Journal of Modern Physics, 2013, 4, 1213-1380
http://dx.doi.org/10.4236/jmp.2013.410165 Published Online October 2013 (http://www.scirp.org/journal/jmp)
Stochastic Quantum Space Theory on Particle Physics
and Cosmology
—A New Version of Unified Field Theory
Zhi-Yuan Shen
Received May 14, 2013; revised June 14, 2013; accepted August 1, 2013
Copyright © 2013 Zhi-Yuan Shen. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Stochastic Quantum Space (SQS) theory is a new version of unified field theory based on three fundamental postula-
tions: Gaussian Probability Postulation, Prime Numbers Postulation, Vacuon Postulation. It build a framework with
theoretical results agree with many experimental data well. Main conclusions of SQS theory are: 1) The 3-dimensional
Space with face-centered lattice structure attached to Gaussian probability is stochastic in nature. 2) SQS theory is
background independent at three levels. 3) Quarks with same flavor and different colors are different elementary parti-
cles. There are 18 quarks in three generations based on 18 prime numbers. 4) There are only three generations of ele-
mentary particles. 5) SQS theory elementary particles table listed 72 particles including 13 hypothetic bosons. Vacuon
is the only elementary particle at the deeper level. 6) Photon has dispersion and special relativity is revised accordingly.
7) Graviton has zero spin. 8) Entangled particles are connected by physics link with limited distance and non-infinite
superluminal speed. 9) All physical events are local, no “spooky action at a distance”. 10) Elementary particle is repre-
sented by discrete trajectories on geometrical model with genus number from 0 to 3. 11) Characteristic points and re-
lated triangles in elementary particle’s model provide its physics parameters. 12) Particles internal movements in a tra-
jectory are deterministic and uncertainty only comes from jumping trajectories. 13) Fermion’s mass exceeds
2
/973.4 cGeVMMax  must pair with anti-fermion serving as a boson state. 14) Fine structure constant is a running constant
2
)71/2(   based on a mathematic running constant

. 15) Converting rules based on Random Walk Theorem are
introduced to deal with hierarchy problems. 16) Logistic recurrent process and grand number phenomena play impor-
tant rules for converting factor in the transmission region between P
L71 and Compton length. 17) Based on
face-centered space structure, 36 symmetries )(rO , )(rC with P
Lr 3 are identified as the intrinsic symmetries serving
as the origin of all physics symmetries. 18) Heisenberg uncertainty principle is generalized with less uncertainty at
sub-Planck scale. 19) There are close correlations between elementary particle theory and three finite sporadic Lie
groups: M, B, Suz. 20) Cosmic structure and evolution are intrinsically correlated to elementary particles and prime
numbers. 21) A part of dark matters is 2-dimensional membranes left over from cosmic inflation driven by e-boson as
the inflaton. 22) After the big bang and before the current cosmic period, there are two cosmic periods with
3
1
1-dimensional space and 2-dimensional space. 23) A cyclic universe model is based on positive and negative prime
numbers. 24) A multiverse includes 22 10~ member universes organized in two levels. 25) The limited anthropic prin-
ciple is introduced. 26) A super-multiverse includes 44
10~ member multi-universes organized in two levels; Total
number of universes in the super-multiverse is 66 10~ . 27) Based on Poincare theorem, SQS theory introduces the ab-
solute black hole without any radiation. 28) A GUT including all four types of interactions occurs at 71 Planck lengths.
29) SQS theory primary basic equations are established based on Einstein equations for vacuum and redefined gauge
tensors attached to Gaussian probability. SQS Theory provides 25 predictions for experimental verification.
Keywords: Unified Field Theory; Space Structure; Elementary Particles; Gaussian Probability; Prime Numbers;
Sporadic Groups; GUT; Dark Matter; Dark Energy; Cosmos Inflaton; Multiverse; Anthropic Principle;
General Relativity; Primary Basic Equations

Z. Y. SHEN 1214
Section 1. Introduction
This paper is the continuation and extension of the au-
thor’s previous paper [1], which was published in Chi-
nese. For people not familiar with Chinese language, a
brief review of the previous paper is included in this paper.
Stochastic Quantum Space (SQS) theory initially was
intended to be a theory of space. It turns out as a unified
field theory including particle physics and cosmology.
In essence, SQS theory is a mathematic theory. Its re-
sults are interpreted into physics quantities by using three
basic physics constants, h, c, G or equivalently P
L,
P
t, )(PP ME . In principle no other physics inputs are
needed.
SQS theory is based on three fundamental postulations,
Gaussian Probability Postulation, Prime Numbers Postu-
lation, Vacuon Postulation, which serve as the first prin-
ciple of SQS theory.
Based on three fundamental postulations, SQS theory
builds a framework.
Based on Einstein’s general relativity equations for
vacuum and redefined gauge tensors attached to prob-
ability, SQS theory established the basic equations in-
cluding two parts. The microscopic part is the primary
basic equations for elementary particles, interactions and
things on upper levels. The macroscopic part as the av-
eraged version includes two sets of basic equations, one
set for gravity and the other set for electromagnetic force.
SQS theory provides twenty five predictions for veri-
fications.
The basic ideas of SQS theory are summarized as the
following:
1) Space is a continuum with grainy structure. It is
stochastic in nature represented by Gaussian probability
distribution functions at discrete points. Elementary par-
ticles and interactions are different types of movement
patterns of the space.
2) Cosmology and particle physics are intrinsically
correlated with mathematics, in which prime numbers
play the central role.
3) The correct way to unify general relativity theory
with quantum theory is to introduce probabilities to Ein-
stein’s original equations for vacuum.
SQS theory laid down the foundations and built a
framework. There are many open areas for physicists and
mathematicians to explore and contribute.
Section 2. Gaussian Probability Assignment
According to Stochastic Quantum Space (SQS) theory,
space is stochastic and continuous with grainy structure
in Planck scale.
The Planck length is:
35
31061625.1
2

c
hG
LP

m, (2.1a)
Based on P
L, Planck time P
t, Planck energy P E
and Planck mass P
M are defined as:
s
c
hG
c
L
tP
P44
51039123.5
2



, (2.1b)
J
G
hc
L
hc
E
P
P10
5
1022905.1
2

, (2.1c)
kg
G
hc
cL
h
M
P
P7
10367498.1
2


. (2.1d)
In which h, c and G are Planck constant, speed of
light in vacuum and Newtonian constant of gravitation,
respectively.
Postulation 2.1A. Gaussian probability postulation.
The relation between different points in space is stochas-
tic in nature. Gaussian probability distribution function is
assigned to each discrete point i x separated by Planck
length. In 1-dimensional case, the Gaussian probability at
point
x
is:


2
2
2
2
1
;


i
xx
iexxp


;







,,0,,x;








,,2,1,0,1,2,,
i
x. (2.2)
The distance between adjacent discrete points is nor-
malized to 1
P
L.
Explanation: The Gaussian Probability Postulation
serves as the first fundamental postulation of SQS theory.
It represents the stochastic nature of space and also
represents the quantum nature of space without sacrific-
ing space as a continuum. The

i
xxp ; serves as the
value at point
x
from the Gaussian probability distribu-
tion function centered at discrete point i x. Postulation
2.1 is for 1-dimensional case as the foundation for
3-dimensional case.
The Standard Deviation (SD)

of Gaussian prob-
ability is selected to let the numerical factor in front of
exponential term in (2.2) equal to 1:
014333989422804.0
2
1


. (2.3)
The reason of selecting such specific value for

will
be explained later.
Substituting (2.3) into (2.2) yields:



2
;i
xx
iexxp


;



 ,,0,,x;

 ,,2,1,0,1,2,,
i
x. (2.4)
Postulation 2.1B. In the 3-Dimensional Case, (2.2) is
Extended as


  
2
222
2
3
2
3
2
1
,,;,,


kji zzyyxx
kjiezyxzyxp


;

Z. Y. SHEN 1215




,,0,,,, zyx ;


 ,,2,1,0,1,2,,,, kji zyx . (2.5)
The values of

are determined by the roots of the
following equation:
01)2( 32/3  
. (2.6)
Equation (2.6) has three roots:


2
1
'0,



2
'
3/2
1
i
e
,



2
'
3/4
2
i
e
. (2.7)
Substituting
 2/1'0 into (2.5) yields:

   
222
,,;,, kji zzyyxx
kji ezyxzyxp 


;



 ,,0,,,,zyx ;


 ,,2,1,0,1,2,,,, kji zyx . (2.8)
In (2.8), only the real root of
 2/1'0 is used.
The meaning of all three roots will be discussed later.
Definition 2.1: The Gaussian sphere centered at
),,( kji zyx is defined as its surface represented by the fol-
lowing equation:
2222 )()()(Rzzyyxx kji  . (2.9a)
The radius of Gaussian sphere is defined as:
932743535533905.0
22
1R. (2.9b)
Explanation: The 3-dimensional Gaussian probability
distribution of (2.8) has spherical symmetry like a sphere
with blurred boundary. The Gaussian sphere is defined
with a definitive boundary. It plays an important role for
the structure of space as shown in Section 22.
For the 1-dimensional case, according to (2.4), the un-
itarity of probability distribution function ),( i
xxp with
respect to continuous variable
x
is satisfied for any dis-
crete point i
x:


1; 2  



 dxedxxxp xx
ii

. (2.10)
In general, the unitarity of probability ),( i
xxp with
respect to discrete variable i x is not satisfied.
Definition 2.2: S-Function. Define the summation of
),( i xxp with respect to i
x as the
 xS-function:
 








i
i
ix
xx
xiexxpxS 2
;

