f(3) LD↗−9, 17/6 D(3)
D(2) 3/4 −1, 4↖LD,f(2) LD↗−6, 15/4 D(2)
D(1) 1/20,2↖LD,f(1) LD↗−3, 13/2 D(1)
fissions D
1 0
D(3) 0−2, 1↗LD,i(3) LD↖−9, 62D(3)
D(2) 0−1, 1↗LD,i(2) LD↖−6, 42D(2)
D(1) 00,1↗LD,i(1) LD↖−3, 22D(1)
Fig8other spin particles are releasedby fissions of Higgs Boson and Graviton
LD,f(3) ≡DR,3
⊙−2, 6; −9, 1I0DR,3 −2, 6; −9, 1(38.1)
LD,i(3) ≡DR,3
⊙−2, 1; −9, 6I0DR,3 −2, 1; −9, 6(38.2)
LD,f(2) ≡DR,2
⊙−1, 4; −6, 1I0DR,2 −1, 4; −6, 1(39.1)
LD,i(2) ≡DR,2
⊙−1, 1; −6, 4I0DR,2 −1, 1; −6, 4(39.2)
LD,f(1) ≡DR,1
⊙0,2; −3, 1I0DR,1 0,2; −3, 1(40.1)
LD,i(1) ≡DR,1
⊙0,1; −3, 2I0DR,1 0,1; −3, 2(40.2)
LD8I0(41)

5) Conclusions
LA(l)andLB(l) in paragraph 3), which construct self-actions of zero spin
particles, could be thought of as the Lagrangian function of Higgs Boson in
quantum quantumfield.Further research could show that such kind of
mechanismmaylead to the change ofsymmetry breakingin the StandardModel.
LC(l)LD(l) in paragraph 4),which construct creation and annihilation
between 0zero spin particlesand 2spin particle,may be able to
dectect the existent of graviton from the ‘particulate’ nature of gravitation
experimentally, comparative study, wavelike propertiesofgravitation havebeen
exhibited [5]
References
[1]Higgs, Peter
Broken Symmetries and the Masses of Gauge Bosons Physical
Review Letters 13 (16) 508–509
Bibcode:1964 PhRvL..13..508H. doi:10.1103/PhysRevLett. 13.508
(1964)
European particle physics laboratory, CERN, (2012)
[2]ShaoXu Ren
Advanced Non-Euclidean Quantum Mechanics
ISBN 978-7-80703-585-4 (2006)
The Third Kind of Particles
ISBN 978-7-900500-91-5 (2011)
ISBN 978-988-15598-9-0 (2012)
ISBN 988-3-659-17892-4 (2012)
The Third Kind of Particles
Journal of Modern Physics, 5, 800-869
http:/dx.doi.org/10.4236/jmp.2014.59090
[3]ShaoXu Ren
The Origins Of Spins Of Elementary Particles
ISBN 978-988-13649-7-5 (2014)
The Origins of Bosons and Fermions
Journal of Modern Physics, 5, 1848-1879
http:/dx.doi.org/10.4236/jmp.2014.517181
[4]ShaoXu Ren
Interaction of the Origins of Spin Angular Momentum
ISBN 978-988-14902-0-9 (2016 2ndedition)
[5]LIGO, LSC (2016)
[6]ShaoXu Ren
Faster Than Velocity Of Light (Infinite Dimensional Lorentz Group
Of TKP )ISBN 978-988-12266-2-4 (2013)