. (2.11)
Theorem 2.1: S-function

 xS satisfies periodic con-
dition:
 xSxS  1. (2.12)
Proof: According to (2.11):

 

)(1
2
22 )1(1 xSeeexS
j
j
i
i
i
i
x
xx
x
xx
x
xx  










.
QED
The values of

 xS in the region 10  x are listed
in Table 2.1 and shown in Fig. 2.1.
Table 2.1: The values of
 xS in region 10


x.
In the region

 1,0 , except two points at 25.0

x and
75.0

x, in general

 xS defined by (2.11) does not
satisfy unitarity requirement, which has important impli-
cations.
0.9
0. 92
0. 94
0. 96
0. 98
1
1. 02
1. 04
1. 06
1. 08
1.1
00.1 0.20.3 0.40.50.60.7 0.80.91
x
S(x)
Figure 2.1.

 xS curve in region 10

x.
Theorem 2.2:

 xS satisfies the following symmet-
rical condition:



 xSxS


1, 10 x. (2.13)
Proof: According to (2.11):




)(1
2
22 )1(1 xSeeexS
j
j
i
i
i
i
x
xx
x
xx
x
xx   










. QED
Def inition 2.3: S

-Function. Define the S

- func-
tion as:

 1)( 


xSxS . (2.14)
Numerical calculation found that
 xS satisfies the
following approximately anti-symmetrical condition:



 xSxS





5.0 , 5.00 x. (2.15)
The symmetry of

 xS with respect to 5.0

x in
region

 1,0 given by (2.13) is exact. The anti-symmetry
of

 xS

with respect to 25.0x in region

 5.0,0
given by (2.15) is approximate with a deviation less
than 5
10. The deviation is tiny, but its impact is signifi-
cant. It plays a pivotal role for SQS theory, which will be
shown later.

Z. Y. SHEN 1216
Numerical calculation found that at the center 25.0

x
of the region
 5.0,0 :
 152889999930253.025.0 S. (2.16)
(2.16) indicates that,
 25.0S has a deviation of
6
107~   from 1 required by the unitarity. Numerical
calculation found a point c
xin region [0, 0.5] satisfying
unitarity:
 1
c
xS, (2.17)

 
5.0
0
0)(11)(
c
c
x
x
dxxSdxxS , (2.18)
73026452499871562.0
c
x. (2,19)
On the x-axis, c
xis located at the left side of
25.0x. It extends the region of
 1xS and shrinks
the region of
 1xS . The special point c
x has a pro-
found effect on elementary particles and unifications of
interactions, which will be given in later sections.
Definition 2.4: Based on c
x, three other special points
a
x,b
x,d x are defined:
3726973550.250012845.0  cd xx . (2.20)
   
 
25.0
0
011
c
a
x
x
dxxSdxxS .
10
10882111819946879.5  
a
x. (2.21)
   
 
25.0
0
0
c
b
x
x
dxxSdxxS ,
-5
10918477191.18218617 
b
x. (2.22)
The physics meaning of four special points, a
x, b
x,
c
x, d x, will be given later.
In 3-dimensional case, according to (2.8), the unitarity
of


kjizyxzyxp ,,;,, with respect to continuous vari-
ables x, y, z is satisfied for any discrete point


kji zyx,, :

   
.1
,,;,,
2
2
2
222
)(
)(
)( 

























dzedyedxe
edzdydxzyxzyxdzpdydx
k
j
i
kji
zz
yy
xx
zzyyxx
kji

(2.23)
In general, the unitarity of probability


kji zyxzyxp ,,;,,
with respect to discrete variables kji zyx ,, is not satis-
fied.
Definition 2.5: Define the summation of the probabil-
ity


kjizyxzyxp,,;,, with respect to kji zyx ,, as:
 
   
 













ijk
kji
ijkxyz
zzyyxx
xyz kji ezyxzyxpzyxS 222
,,;,,,,
3

.
(2.24)
Theorem 2.3:


zyxS,,
3 can be factorized into three
factors:






).()()(
,,
2
2
2
222
)(
)(
)(
3
zSySxSeee
ezyxS
k
k
j
j
i
i
ijk
kji
z
zz
y
yy
x
xx
xyz
zzyyxx










































(2.25)
Proof: The three-fold summation in (2.25) includes
terms for all possible combinations of 2
)( i xx
e , 2
)( j yy
e ,
2
)( k zz
e . The three multiplications in (2.25) include the
same terms. They are only different in processing, the
results are the same. QED
By its definition and (2.12), (2.25),
 zyxS,,
3 satisfies
the following periodic conditions:

 
zyxSzyxS,,,,1 33


, (2.26a)

  zyxSzyxS,,,1, 33


, (2.26b)

  zyxSzyxS ,,1,, 33  . (2.26c)
Definition 2.6: Planck cube is defined as a cube with
edge lengths 1
P
L and with discrete point ),,( kji zyx at
its center or its corner.
The values of

 zyxS,,
3 at 125 points in a Planck
cube with discrete points at its corner are calculated from
(2.24) and listed in Table 2.2.
Table 2.2.

 zyxS ,,
3 values at 125 points in a Planck cu-
be (Truncated at

1000).
Theorem 2.4: Probability Conservation Theorem.
The average value of

 zyxS,,
3 over a Planck cube
equals to unity:
  1,,,, 3
1
0
1
0
1
0
3

zyxSdzdydxdvzyxS
PlanckCube
. (2.27)
Proof: Substitute (2.24) into left side of (2.27):
 






























ijk
kji
ijk
kji
xyz
zzyyxx
xyz
zzyyxx
PlanckCube
edzdydx
edzdydx
zyxSdzdydxdvzyxS
1
0
1
0
])()()[(
1
0
])()()[(
1
0
1
0
1
0
3
1
0
1
0
1
0
3
.
,,,,
222
222


(2.28)
Change variables as:

Z. Y. SHEN 1217
,'xxx i ,' yyy j zzz k'. (2.29)
Substituting (2.29) into (2.28) and changing integra-
tions’ upper and lower limits accordingly yield:
 


.1'''
'''
,,,,
222
222
)'()'()'(
11
)'()'()'(
1
3
1
0
1
0
1
0
3





















kji
ijk
i
i
j
j
kji
k
k
zzyyxx
xyz
x
x
y
y
zzyyxx
z
z
V
edzdydx
edzdydx
zyxSdzdydxdvzyxS


QED
Probability Conservation Theorem is important. It
proved that, even though in general
 zyxS ,,
3 does not
satisfy unitarity requirement, but it does satisfy unitarity
requirement in terms of average over a Planck cube. The
conservation of probability means that, the event carriers
of probability are moving around but they cannot be cre-
ated or annihilated.
Lemma 2.4.1: The average value of
 xS over re-
gion [0,1] equals to unity:


1
0
1dxxS . (2.30)
Proof: Substitute (2.11) into the left side of (2.30):


 




 
i
i
i
i
x
xx
x
xxedxedxxS
1
0
)(
1
0
1
0
2
2
 . (2.31)
Change variables as:
xxx i'. (2.32)
Substituting (2.32) into (2.31) and changing integra-
tion’s upper and lower limits accordingly yield:


 



 
 1' 22 '
1
'
1
0
dxedxedxxS x
x
x
x
x
i
i
i

. QED
Lemma 2.4.2: Planck cube with volume 1

V (leng-
th normalized to 1
P
L) is divided into two parts 1 V
and 2
V:
1
21  VVV , (2.33a)
,1

V
Vdv ,
1
1

V
Vdv .
2
2

V
Vdv (2.33b)
Theorem 2.4 leads to the following equation:
  




12
,,11,, 33
VV
dvzyxSdvzyxS. (2.34)
Proof: According to (2.27) and (2.33):
 


212112
21333 1),,(,,,,
VVVVVV
d
v
dvVVVdvzyxSdvzyxSdvzyxS.
Moving the terms on left and right sides yields (2.34).
QED
Section 3. Unitarity
Unitarity is a basic requirement of probability. As shown
in Section 2, the unitarity with respect to discrete vari-
ables and continuous variables for Gaussian probability
are contradictory. In this section, three schemes are pre-
sented to solve the unitarity problem.
Scheme-1. To Treat All Points in Space Equally
For Scheme-1, Gaussian probabilities are not only as-
signed to discrete points but to every point in the con-
tinuous space. The summations in
 zyxS ,,
3 of (2.24)
introduced in Section 2 become integrations:
  
 
 .1'''
'''',',';,,,,;,,,,
'')'(
3
2 2
























zSySxSdzedyedxe
dzdydxzyxzyxpzyxzyxpzyxS
zzyyxx
xyz kji
ijk

(3.1)
The unitarity problem is solved.
For Scheme-1, the discrete points are no longer special
and all points in the space are on an equal footing. But
Scheme-1 does not represent the space for SQS theory,
instead, it represents the space for quantum mechanics.
To introduce Scheme-1 is for comparison purpose. It
indicates that, the uniform space does not have unitarity
problem. The unitarity problem is caused by space grainy
structure, which SQS theory must deal with.
Let’s go back to the grainy space proposed by SQS
theory. In Appendix 1, A Fourier transform is applied to
probability


xp of (2.4) to convert it into k-space. Ac-
cording to (A1.2), the corresponding Gaussian probabil-
ity function

 kP in k-space is:



4
2
2
1k
ekP   . (3.2)
The standard deviation of


kP is .2  
k Mul-
tiplying (A1.6) with yields:

   xpxk 
. (3.3)
In (3.3), x

and x
pare 1-dimensional displacement
and momentum difference, respectively. The  on right
side is two times greater than the minimum value 2/
from Heisenberg uncertainty principle. The increased
uncertainty is due to the asymmetry of
 2/1 and
 2
k.
The wave function corresponding to

i
xkP ; of (A1.1)
is:

i
ikx
k
iieexkQkPxk



8
2
2
1
);()(;

 ;




,,0,, k;



,,2,1,0,1,2,, 
i
x. (3.4)
Notice that, the wave function (3.4) has following fea-
tures:
1) The relation between

i
xkP ; and

i
xk;

is con-

Z. Y. SHEN 1218
sistent with quantum mechanics:
 
iii xkxkxkP;;;


. (3.5)
2) Only discrete points i
x appear in the phase func-
tion i
ikx
e.
3).

i
xk;

is not an eigenstate of k. The magni-
tude )(kP of

i
xk;

serves as distribution function
for k.
Before explore other schemes, a discussion for the es-
sence of probability unitarity is necessary. Probability is
associated with events. In Section 2, Table 2.2 data show
that, in the vicinity of Planck cube’s center


5.0,5.0,5.0 zyx ,
the sum of probabilities
 1,,
3zyxS. Because the set of
events at these points are incomplete; some events are
missing. These missing evens cause the sum of local
probabilities less than one. In the vicinity of the Planck
cube corners


kji zyx ,, ,
 1,,
3zyxS , because the set
of events over there includes some events belong to other
places. These excessive evens cause the sum of local
probabilities greater than one. In other words, events
associated with their probabilities move around inside
Planck cube causing the unitarity problem. To move
these events back to where they belong will solve the
discrete unitarity problem. But it distorts the Gaussian
probability distribution and jeopardizes the unitarity with
respect to continuous variables based on Gaussian prob-
ability distribution.
To solve the problem requires some new concept. Tra-
ditionally, unitarity is local, which requires the sum of
probability equals to unity at each point in space.
 1,,
3zyxS is caused by events moving around. The
foundation for local unitarity no longer exists. A gener-
alized unitarity is proposed:
1) Recognize the fact that events associated with
probabilities move around;
2) Follow the moving events for probability unitar-
ity.According to Theorem 2.4, the Probability Conserva-
tion Theorem, generalized unitarity is not contradictory
to the traditional unitarity for the Planck cube as a whole
entity. But it does change the rules inside the Planck cube.
For the microscopic scales, as the events inside Planck
cube are concerned, generalized unitarity is necessary.
For the macroscopic scale including many Planck cubes,
the local unitarity is still valid in the average sense.
The following two schemes are based on generalized
unitarity.
Scheme-2. Unitarity via Probability
Transportation on Complex Planes
The complex planes are inherited from the
3-dimensional Gaussian probability. Consider a Planck
cube centered at a discrete point )0,0,0(

 kji zyx as
shown in Figure 3.1. According to (2.5), the
3-dimensional Gaussian probability is:


2
222
2
3
2
3
2
1
0,0,0;,,


zyx
ezyxp


. (3.6)
Normalize the three values of the standard deviations
0
'

,1
'

, 2
'

of (2.7) as:
1'200  
,
3/2
11 '2


i
e ,
3/2
22 '2


i
e . (3.7)
In which two of them 1

and 2

are complex num-
bers. To keep the probabilities as real numbers related to
1

and 2

, it is necessary to extend the x-axis, y-axis,
z-axis into three complex plans x -plane, y
-plane, z
-
plane, respectively.
Figure 3.1. The Planck cube with center at a discrete point
)0,0,0(



kji zyx.
Definition 3.1: Define parameters to build three com-
plex planes associated with x-axis , y-axis, z-axis:

 
 i
exx  2,12,1
, (3.8a)

 

 i
eyy   2,12,1
, (3.8b)

  
i
ezz   2,12,1
, (3.8c)

120
3
2


.
(3.8d)
In which



,, and



,, are real parameters.
Explanation: In (3.8a), 2,1

 represents two straight
lines on complex x -plane intercepting to the real x-axis
at


x
with angles of 
120 
. Continuously

Z. Y. SHEN 1219
change the value of

, 1

 and 2

 sweep across x-axis
to construct the complex x
-plane. Every point on the
complex x
-plane is the intersection of two straight lines
defined by (3.8a). They
-plane and z -plane associated
with y-axis and z-axis are constructed in the same way.
On the complex x
-plane, three straight lines 1

, 2


and x-axis intercept at 0



x with 3-fold rotational
symmetry as shown in Figure 3.2. The 3-fold rotational
symmetry has its physics significance, which will be
discussed later.
Figure 3.2. Three straight lines with 3-fold rotational sym-
metry on complex x
-plane.
Rule 3.1: In order to keep the values of


i
xxp ;,


kji zyxzyxp ,
,;,, and


xS ,
 zyxS ,,
3as real numbers,
in the Gaussian probability exponential part, spatial va-
riables
x
y
x
,, in the numerator choice their path ac-
cording to (3.8) matching the

value in denominator
to keep these values always equal to real numbers.
Explanation: The validity of Rule 3.1 to


i
xxp;,


kji zyxzyxp ,
,;,, is obvious. Its validity for

 xS ,
 zyxS ,,
3 needs explanation. According to the definition
of


xS :


 





i
i
ix
xx
xiexxpxS2
;

. (2.11)
All terms of
 




i
xi
xxpxS ; except

0;xp have
their “tail” in region ]5.0,5.0[ , which are equivalent to
the “tails” of


0;xp in regions of ]5.0,[

 and
],5.0[  :

 

 


 




x
x
x
x
x
x
xx etoequivalente
i
i
i
5.0
,5.0
)0(
5.05.0
;0
)( 2
2
 
. (3.9)


xS in region [−0.5,0.5] can be viewed as a single
probability distribution function
 0;xp with “mul-
ti-reflections” at the two boundaries of region [−0.5,0.5].
Figure 3.3 shows an example for the 0
i
x term along
with two adjacent terms 1
i
x and 1
i
x with their
“tails” in region [−0.5,0.5]. In essence, probability trans-
portation via complex plane for

i
xxp ; is also valid for

 xS . According to Theorem 2.3,


)()()(,,
3xSySxSzyxS ,
the same argument is valid for
 zyxS,,
3 as well.
Figure 3.3. Three adjacent Gaussian probability distribu-
tion functions show the “tails”.
For double check, let’s look it the other way, consider
the


i
xxp; term in


xS :

2
2
)(
;


i
xx
iexxp


. (3.10)
In which i x is a real number and
x
is a complex
number. As long as the point corresponding to


i
xx is
on the lines defined by (3.8a),

i
xxp; is a real number,
and so is

 xS ..
In the Planck cube centered at discrete point
)0,0,0(  kji zyx as shown in Fig. 3.1, a closed sur-
face S is defined by




01,,
3zyxS , which divides
Planck cube in two parts 1
V and 2 V. In the inner region
1
V,



 01,, 1
3

V
zyxS ; in the outer region 2
V,



 01,, 2
3 V
zyxS. By means of probability transportation,
the excessive events associated with probabilities in the
inner region 1
V transport to the outer region 2 V. Ac-
cording to Theorem 2.4 and lemma 2.4.2,


zyxS ,,
3
satisfies generalized unitarity.
Since



 zSySxSzyxS

1113,, and

 xS ,

 yS ,

 zS have the same type of exponential expression, ex-
ploring one,

 xS , is sufficient. (2.13) shows that, in
region [0,1],


xS is symmetry with respect to5.0

x,
to explore

 xS in the half region [0,0.5] is sufficient. In
region [0,0.5], (2.15) shows that
 1)(

 xSxS is
approximately anti-symmetry with respect to 25.0

x.
In the meantime, let’s treat it as exactly anti-symmetry
and consider the difference later.
In Scheme-2, probabilities along with events transport
back and forth to satisfy the discrete and continuous uni-
tarity requirements alternatively.
Fermions and bosons are essentially different particles
with different properties. Their probability transporta-
tions are different. It turns out that, bosons without mass

Z. Y. SHEN 1220
take the straight real path along the real axis; while Dirac
type fermions take the zigzagging path on the complex
plane.
The following rules of probability transportation are
for Dirac type fermions.
Rule 3.2: The probability transportation rules for fer-
mions are as follows.
1) Consider two points

 25.00, 11 xx , 12 5.0xx


)5.025.0( 2 x along the real x-axis, as shown in Fig.
3.4. The excessive probability
 1
1xS at 1
x trans-
ports along a set of complex lines 1

 and 2

 to 2
x
where probability having deficient
 1
2xS. The path
length is:

12
2xxl  . (3.11)
The factor 2 in (3.11) comes from:
  2120cos/1cos/1  
. (3.12)
The probability transportation makes

1
1

xS and
 1
2xS to satisfy unitarity with respect to discrete va-
riable i x. But it distorts the Gaussian probability with
respect to continuous variable
x
.
Figure 3.4. Transporting paths with the same loop lengths
and different routs on complex plane.
2) To reinstall the Gaussian probability distribution, it
transports back from 2
x to 1 xalong another set of
complex lines 1

 and 2


via another path with the
same path length

12
2xxl as shown in Fig. 3.4.
The two paths form a closed loop with loop length:

12
42 xxlL  . (3.13)
The probability following its event goes back and forth
between 1
x and 2 x around closed loops.
3) The path length of (3.11) and the loop length of
(3.13) are valid for all zigzagging paths shown in Fig.3.4.
The multi-path nature has its physics significance, which
will be discussed in later sections
The repetitive probability transportations along closed
loops cause oscillating between two points 1 x and 2
x.
In this way, the two types of local unitarity are satisfied
alternatively, and the generalized unitarity is always sat-
isfied. It provides a kinematic scenario for the oscillation.
The dynamic mechanism and driving force of the oscilla-
tion will be discussed in Scheme-3.
As mentioned in Section 2, the anti-symmetry of



 xSxS





5.0 is only an approximation. In gen-
eral, the unitarity by probability transportation is not ex-
act. The tiny difference between 
1
xS and




2
xS
provides a slight chance for probability transportation
path to go off loop. The off loop path goes to other places
with different values of 1
x and 2 x corresponding to
other particles, which provide the mechanism for interac-
tions between particles and transformation of particles.
This is the scenario of probability transportation on the
complex x
-plan associated with x-axis. The same is for
the complex y
-plane and
z
-plane associated with
y-axis and z-axis.
For Scheme-2, the three real axes in 3-dimensional
real space are extended to three complex planes with 6
independent variables instead of 3. The extended space
with three complex planes is an abstract space. For SQS
theory, the real space is 3-dimensional. The essence of
complex plane is to add the phase angle to real spatial
parameters. The physics meaning of the phase angle will
be discussed later.
Scheme 3. Unitarity in Curved 3-Dimensional
Space
According to general relativity, in 3-dimensional
curved space, the distance between point ),,( zyxP and
discrete point ),,( kjid zyxP is the geodesic length:




kjiGdG zyxzyxLPPL ,,;,,;

. (3.14)
According to (A2.2) in Appendix 2, geodesic length


dGppL ; is determined by following differential equa-
tion:
.,0
2
2
d
cb
a
bc
aPtoP
ds
dx
ds
dx
ds
xd (3.15)
In which, ab
g is the gauge tensor and a
bc
 is Chris-
toffel symbol of second type. Taking

dG PPL ;
2 to re-
place


222 )()()( kji zzyyxx  in (2.24) yields:











ijk
kjidG
xyz
zyxPzyxPL
ePS ),,();,,(
3
2

. (3.16)
As mentioned previously, at point ),,( 1111 zyxP in 1
V
shown in Fig. 3.1, there are excessive events associated
with

 01)( 1
13

V
PS; at point ),,( 2222zyxP in 2
V, there

Z. Y. SHEN 1221
are deficient events associated with
 01)( 2
23

V PS . For
scheme-3, the probability transportation from 1
P to 2 P
takes its geodesic path:
 
),,();,,(, 22221111212121 zyxPzyxPLPPL . (3.17)
To adjust gauge tensor
 zyxgab,, along the path

2121 ,PPL in curved space, the unitarity of probability
 01)(1
13V
PS at 1
P and
 01)( 2
23  V
PS at 2
P are satis-
fied. But the Gaussian probability is distorted. Then the
gained probability at2
P transports back to1 P takes the
geodesic path:
  ),,();,,(, 11112222121212 zyxPzyxPLPPL. (3.18)
It goes back to1
P to reinstall Gaussian probability.
The transportations via


2121 ,PPL and

1212 ,PPL  fin-
ish one cycle of oscillation. The process goes on and on.
In this way, the local unitarity requirement with respect
to discrete variables and continuous variables of Gaus-
sian probability are satisfied alternatively, and the gener-
alized unitarity is always satisfied. This is the scenario of
probability oscillation in 3-dimesional curved space.
Hypothesis 3.1: To adjust the gauge tensor
 zyxgab,, properly makes geodesic paths


2121 ,PPL
such that
 01)(1
13 V
PS and

 01)( 2
23  V
PS are satis-
fied. To adjust the gauge tensor
 zyxgab,, properly
makes geodesic paths

1212 ,PPL  such that the Gaussian
probability is reinstalled. The adjusted
 zyxgab ,, de-
termines the space curvature inside the Planck cube.
Explanation: According to Hypothesis 3.1, the alter-
native unitarity of Gaussian probability with respect to
discrete variables and continuous variables is not only
the driving force for probability oscillation, but also
serves as the driving force to build the curved space in-
side Planck cube. This is the expectation from SQS the-
ory.
Let’ go back to the 1-dimension case.
Definition 3.2: S-equatio n. Define the S-equation
along the x-axis as:
011)( 2
)(  



i
i
x
xx
exS

. (3.19)
Explanation: S-equation is the origin of a set of sec-
ondary S-equations serving as the backbone of SQS the-
ory. It plays a central role to determine particles parame-
ters on their models, which will be discussed in later sec-
tions.
Theorem 3.1: Along the x-axis, the 1-dimensional un-
itarity requires:
011)( 2
))(( 



i
i
x
xxx
exS

for all
x
. (3.20)
The only way to satisfy01)( xS for all
x
is that
)(x


 is a function of
x
as a running constant.
Proof: In Section 2, (2.17) show that,
1)73026452499871562.0()( 

SxSc.
For allother points in region [1,0.5],
 1 c xxS . In
order to satisfy


01


xS for all x, something in the


xS must be adjustable. There are only two constants
e and

in

 xS. In which e as a mathematical
constant does not depend on geometry, while

does.
Therefore, the only way to satisfy unitarity of

 1

xS
for all x is that )(x



is a function of
x
as a run-
ning constant. QED
Explanation: For SQS theory, Theorem 3.1 plays a
central role for the models and parameters of elementary
particles, which will be demonstrated in later sections.
In the 1-dimensional case, what does

 x

mean?
The answer is:

 x

carrying information in curved
3-dimensional space around point
x
,




x indi-
cates space having positive curvature corresponding to
attraction force;




x indicates space having nega-
tive curvature corresponding to repulsive force. The real
examples will be given later.
In Table 3.1, the values of
 x

calculated from
(3.20) are listed along with the types of space curvatures
and corresponding forces.
Table 3.1.

 x

as a function of
x
calculated from (3.20)
(i
x truncated at 1000000

).
Notes: *The precision of values for


5
102.1 x 
is limited by 16-
digit numerical calculation. The lower limits are listed.
The attraction force is the ordinary gravitational force.
The repulsive force means that, in the vicinity of discrete
point gravity reverses its direction. This is one of predic-
tions provided by SQS theory, which is important in
many senses. For one, the repulsive force prevents form-
ing singularity, which solves a serious problem for gen-
eral relativity. For another, without repulsive force to
balance the attraction force, space cannot be stable. The
others will be given later.
Theorem 3.2: At discrete points i
xx , the unitarity
equation of (3.20) requires:



i
x 
, for





,,2,1,0,1,2,,
i
x. (3.21)

Z. Y. SHEN 1222
Proof: Consider the opposite. If

i
x 
is not infinity,
When the summation index 
i
x,



 
i
ii
x
xxx
iexS 2
))((
)(

.
Equation (3.20) cannot be satisfied. The opposite, i.e.
 
i
x 
must be true. QED
Theorem 3.2 is a mathematic theorem with physics
significance, which will be presented later.
For Scheme-2, probability oscillation is to satisfy al-
ternative unitarity, which does not provide the dynamic
mechanism and the driving force. For scheme-3, the re-
pulsive and attraction forces provide the dynamic me-
chanism and the driving force for oscillation. At 1
xx

where 1)( 1x

, the repulsive force pushes the event
associated with its probability towards 2
x. When it ar-
rived 2
xx  where 1)(2x

, the attractive force pulls
it back to 1
x. In this way, the oscillation continues. The
dynamic scenario provides the mechanism of oscillation,
which is originated from space curvature.
As mentioned in Scheme-2, the approximation nature
of anti-symmetry of (2.15) provides a slight chance for
transportation off loop representing interactions, which is
also valid for Scheme-3.
For Scheme-3, the curvature patterns make the Planck
scale grainy structure.
As a summary, Table 3.2 shows a brief comparison of
three schemes.
Table 3.2. Summary of the features for three schemes.
The three schemes are three manifestos of the vacuum
state. Scheme-1 corresponds to the quantum mechanics
vacuum state. Schemes-2 and Scheme-3 are SQS vacuum
states at a level deeper than quantum mechanics.
The probability oscillation in Scheme-2 is the same as
in Scheme-3. It implies that Scheme-2 is equivalent to
Scheme-3. Moreover, in Scheme-2, three complex planes
have 6 independent real variables; in Scheme-3, the
symmetrical 33 gauge matrix of ab
g also has 6 in-
dependent components. The correlation indicates that,
the complex planes of Scheme-2 are closely linked to
curved space of Scheme-3. It confirms that, the three
complex planes associated with three real axes are some
type of abstract expression of the curved 3-dimensional
real space. For SQS theory, there is no additional dimen-
sion or dimensions beyond the real 3-dimensional space
in existence.
In reference [2], Penrose demonstrated the correlation
between Riemann surface and the topological mani-
fold—torus. According to Penrose, 0

, 1

, 2

of (3.7)
are three branch points of the complex function 2/13 )1( z
on the Riemann surface:
1
0
 z, 3
2
1


i
ez ,3
4
2
2


i
ez  .
(3.22)
As shown in Figure 3.5(a), The Riemann surface for
2/13 )1( z has branch points of order 2 at 1,

, 2

and another one at

. Penrose showed that, for Rie-
mann surface’s two sheets each with two glued slits, one
from 1 to

and the other from

to 2

, these are
two topological cylindrical surfaces glued correspond-
ingly giving a torus as shown in Figure 3.5(b). On the
torus surface, there are four tiny holes 1
h, 
h,

h, 2

h
representing 1,

,

, 2

on the Riemann surface,
respectively. The four tiny holes on torus have important
physics significance, which well be discussed in later
sections.
Figure 3.5. (a): Four branch points and two glued cuts on
two sheets of Riemann surface; (b): Four tiny holes on torus
surface.
For SQS theory, the correspondence of Riemann sur-
face and torus is very important. It plays a pivotal rule
for constructing the topological models for quarks, lep-
tons, and bosons with mass and much more, which will
be discussed in later sections.
Section 4. Random Walk Theorem and
Converting Rules
Random walk process is based on stochastic nature of
space. It plays an important role for SQS theory. In this
section, the Random Walk Theorem is proved and con-
verting rules are introduced serving as the key to solve
many hierarchy problems.