6) Appendix: Higgs Boson Wave Differential Equation of First Order
and Klein–Gordon Wave Differential Equation
e) Usingmathelements j,kin STS[4], the Hamiltoniansof the first
order and the second order linearwave differential equantions of 0spin
zero particles, (Higgs Boson) are written as the following：
Hj,jll{Hj,jll3m} (A–1)
Hj,jl;k,kllHj,jllHk,kll(A–2)
Hj,jllin (A–1) is kinectic energy. Thereare many combinations in
(A–2),which made by various choise of jand k.
For clarity, here l1 andomittingthemark " 1" in aboveexpressions.
Then, taking jk−1 in casesof (A–1) and (A–2). getting below：
For (A–1)H−1,0 1−1,0 P(A–3)
and first order wave differential equantions of 0spin zero particle
0{i∂t−1−1,0 P−3m}−1,0 0 (A–4)
For (A–2)H−1,0;−1,0 H−1,0
2m2(A–5)
and second order wave differential equantions of 0spin zero particle
0{∂tt
2H−1,0 2m2}−1,0 0 (A–6)
To make it clearer, we consinder the diagonal terms of (A–5) and have：
diagonal {H−1,0
2} (A–7)
diag{, –25/2, –16/2, –9/2, –4/2, –1/2,0,–1/2, –4/2, –9/2, –16/2,–25/2, }P1
2
diag{, –25/2, –16/2, –9/2, –4/2, –1/2,0,–1/2, –4/2, –9/2, –16/2,–25/2, }P2
2
diag{,25, 16,9, 4, 1,0, 1, 4, 9, 16,25,}P3
2
(A–8)
{−1
2{P1
2P2 2}P3 2}0 20(A–9)
0
20diag{, 52,4
2,3 2,2 2,1 2, 02,1 2,2 2,3 2,4 2,5 2, } (A–10)
(A–9) indicates
{−1
2{P1
2P2 2}P3 2}⊂diagonal {H−1,0 2;0spin }(A–11)
and{∂tt
21
2{∂xx
2∂yy 2}−∂
zz
2m2}diagonal; −1,0 0 (A–12)

f) The Hamiltonians of the first order and the second order linear
differential equantions of /2 spin Fermion particles are written as
following：
Hj,j−2ll{Hj,j−2ll3m} (A–13)
Hj,j−2l;k,k−2llHj,j−2llHk,k−2ll(A–14)
Accordingly, taking jk0 in case of (A–13) and (A–14), we get：
For (A–13)H0,−2210,−2P(A–15)
and first order wave differential equantions of /2 spin particle
/2{ i∂t−210,−2P−3m}0,−20 (A–16)
For (A–14)H0,−2;0,−2H0,−2
2m2(A–17)
and second order wave differential equantions of /2 spin particle
/2{ ∂tt
2H0,−2 2m2}0,−20 (A–18)
Taking out the diagonal terms from(A–17) and have：
diagonal {H0,−2
2} (A–19)
diag{,–39, –23, –11, –3,1,1, –3, –11, –23, –39, –59, }P1
2
diag{,–39, –23, –11, –3,1,1, –3, –11, –23, –39, –59, }P2
2
diag{,81, 49, 25, 9, 1,1, 9, 25, 49, 81,121,}P3
2
(A–20)
Now, we see the two terms (A–21) in the center part of diagonal
{H0,−2
2} (A–20),is just the square sumHDirac 2of kinectic energy of
well-known Dirac equation of second order.
{HDirac
2}diag{..., 1,1, ...}{P1
2P2 2P3 2}⊂diagonal {H0,−2 2}
(A–21)
Or{HDirac
2}{HKG}−∇2⊂diagonal {H0,−2 2;/2spin }(A–22)
And contrast with (A–11), we get
{HDirac
2}{HKG}−∇2diagonal {H−1,0
2;0spin }(A–23)
Formula (A–22) and (A-23) mean：−∇2is a subset of H0,−2
2,not a set
ofH −1,0
2.So Klein-Gordon Equation
{2∂tt
2−c22∇2m2c4}KG 0 (A–24)
{−2}KG 0 (A–25)
∇2−∂
tt
2/c2,mc/(A–26)
is closer to /2 spin Fermion particle,rather than0spin Boson particle.
It is more reasonable to use equation (A–4), equation (A–6) to describe
zero spin Boson particle (Higgs Boson) than touse Klein Gordon equation.