Z. Y. SHEN 1223
Definition 4.1: Short Path and Long Path.
In 3-dimensional space, there are two types of paths
between two discrete points. The “short path”
L
from
point ),,( kji zyx to point ),,( '''kji zyx is defined as the
straight distance between them.
  
2
'
2
'
2
'kkjjiizzyyxxL . (4.1a)
The “long path”
L
ˆ from point ),,( kji zyx to point
),,( ''' kji zyx is defined as step-by-step zigzagging path in
lattice space with Planck length P
L as step length
Pi Ll.
P
N
iiNLlL  1
ˆ, nmlN 

. (4.1b)
The random walk from point ),,( kji zyx to point
),,( '''kjizyx takes l, m, n steps along
x
,
y

, z

directions, respectively.
Theorem 4.1: Random Walk Theorem. Short path
L
and long path L
ˆ are correlated by the random walk
formula:
2
ˆ
L
L
; or LL ˆ
. (4.2)
L
andL
ˆ are normalized with respect to Planck
lengt P
L, both are numbers.
Proof: According to (2.8), the probability from point
),,( kjizyx to point ),,('''kji zyx is:

   
2
2
'
2
'
2
'
''',,;,, L
zzyyxx
kjikji eezyxzyxp kkjjii



 . (4.3)
Take a random walk from


kji zyx ,, to


''' ,,kjizyx
with l, m, nsteps along
x

, y, z directions,
respectively. The probability of reaching the destination
is:


 L
lmn
nml
kjikji eeeeezyxzyxp ˆ
111
'''
222
,,;,,
  
nmlL 
ˆ. (4.4)
Combining (4.3) and (4.4) yields 2
ˆ
L
L
. QED
Obviously, Random Walk Theorem is based on Gaus-
sian Probability Postulation introduced in Section 2. As a
precondition, the standard deviation

of
3-dimensional Gaussian probability must take the values
to make the factor in front of exponential term equal to 1.
Otherwise, Random Walk Theorem does not hold. It
means that, the only parameter

in the first fundamen-
tal postulation of SQS theory is determined.
Random Walk Theorem provides the foundation for
conversions, which are governed by a set of converting
rules. Physics quantities can be converted by applying
these converting rules, which serve as the way to dealing
with hierarchy problems.
Definition 4.2: The converting factor for short path
and long path is defined as:
P
LLN /. (4.5)
Lemma 4.1:
L
, L
ˆ and
N
are related as:
P
LNNLL 2
ˆ . (4.6)
Proof: According to Theorem 4.1, the lengths
L
and
L
ˆ in (4.2) are normalized with respect to P
L. Let P L
appears in (4.2):
PPPPPL
L
N
L
L
L
L
L
L
L
L


















2
ˆ. (4.7)
Multiplying P
L to both sides of (4.7) yields:
NLL
ˆ. (4.8a)
According to (4.5), substituting P
NLL  into (4.8a)
yields:
P
LNL 2
ˆ. (4.8b)
(4.8a) plus (4.8b) is (4.6). QED
The basic unit of length in Theorem 4.1 and Lemma
4.1 as well as the step length of random walk is P
L,
which indicate the importance of Planck length.
According to SQS theory, physics quantities at differ-
ent scales have different values determined by converting
factors, which are governed by converting rules origi-
nated from Random Walk Theorem.
Definition 4.3: The conversion factors for general
purpose are defined as follows.
1) For bosons without mass:
P
LN /


. (4.9)

is the wavelength of the boson.
2) For particle s with mass:
PC LN/
 . (4.10)
C

is the Compton wavelength of the particle:
M
c
h
C

. (4.11)
M
is the mass of that particle and c is the speed of
light in vacuum.
Conversion rules for general purpose are given as fol-
lows.
1) For length:
P
LNNLL2
ˆ . (4.12)
L
ˆ, L, and P
L are long path, short path, and Planck
length, respectively.
2) For time interval:
P
tNNtt 2
ˆ. (4.13)
t
ˆ, t, and P
t are long path time interval, short path
time interval, and Planck time, respectively.
3) For energy and mass:
2
//
ˆNENEE P
 . (4.14)
2
//
ˆNMNMM P
 . (4.15)
E
ˆ,
E
, and P
E are long path energy, short path energy,

Z. Y. SHEN 1224
and Planck scale energy, respectively.
M
ˆ,
M
, and P
M
are long path mass, short path mass, and Planck mass,
respectively.
Take the ratio of electrostatic force to gravitational
force between two electrons as an example to show how
converting rules work.
According to Coulomb’s law, the electrostatic force
between two electrons separated by a distance
r
is:
2
0
2
4r
e
fE

. (4.16)
In which, e is the electrical charge of electron, 0

is permittivity of free space.
According to Newton’s gravity law, the gravitational
force between two electrons separated by a distance
r
is:
2
2
r
M
Gf e
G. (4.17)
In which, Gis Newtonian gravitational constant, e
M
is electron mass.
According to (4.16) and (4.17), the ratio of electro-
static force to gravitational force between two electrons
is:
2
0
2
/4eG
E
GE GM
e
f
f
R

 (4.18)
According to (4.15) and (2.1d):
ePe NMM/, (4.15)
G
hc
MP
 2
, or 2
2
P
M
hc
G

. (2.1d)
e
N is the converting factor for electron. P
M is Planck
mass. Substituting (4.15) and (2.1d) into (4.18) yields:
2
2
2
0
2
22
0
2
/424
1
4e
e
P
eG
E
GE N
M
M
hc
e
GM
e
f
f
R










 (4.19)
In (4.19),

is the fine structure constant. At elec-
tron mass scale:
05999084.137
1
2
)(
0
2
hc
e
Me


. (4.20)
In which, )51(035999084.137/1

is cited from
2010-PDG (p.126) according to references [3] and [4].
Electron converting factor is:
23
10501197.1 
e
P
eM
M
N. (4.21)
Substituting (4.20) and (4.21) into (4.19) yields:
42
/10164905.4  GE
R. (4.22)
GE
R/ is one of many hierarchy problems in physics.
By applying conversion rules not only solves the hierar-
chy problem but also reveals its origin and mechanism.
On the right side of (4.19), the first factor is electrically
originated:
4
21084811744.1
4




. (4.23)
The second factor 2
e
N is mass originated:

46
2
23
2
210253593.210501197.1 







e
P
eM
M
N. (4.24)
According to Random Walk Theorem and Lemma 4.1,
converting factor e
N is equal to the ratio of long path
over short path. Keep this in mind, the 46210~
e
N factor
can be explained naturally. For a pair of electron, the
electrostatic force is inversely proportion to the square of
the straight distance
r
(short path) between them;
while the gravitational force actually is inversely propor-
tional to the square of the zigzagging long path rNr e
 ˆ
between them. In terms of force mediators, photon takes
the short path, while graviton takes the long path. Ac-
cording to SQS theory, this is the mechanism of tremen-
dous strength difference between electrostatic and gravi-
tational forces, which is originated from random walk.
It is the first time to show that Random Walk Theorem
and the long path versus short path as well as the con-
versing rules are real and useful. There are more exam-
ples along this line in later sections.
Once the mechanism is revealed, there are more in-
sights to come.
Rule 4.1: Electron’s converting factor )(lNe is a run-
ning constant as a function of length scale l(in this case,
l is the distance between two electrons) with different
behaviors in two ranges.
Range-I: For the length scale Ce
l,  :
.)(,const
LM
M
NlN
P
Ce
e
P
ee 

for Ce
l,  . (4.25a)
Range-II: For the length scale eCP lLl


71
min :
PCee
P
Ce
ee L
ll
M
Ml
NlN 







,,
)(

for eCPlLl




71
min . (4.25b)
In (4.25), Ce,

is the Compton wavelength of electron,
eCP lLl
  71
min is the lower limit of l in Range-II,
which will be given in Section 16.
Explanation: The reason for .)(,constlNCee



in
Range-I is obvious. Otherwise, if)(,Cee lN

 is not a
constant, then electron mass in macroscopic scale varies
with distance, which is obviously not true. Range-II
needs some explanation. According to Lemma 4.1,
P
LLN /

, in this case Pe LlN /, (4.25b) is explained.
Figure 4.1 shows the variation of )(lNe, the )(lNe
versus l profile is made of two straight lines. In Range-
I, )(lNe is a flat straight line with zero slop. In Range-II,
)( lNe is a straight line with slop 1/1
P
L. Two strai-

Z. Y. SHEN 1225
ght lines intersect at Ce
l,

. It shows a peculiar behav-
ior of )(lNe. Most physics running constants vary as-
ymptotically toward end. This one is different. The
straight line with slop 1/1
P
L on left suddenly stops at
Ce
l,

 and changes course to the flat straight line on
right. At two straight lines’ intersecting point, the first
derivative is not continuous. The mechanism of such
peculiar behavior will be explained in Section 16.
There is another factor)4/( 2

in GE
R/, in which
)( M


 is a running constant. The variation of
)(M

makes )( /lRGE different from
 lNe
2. It rounds
the corner of )(
/lR GE versus l curve at intersecting
point show in Figure 4.1.
The )(
/lR GE for two electrons given by (4.19) is just
an example. It can be extended to other charged particles.
For instance, two protons separated by a distance
r
, the
ratio of electrostatic force to gravitational force is:
2
2
2
0
2
22
0
2
/4
)(
24
1
4prot
prot
prot
prot
prot
GE N
M
N
hc
e
GM
e
R



 ;
protPprotMMN /. (4.26)
In which, prot
M, prot N, and )( prot M

are mass, con-
verting factor and fine structure constant at proton energy
scale, respectively.
Figure 4.1. )(lNe and )(
/lRGE versus distance l curves.
(Scales are not in proportion.)
Substituting data into (4.26) and ignoring the differ-
ence between )( prot
M 
and )( e
M

of (4.20) yields
the ratio for protons:
36
/10235343.1 
prot
GE
R. (4.27)
The conversion rules introduced in this section are
subject to more verifications. Other applications of con-
verting rules will be presented in later sections.
Section 5. Apply to Quantum Mechanics and
Special Relativity
In this section, the converting rules introduced in Section
4 are applied to some examples in quantum mechanics
and special relativity.
According to Feynman path integrals theory [5], the
state


22 ;tx

at point 2
x and time 2 t is related to
the initial state


11 ;tp

at point 1
x and time 1
t as:
 


2
1
1122;1,2;
x
x
dltxKtx 

, 12tt ; (5.1a)

 



 allpaths
xx
allpaths
xx
dttxxL
h
i
txS
h
it
t
AeAeK
2121
2
1
;,
1,2
. (5.1b)
In which A is a constant,

)(txS is action,

 txxL;,  is
Lagrangian, 1
x, 2 x,
x
and )(tx are 3-dementional
coordinates with simplified notations. The integral in
(5.1a) and summation in (5.1b) include “all possible
paths” from point 1
x to point 2
x.
Assuming the particle is a photon with visible lights
wavelength of m
7
10~  , it travels with speed c from
1
x to 2 x separated by distancemL 1. The photon
traveling through mL 1

once takes time
scLt9 103.3/   .
The obvious question is: How does photon have time to
travel so many times through “all possible paths” be-
tween 1
x and 2 x? Theorem 4.1 and Lemma 4.1 pro-
vide the answer. According to (4.9), the converting factor
for photon with wavelength m
7
10~ 
is:
28
10~/ P LN  . (4.9)
According to (4.12), the photon’s long path wave-
length is:
lightyearsmN 52128 10~10~10~
ˆ
 . (5.2)
The 28
10~N tremendous difference between long
path wavelength

ˆ and wave length

is originated
from the Random Walk Theorem. From SQS theory
viewpoint, the “all possible paths” in (5.1) of Feynman
path integrals theory are covered by photon’s long path
wavelength m
2128 10~10~
ˆ

. It is sufficient for the
photon to cover through “all possible paths” via many
billions of billions different routes from 1
x to 2
x. This
is the explanation of Feynman path integrals theory for
SQS theory.
But there is a question. If the photon with wavelength
m
7
10~ 
really travels through the long path
m
2128 10~10~
ˆ

, for a stationary observer, it only
takes time interval of st 9 103.3   . The question is:
What is photon speed seen by a stationary observer? If
the stationary observer sees the long path, the speed is
indeed superluminal. For example, photons with wave-
length m
7
10~ 
, it travel along the long path with su-

Z. Y. SHEN 1226
perluminal speed:
smsmcc
L
Ncv
P
/103/1031010~ 3682828 









 .
(5.3)
Question: Does the stationary observer sees the tre-
mendous superluminal speed? According to SQS theory,
the wave pattern of a particle such as photon is estab-
lished step by step with step length P
L during its zig-
zagging long path journey. The short path is the folded
version of the long path. For an ordinary photon, the
folded long path is hidden in its wave pattern. The sta-
tionary observer sees neither the hidden long path nor the
superluminal speed. In case the photon’s long path shows
up from hiding that is another story. It will be discussed
later.
The superluminal speed cNcv 


enhances the
explanation of Feynman path integrals theory for SQS
theory.
The explanation for Feynman’s path can be used to
explain other similar quantum phenomena such as the
double-slot experiment for a single particle and quantum
entanglements.
Take the double slots experiment for a single photon
as an example. Experiments have proved that, when the
light source emits one phone at a time, the interference
pattern still shows up. As mentioned previously, a photon
with wavelength m
7
10~  
has its long path wave-
length m
212810~10~
ˆ
 and superluminal speed for
vacuons (in Section 18, vacuon is defined as a geometri-
cal point in space) to travel along the long path, which
provide the condition to let the vacuons pass through two
slits enormous times to form the interference pattern.
Figure 5.1 shows the double-slit interference pattern for a
single photon.
Figure 5.1. The double-slit interference pattern for a single
photon.
The single photon’s long path builds the wave pattern
step by step in the space between the plate with double-
slit and the screen. The two waves come from two slits to
form the interferential pattern on the screen just like the
regular double-slit interference pattern. The single pho-
ton strikes on the screen at a location according to prob-
ability determined by the interference wave pattern’s
magnitude square. When more photons strike on screen,
the interference pattern gradually shows up. The long
path provides the condition for a single photon’s wave
pattern to interfere with itself. It is possible because of
the long path’s extremely long length and vacuons’ su-
perluminal speed, which allow the vacuons pass through
two slits so many times. In this sense, a single photon
does pass through two slits.
According to the converting factor
PPfLcLN // 


based on the Random Walk Theorem, as photon’s fre-
quency f and energy increase, P
fLcN / decreases.
The difference between long path and short path de-
creases accordingly. As a result, the wave pattern be-
comes coarser and more random. In other words, the
wave-particle duality is a changing scenario with energy,
the particle nature is enhanced and the wave nature is
diluted with increasing energy.
The tremendous difference between short path and
long path is related to special relativity. A stationary ob-
server sees the photon having wavelength

. The pho-
ton travels along its short path with a speed v less than
c and very close to c, according to Lorentz transfor-
mation:
2
)/(1
ˆcv

,

2
/1
1
ˆ
cv



. (5.4)
From SQS theory perspective,

and

ˆ are pho-
ton’s short path wavelength and long path wavelength
originated from Random Walk Theorem. From special
relativity perspective,

versus

ˆ is the result of Lo-
rentz transformation. These two apparently different
scenarios are two sides of the same coin. The key con-
cept is to recognize photon traveling along its short path
with a speed
v
less than c and very close to c. It is
a deviation from special relativity.
Substituting
 /
ˆ
N into (5.4) yields:







22 1
1
/1
1
cv
N. (5.5)

and

are the standard notations in special relativity.
As shown by (5.5), converting factor
N
is closely re-
lated to

and

of special relativity.
Substituting photons’ converting factor P
LN/


from (4.9) into (5.5) yields:

2
/1
1
cv
fL
c
LPP


. (5.6)
Solving (5.6) for photon’s speed
v
as a function of
frequencyfor wavelength

yields:

Z. Y. SHEN 1227

2
/1)(cfLcfv P  , (5.7a)

2
/1)( 
P
Lcv  . (5.7b)
Photon’s speed varying with its frequency or wave-
length means dispersion. (5.7) is the dispersion equation
of photon. The speed of photon decreases with increas-
ing frequency. The constant c is not the universal
speed of photons, instead, it is the speed limit of photon
with frequency approaching zero. This is a modification
of special relativity proposed by SQS theory.
According to the Gaussian Probability Postulation,
space has periodic structure with Planck length P
L as
spatial period. It is well known that, wave traveling in
periodic structure has (5.7) type dispersion. Look at it the
other way: Dispersion is caused by the fact that photon
interacts with space. For SQS theory, space is a physics
substance.
The dispersion effect of visible lights is extremely
small. It is negligible in most cases. According to (5.7),
the speed
v
of a photon with wavelength m
7
10~  
deviates from c in the order of 56
10~ . On the other
hand, for

- ray with extremely high energy, the disper-
sion effect is detectable. It serves as a possible way for
verification.
On May 9th, 2009, NAS A’s Fermi Gamma-Ray Space
Telescope recorded a

-ray burst from source GRB-
090510 [6-8]. The observed data are given as follows.
Low energy

-ray
Energy: JeVE 154
110602.1101 
,
Wavelength: m
10
11024.1 
 
.
High energy

-ray
Energy: JeVE910
210967.4101.3 
 ,
Wavelength m
17
210999.3 
 
.
Distance to

-ray source:
mlyLO259 10906.6103.7  .
Observed time delay (after CBM trigger) for the high
energy

-ray: stO829.0 .
According to (5.7b), the SQS theoretical value for time
delay is:
 






































2
1
2
2
2
2
2
1
2
2
2
1
21
21
12 2
/1/1
/1/1
11



PP
PP
PP
TLL
c
L
LL
LL
c
L
vv
vv
L
vv
Lt .
(5.8)
The approximation is due to1/,1/ 21




PP LL .
Substituting observed data and O
LL  into (5.8) yields:
.10881.1
2
20
2
1
2
2
1s
LL
c
L
tPPO
T




