g) For Vacuum Spin particle, VSP, –/2 negative one-second fermion
particle, its Hamiltoniansof the first order and the second order linear wave
differential equantions are written asthe following：
Hj,jl{Hj,jl3m} (A–27)
Hj,j;k,klHj,jlHk,kl(A–28)
Taking jk0 in case of (A–27) and (A–28), we get：
For (A–27)H0,0 210,0 P(A–29)
and first order wave differential equantions of –/2 spin particle
–/2{ i∂t−210,0 P−3m}0,0;VSP 0 (A–30)
For (A–28)H0,0;0,0 H0,0
2m2(A–31)
and second order wave differential equantions of –/2 spin particle
–/2{ ∂tt
2H0,0 2m2}0,0;VSP 0 (A–32)
Taking out the diagonal terms from(A–31) and have：
diagonal {H0,0
2} (A–33)
diag{, –61,–41, –25, –13, –5, –1, –1, –5, –13, –25, –41, }P1
2
diag{, –61,–41, –25, –13, –5, –1, –1, –5, –13, –25, –41, }P2
2
diag{,121,81, 49, 25, 9, 1,1, 9, 25, 49, 81,}P3
2
(A–34)
Now turn to the two terms (A–35)in the center part of diagonal {H0,0
2}
{HVSP }diag{..., 1,1, ...}{–P1
2−P2 2P3 2}⊂diagonal {H0,0 2}
(A–35)
and have wave equation of VSP, (–/2 spin fermion particle)
{∂tt
2∂xx 2∂yy 2−∂
zz
2m2}VSP 0 (A–36)
h) Next we shall discussthe solutions 0≡j,0
m0of zero mass particle
differential equantions of First Order,which are based on free −/2 VSP
particle (A–30),free 0zero spin particle (Higgs Boson) (A–4),free /2
Dirac spin particle (A–16).Which are given as below：
−/2{ i∂t−20,0 P}0,0;VSP
00 (A–37)
0{i∂t−−1,0 P}−1,0;Higgs Boson
00 (A–38)
/2{ i∂t−20,−2P}0,−2;Dirac
00 (A–39)

Notation：
EE p,pp1
2p2 2p3 2(A–40)
h1) For free −/2 VSP zero mass particle (A–37)
0,0;VSP
0F0,0 e−iEt (A–41)
Getting Fp；0,0 e−iEtF−p；0,0 e−iE−t

−p−p35/p
5
p−p34/p
4
−p−p33/p
3
p−p32/p
2
−p−p31/p
1
p−p30/p
0
pp30/p−
0
−pp31/p−
1
pp32/p−
2
−pp33/p−
3
pp34/p−
4
−pp35/p−
5
e−iEt
pp35/p
5
pp34/p
4
pp33/p
3
pp32/p
2
pp31/p
1
pp30/p
0
p−p30/p−
0
p−p31/p−
1
p−p32/p−
2
p−p33/p−
3
p−p34/p−
4
p−p35/p−
5
e−iE−t
(A–41.1) (A–41.2)
Fp;0
−tan5/2 e−i5
tan4/2 e−i4
−tan3/2 e−i3
tan2/2 e−i2
−tan1/2 e−i
ei0
F−p;0
cot5/2 e−i5
cot4/2 e−i4
cot3/2 e−i3
cot2/2 e−i2
cot1/2 e−i
ei0
Fp;0
ei0
−cot1/2 ei
cot2/2 ei2
−cot3/2 ei3
cot4/2 ei4
−cot5/2 ei5
F−p;0
ei0
tan1/2 ei
tan2/2 ei2
tan3/2 ei3
tan4/2 ei4
tan5/2 ei5
(A–41.3) (A–41.4)
VSP particleFp；0,0 Fp;0Fp;0(A–41.5)
F−p；0,0 F−p;0F−p;0(A–41.6)

h2) For free 0zero spin zero mass particle (Higgs Boson) (A–38)
−1,0;HB
0F−1,0 e−iEt (A–42)
Getting
p；−1,0;HB
0−p；−1,0;HB 0