(5.9a)
Substituting observed data and

OPOOLLNLLL /
ˆ
 
into (5.8) yields:
s
LL
c
LL
N
L
N
c
L
tPP
O PP O
T047.0
2212
2
1
1
2
2
22  
































. (5.9b)
O
L
ˆ is the long path of O
L, P
LN/
11

 and P
LN /
22
 
are converting factors for 1

and 2

, respectively. The
dispersion equation corresponding to (5.9b) according to
some other theories is：



 cfLcLcfvPP/1/1  
. (5.10)
(5.7) and (5.10) can be expressed as one equation：
 
n
P
n
PcfLcLcfv /1/1 

;
1

n, 2

n. (5.11)
In which, 1

n is for (5.10) and 2n is for (5.7).
Superficially, the observed data seem to favor the re-
sult of (5.9b) and 1

n for (5.11). Actually it is not true.
After extensive analysis, the authors of [6-8] concluded:
“… even our most conservative limit greatly reduces the
parameter space for 1

n models. … makes such theo-
ries highly implausible (models with 1n are not sig-
nificantly constrained by our results).”
The observation data from GRB090510 neither con-
firm nor reject dispersion Equation (5.7). In fact, for the
distance of lyLO9
103.7~ , to verify (5.7) directly re-
quires the high energy

-ray burst with energy level
around eVE 20
210~, which is a very rare event.
Quantum mechanics supports non-locality. For a pair
of entangled photons separated by an extremely long
distance, their quantum states keep coherent. Measure
one photon’s polarization, the other one “instantane-
ously” change its polarization accordingly. Einstein
called it: “Spooky action at a distance.”
SQS theory does not support non-locality. For a pair
of entangled photons, SQS theory provides the following
understanding and explanation.
1) There is a real physical link between entangled
photons. They are linked by the long path. In case of en-
tanglement, the long path shows up from hiding with
energy extracting from entangled photons.
2) The transmission of information and interaction
between two entangled photons does not occur instant-
neously, instead, it takes time. Even though the time in-
terval is extremely short, but it is not zero. For ordinary
photons, the long path is folded to form photon’s wave
pattern, the stationary observer only sees the short path
with photon speed of c
v

given by (5.7). For a pair of
entangled photons, the long path shows up serving as the
link. A stationary observer now sees the long path and
superluminal speed. The speed of signal transmitting
along the long path between two entangled photons is
cc
L
NcNvv
P









ˆ. (5.12)

Z. Y. SHEN 1228
For visible light with wavelength m
7
10~  
, accord-
ing to (5.12), cNcv 28
10~
ˆ. This is why territorial
entanglement experimenters found that the interaction
seems instantaneous. Actually it is not. The interaction
between entangled photons is carried by a signal trans-
mitting alone the long path with superluminal speed of
(5.12). Recently, Salart et al report their testing results:
the speed exceeds c
4
10 [9]. Indeed, it is superluminal.
3) In the entanglement system, two entangled photons
and the link connecting them have the same wavelength
to keep the system coherent.
Entanglement provides a rare opportunity to peep at
the long path. It is worthwhile to take a close look.
According to (4.6) of Lemma 4.1 based on the Ran-
dom Walk Theorem, the relations of photon wave-
length

, long path wavelength

ˆ, converting factor
N
and Planck length P
L are:
P
NL

, P
LN/

, (5.13a)
PP LLNN /
ˆ22
  . (5.13b)
The relations given by (5.13) serve as the guideline to
deal with photons entanglement.
Postulation 5.1: For a pair of entangled photons, the
entanglement process must satisfy energy conservation
law and (5.13) relations. Under these conditions, a pair of
entangled photons’ original wavelength 0

changes to
0


 and the original long path wavelength 0
ˆ
 chan-
ges to 0
ˆˆ
  according to the following formulas:

link
NL 2, (5.14a)

link
NLd  2/,

/dNlink , (5.14b)
P
LN /

, (5.14c)
PP LLNN /
ˆ22
  . (5.14d)
Explanation: The distance between two entangled
photons is d. The link has two tracks, one track for one
direction and the other for opposite direction. The total
length of two tracks is dL 2. According to SQS theory,
photon’s geometrical model is a closed loop with loop
length of P
L2. In the entanglement system, two entan-
gled photons and the link connecting them share a com-
mon loop. The link’s double-track structure is necessary
to close the loop. P
LN /

 is converting factor for
photons with wavelength

,

/dNlink  is the number
of wavelengths in one track. Conservation of energy re-
quires total energy for entanglement system kept con-
stant:

hc
NN
N hchc
link
link 222
0
, (5.15a)

1
/1
111
0link
NN
 . (5.15b)
In which, h is Planck constant. The term on (5.15a)
left side is the energy of two photons with original wa-
velength 0

. On (5.15a) right side, the first term is the
energy of two photons with elongated wavelength
0


, the second term is the energy extracted from two
photons to build the link. Substituting link
N and
N
from (5.14) into (5.15b) yields the formula to determine
the elongated wavelength

:
d
dLP

 

ˆ
1
1
1
1
1
12
0


 . (5.16)
A 16-digit numerical calculation is used to solve (5.16)
for

as a function of d for mmm 3 0101 
 
. The
resultsare listed in Table 5.1.
Table 5.1. Entangled photons wavelength )( d

as a function
of d for m
3
010
 
.
The data listed in Table 5.1 show some interesting
features.
1) Maximum entanglement distance: Solve (5.16)
for d:














P
L
d2
0
0
0
0
2
ˆ
2






. (5.17)
It shows that, the distance d between two entangled
photons increases with increasing wavelength

. At the
wavelength 0
2



, the distance d. It seems
no limitation for d. But that is not the case. Because
another requirement is involved: The link as an inte-
grated part of entanglement system must have the same
wavelength of two photons. Otherwise, there is no co-
herency. In this case, 0
2   corresponds to
2/
0
hfhf

. A half (2/12/11 

) of each photon’s
energy is extracted out to build the link. According to
SQS theory, photon’s model is a closed loop with loop
length of P
L2, which corresponds to two wavelengths
and two long path wavelengths inside the photon to build
its wave pattern. The half energy extracted from two en-
tangled photons is only sufficient to provide two wave-
lengths and two long path wavelengths for the link. Un-
der such circumstance, the only way to build the link
with infinite length is to infinitively elongate the long

Z. Y. SHEN 1229
path wavelength as well as the wavelength in the link,
which make them very different from two entangled
photons’. It is prohibited by violating coherency re-
quirement. So the entanglement distance ddoes have its
limit. In fact, only one case satisfies both requirements:
energy conservation and quantum coherency. The unique
case is 
ˆ
 d. According to (5.16),

ˆ
d yields
000 2
3
)
2
1
1(5.1  
corresponding to
0
3
2hfhf .
A third )3/13/21(  of each photon’s energy is
extracted out to build the link. The total energy is just
right to make the original two wavelengths and two long
path wavelengths in each photon becoming three wave-
lengths and three long path wavelengths for the entan-
gled system, in which two are kept for each photon itself
and one extracted out to build one track with length

ˆ
d. In this way, both requirements are satisfied and
self-consistent. Hence, there is a maximum distance
max
dbetween two photons to keep entangled, which is
determined by (5.17) with
005.1
2
3
  :
000
2
2
5.1
0
0
max ˆ
25.2
4
1
2
ˆ
2
3
ˆˆ
2
0






































P
L
d.
(5.18)
When max
dd , the link breaks down and two en-
tangled photons are automatically de-coherent even
without any external interference. The data for
m
3
010
 
are listed in the bottom row of Table 5.1.
2) Energy balance: At the maximum distance
max
dd,
2
3
2
1
1/0 
corresponds to
0
3
2hfhf.
It indicates that, one third of each photon’s energy is
extracted out to build the link. Because the double-track
link extracts energy from tow photons,
003
2
)
3
2
1(2 hfhf ,
the link has the same energy of each photon’s energy.
The link acts like another photon with the some energy
and the same wavelength of each entangled photon. In
other words, at the maximum entanglement distance
max
dd, the entanglement system is seemingly made
of three photons, in which two are entangled real photons
and the third one makes the link to connect them. It
serves as evidence that, the link is a physics substance
with energy. At shorter distance max
dd , the extracted
energy gradually increases to build the link and to push
the link for expansion. At distance beyond maximum
distance, max
dd , the entanglement system breaks
automatically, because it lacks sufficient energy to main-
tain the over expanded link. In this way, both require-
ments are satisfied and everything is consistent. The key
is to recognize the long path serving as the physics link
for entanglement.
3) Entanglement red shift: The wavelength )(d

increases with increasing distance d. The red shift is
caused by the fact that, a portion of the entangled pho-
tons energy is extracted out to build the physics link. It is
the energy conservation law in action. According to
(5.16), the red shift continuously increases with increas-
ing distance. The maximum red shifted wavelength at
max
dd

is.
00max5.1)
2
1
1(  . (5.19)
The entanglement red shift happens gradually. For a
pair of photons separated by distance much shorter than
the maximum distance, the tiny red shift is very difficult
to detect. As listed in Table 5.1, for a pair of photons
with wavelength m
3
10   
at distance 6
10dm,
the relative red shift is only 16
10~ . For entangled pho-
tons with visible light wavelength m
7
10~  
, the red
shift is many orders of magnitude less than 16
10~ .
This is why entanglement experiments with limited dis-
tance haven’t found the red shift effect yet. But it is out
there. Otherwise, the physics link energy has nowhere to
come from.
4) De-coherent blue shift: When a pair of entangled
photons is de-coherent, the outcomes depend on the
de-coherence location. If the location is right at the mid-
dle 2/d, the physics link is broken evenly and each
photon gets back equal share of the link energy to resume
original wavelength corresponding to a blue shift. Accord-
ing to (5.16), the two de-coherent photon’s wavelength is
shortened from

to



0 causing the blue shift.
d
d
dL
dL
P
P






ˆ
2
ˆ
1
2
1
2
2
0





. (5.20)
At 
ˆ
max  dd , the blue shift is:
3
2
12
11
ˆ
0









 


d. (5.21)
In terms of frequency, the blue shift is:
5.1
2
3
ˆ
0
ˆ
0



















d
d
f
f. (5.22)
If the de-coherent location is at the close vicinity of
one photon, this one does not gain energy to change its