−p−p34p/p
5
p−p33p/p
4
−p−p32p/p
3
p−p31p/p
2
−p−p30p/p
1
p∓p30p0/p
0
−pp30p/p−
1
pp31p/p−
2
−pp32p/p−
3
pp33p/p−
4
−pp34p/p−
5
e−iEt
pp34p/p
5
pp33p/p
4
pp32p/p
3
pp31p/p
2
pp30p/p
1
pp30p0/p
0
p−p30p/p−
1
p−p31p/p−
2
p−p32p/p−
3
p−p33p/p−
4
p−p34p/p−
5
e−iE−t
(A–42.1) (A–42.2)

−tan4/2 e−i5
tan3/2 e−i4
−tan2/2 e−i3
tan1/2 e−i2
−e−i
sin 
−ei
cot 1/2 ei2
−cot 2/2 ei3
cot 3/2 ei4
−cot 4/2 ei5
e−iEt
cot4/2 e−i5
cot3/2 e−i4
cot2/2 e−i3
cot1/2 e−i2
e−i
sin 
ei
tan 1/2 ei2
tan 2/2 ei3
tan 3/2 ei4
tan 4/2 ei5
e−iE−t
(A–42.3) (A–42.4)
There are two singularities at 0, andin the above two
expressions.Obviously, some uncertainties of choise of free 0zero spin
zero mass wavefunction should be addressed. Here(A–42.3) and (A–42.4)
are only an investigation.

h3) For free /2 Dirac spin zero mass particle (A–39)
0,−2;Dirac
0F0,−2e−iEt (A–43)
Getting p；0,−2;Dirac
0−p；0,−2;Dirac 0

......
pp34/p
4
pp33/p
3
pp32/p
2
pp31/p
1
pp30/p
0
p−p31/p−
1
p−p32/p−
2
p−p33/p−
3
p−p34/p−
4
p−p35/p−
5
.....
e−iEt,
......
−p−p35/p
5
p−p34/p
4
−p−p33/p
3
p−p32/p
2
−p−p31/p
1
p∓p30/p
0
−pp31/p−
1
pp32/p−
2
−pp33/p−
3
pp34/p−
4
......
e−iE−t
(A–43.1) (A–43.2)

......
cot4/2cos /2e−i4
cot3/2cos /2 e−i3
cot2/2cos /2 e−i2
cot1/2cos /2 e−i
cos /2
sin /2 ei
tan1/2sin /2 ei2
tan2/2sin /2 ei3
tan3/2sin /2 ei4
tan4/2sin /2 ei5
.....
e−iEt,
......
−tan4/2sin /2 e−i5
tan3/2sin /2 e−i4
−tan2/2sin /2 e−i3
tan1/2sin /2 e−i2
−sin /2 e−i
cos /2
−cot1/2cos /2 ei
cot2/2cos /2 ei2
−cot3/2cos /2 ei3
cot4/2cos /2 ei4
.....
e−iE−t
(A–43.3) (A–43.4)
The two elements in the centersof the above expressions are just the
spin wavefunction representation of operatornof /2Diracspinintwo
dimensional spin space intraditional quantum machenics.
p；Dirac
0−p；Dirac 0
cos /2
sin /2 ei,−sin /2e−i cos /2
(A–43.5) (A–43.6)

i) Finally we digress slightly, to tackle the situations of the velocity of
light,because light speed is related to spin angular momentum in STS.
i1) The special case of P1P10 for non-zeromass particle Second
Order differential equantions –/2 (A-32), VSP particle, zerospin particle
(Higgs Boson) (A-6), /2 Dirac spin particle (A–18) are givenbelow：
−/2{ E20,0
2∂zz
2−m2}0,0;VSP 0 (A–44)
0,0
2diag{,121,81, 49, 25, 9, 1,1, 9, 25, 49, 81,}
0{E2−1,0
2∂zz
2−m2}−1,0;Higgs Boson0 (A–45)
−1,0
2diag{, 25,16,9,4,1,0, 1, 4,9, 16, 25,}
/2{ E20,−2
2∂zz 2−m2}0,−2;Dirac 0 (A–46)
0,−2
2diag{,81, 49, 25,9, 1,1, 9,25, 49, 81,121,}
getting [2]
ES|j−k|
 m2c4i,j
2c2P3 2≥m2c4c2P3 2(A–47)
Photon velocity in multi-level universes world is quantized：the limiting
speed of particle with zero massm0,could be greaterthan c
CS|j−k|i,jc1c,2c,3c,4c,...or 1c,3c,5c,7c,...≥c, (A–48)
i2) Lorentz Group Operators are constructed by six44 dimensional
matrices : J1,J2,J3and K1,K2,K3
J1
0000
0000
000−i
00i0
,J2
0000
000i
0000
0−i00
,J3
00 0 0
00−i0
0i00
00 0 0
(A–49)
K1
0i00
i000
0000
0000
,K2
00i0
0000
i000
0000
,K3
000i
0000
0000
i000
(A–50)
K3i∂
∂3L3|30i∂ ∂3
Ch 00Sh 
0100
0010
Sh 00Ch 
|30(A–51)
L3is the familiar expressionof Einstein special relativity.

Using infinite dimensional spin augular momentum operators−2,1
(1: −2,1 ,2: −2,1 ,3: −2,1 )of1spin boson particle, we could get six infinite
dimensional matricesJ1,J2,J3and K1,K2,K3of Lorentz Group Operators.
Among them,matrix K3is shown below
K3K3

0000 5i
0000 4i
0000 3i
0000 2i
000i
0000 
0000 
i000 
2i0000
3i0000
4i0000
5i0000

(A–52)
Or
K3K3

0000 5
00004 
00003 
00002 
0001 
0000 
0000 
−1000 
−2 0000
−3 0000
−4 0000
−5 0000

(A–53)
Take note of (A–52) and (A–53),they are two different types of
Non-Hermitian operators, antisymmetrical matrices,base on them,proceed
as follows
L3L3(A–54)

From (A–57), some curious spectacles that similar to (A–48) are emerged
[6] ：
For (A–52),For (A–53)
L3
00Ch Sh 
Sh Ch ，L3
00Sech −Th 
Th Sech 
L3
01Ch 2Sh 2
Sh 2Ch 2，L3
01Sech 2−Th 2
Th 2Sech 2
L3
02Ch 3Sh 3
Sh 3Ch 3，L3
02Sech 3−Th 3
Th 3Sech 3
..... ..................... ................
(A–55) (A–56)
Then, Einstein Special Relativity is extended to the following so-called：
Worm Hole Special Relativity in Multi-Level Universes World：[6]
x0, j
′
x3, j
′L3
0, jx0, j
x3, j(A–57)
For (A–55), have:,For (A–56), have
x0
′Ch x0Sh x3,x0 ′Sechx0−Th x3
x3
′Sh x0Ch x3,x3 ′Th x0Sech x3
,2,3,... ,,2,3,...
(A–58) (A–59)
and Sh  (A–58.1), Th  (A–59.1)
Ch (A–58.2), Sech (A–59.2)
Th (A–58.3), Sh (A–59.3)
Ch2−Sh21 (A–58.4),Sech2Th21 (A–59.4)
j) Spin Topological Space STS is the space that could discribe and help
people understand how the transitions of particle spins, between various types
of spin particles,are happening.Before this,the concepts of physicsand
math about these transitions were indistinct andblurred.
To appreciate the beauty and subtlety of STS, theFIG.below is essential.