Z. Y. SHEN 1230
wavelength and shows no blue shift. The other one gets
all energy of the link and has the maximum de-coherent
blue shift to the wavelength 0
' 
shorter than the original
wavelength 0

. According to energy balance of (5.15):












dL
hchchc
NN
N
hchc
P
link
link
2
01
22
',
or
dL
dL
dL P
P
P
2
2
2
01
3
1
2
1
' 

 





 .
For photon at distance d from de-coherent location,
its wavelength is shortened to 0
' 
:
d
d
dL
dL
P
P






ˆ
3
ˆ
1
3
1
'
2
2
0





. (5.23)
For de-coherence at locations between 2/d and d,
the blue shift for the far away one is between the two
values given by (5.20) and (5.23). At the maximum en-
tanglement distance

ˆ
max dd, according to (5.23),
the maximum blue shift in terms of frequency is:
2
11
13
'
'
ˆ
0
max,0










 


d
f
f. (5.24)
max,0
'f is the blue shifted frequency of the photon at
the distance max
dd from the de-coherence location.
For de-coherence at locations between 2/d and d, the
blue shift is between the two values given by (5.22) and
(5.24). The de-coherent blue shift happens suddenly with
a large frequency increase, which is relatively easy to
detect, but the problem is the uncertainty of de-coherent
timing.
The above analyses show that, entangled photons are
connected by a physics link, interactions and information
between them are transmitted with superluminal speed
ccLNcNvv P)/(
ˆ

.
It is much faster than c but not infinite. From SQS the-
ory standpoint, the physics link and the non-infinite su-
perluminal speed serve as the foundation for locality.
After all, Einstein was right: No spooky action at a dis-
tance.
Conclusion 5.1: Entanglement has limited distance.
The distance between entangled particles cannot be infi-
nitely long.
Proof: Conclusion 5.1 is not based on Postulation 5.1.
It is based on basic principle. Consider the opposite. If a
pair of entangled particles is separated by infinite dis-
tance, the physics link between them must have nonzero
energy density, energy per unite length. Then the total
energy of the link equals to infinity. That is impossible,
the opposite must be true. QED
Explanation: According to Conclusion 5.1, the max-
imum entanglement distance max
d given by (5.18)
serves only as an upper limit. Whether a pair of entan-
gled photons can be separated up to max
dor not, it also
depends on other factors. For entangled photons with
very long wavelength, their quantum has very low energy.
As the link stretched very long, the energy density be-
comes lower than the vacuum quantum noise. The link
could be broken causing de-coherence with distance
shorter than the maximum distance max
d. The other
factor is external interferences causing do-coherence,
which is well known and understood.
According to SQS theory, photons travel along the
short path with speed of c
v
with dispersion given
by (5.7); the signals between entangled photons transmit
along the long path with superluminal speed
NcNvv


ˆ of (5.12). These are the conclusions derived
from converting rules introduced in Section 4. The key
concept is the long path, which is defined by (4.12) based
the converting factor and originated from the Random
Walk Theorem. If the existence of long path is confirmed,
so are these conclusions as well as its foundation.
If photon’s long-path is confirmed, the non-locality of
quantum mechanics must be abandoned. Moreover, long
path is based on converting rule. If it is confirmed mean-
ing photon does have dispersion. Special relativity
should been revised as well.
The dispersion equation of (5.7) is not the final version.
In Section 26, a generalized dispersion equations will be
introduced, in which the Planck length in (5.7) is re-
placed by longer characteristic lengths. It makes easier
for experimental verifications.
In this section, special relativity is revised. For most
practical cases, the revision for photon’s speed in vac-
uum is extremely small, but its impact are huge such as
the introduction of superluminal speed
 ccLNcNvv P



/
ˆ

.
Is it inevitable? Let’s face the reality: Experiment [9]
carried out by Salart et al proved that, the speed of signal
transmitting between two entangled photons exceeds
10000c. It leaves us only two choices: One is to intro-
duce non-infinite superluminal speed as we did in this
section; the other is to accept “spooky action at a dis-
tance”. Obviously, the second choice is much harder for
physicists to swallow. Therefore, the superluminal speed
is indeed inevitable. Besides, the superluminal speed
introduced in this section is within special relativity
framework. The key concept is that, the long path and the
superluminal speed are hidden, they only show up in
very special cases such as entanglement.
The converting factor seemingly has two different
meanings: One is from random walk; the other is from

Z. Y. SHEN 1231
Lorentz transformation. Actually, they are duality. Such
duality is common in physics. One well known example
is wave-particle duality. In the meantime, the mechanism
of the random-walk versus Lorentz duality is not clear,
which is a topic for further work; and so it the mecha-
nism of the wave-particle duality. In fact, the long path
concept digs into the mechanism of wave-particle duality
down to a deeper level: The vacuons’ movement builds
the wave-pattern step by step.
Section 6: Electron.
Define the DS-function as:
     
 
1
2
1
5.011
2
122 5.0 





 





i
i
i
i
x
xx
x
xx eexSxSxDS 
.
(6.1)
According to definition,
 xDS is symmetrical with
respect to 25.0x in region

 5.0,0 :
 xDSxDS 5.0 ; 5.00  x. (6.2)
 xDS satisfies the periodic condition:
 xDxDS  5.0 . (6.3)
Fig. 6.1 shows


xDS versus
x
curve in region
 25.0,0. The other part in region  5.0,25.0 is the mir-
ror image of this part with respect to 25.0x.
-8
-6
-4
-2
0
2
4
6
8
00.025 0.050.0750.10.125 0.150.1750.20.225 0.25
x
DS(x)x10^6
Figure 6.1.
 xDS versus
x
curve in region

 25.0,0.
Definition 6.1: Define the DS-equation as a member
of the S-equation family:


01
2
122 5.0





  





i
i
i
i
x
xx
x
xxeexDS 
(6.4)
In region
 5.0,0 ,
 0xDS has two roots:
125.0
1x, 375.0 2

x.
According to (3.11), the path length of probability
transportation from 1 x to 2 x via complex x
-plane
is:

PPe LLxxl 5.0212 . (6.5)
In (6.5), P
L appears as the unit length hidden in
(3.11). The reason for the factor 2 in (6.5) has been ex-
plained mathematically in Section 3. Physically, accord-
ing to the spinor theory proposed by Pauli, electron as
Dirac type fermion has two components, which move in
the zigzagging path called “zitterbewegung” phenome-
non [10].
According to (3.13), the loop length corresponding to
path length for 1
x and 2 x is:
Pee LlL

2. (6.6)

 0

xDS means that the probabilities compensation
between excess and deficit is exact. The oscillation be-
tween 125.0
1

x and 375.0
2x does not decay,
which corresponds to a stable fermion. Electron is the
only free standing stable elementary fermion, which nei-
ther decays nor oscillates with other particles. It is the
most probable candidate for this particle.
Assuming the resonant condition for the lowest excita-
tion in a closed loop with loop length e
L is:
cM
h
LCe ˆ  
. (6.7)
In which,
M
ˆ and C

are the mass and Compton
wavelength of the particle, respectively. Substituting (6.6)
into (6.7) and solving for the mass of this particle yield:
.10367498.1
ˆ7kg
cL
h
M
P

 (6.8)
It is recognized that P
MM 
ˆ is the Planck mass.
According to 2010 PDG data, the mass of electron is:
kgMe31 10)45(10938215.9   . (6.9)
M
ˆ is 23
10~ time heavier than e
M, which is one of
the hierarchy problems in physics. It can be resolved by
applying conversion rule. According to (4.10), the con-
verting factor for electron is:
cLM
h
L
N
PeP
Ce
e ,

. (6.10)
The mass of (6.8) after conversion is:


.
/
/
ˆ
e
Pe
P
e
M
cLMh
cLh
N
M
M (6.11)
The particle is identified as electron. Of cause, this is a
trivial case, but it serves as the basic reference for non-
trivial cases given later.
The reason for miscalculating the mass with 23 10~
times discrepancy is mistakenly using Compton wave-
length C

in (6.7). In reality, the resonant condition in
Planck scale closed loop should be:



,3,2,1; mmL P

. (6.12)
PCP LN /

. (6.13)

Z. Y. SHEN 1232
N is the converting factor for that particle. PP L


is
defined as the Planck wavelength. The number m in (6.12)
is related to the spin of particle. For electron, 1

m
corresponds to spin 2/h. In general, the spin of a parti-
cle equals to 2/hm . Odd m corresponds to fermions,
and even m corresponds to bosons. m is the first numeri-
cal parameter introduced by SQS theory.
Electron as a Dirac type fermion, its trajectory has two
types of internal cyclic movements, one contributes to its
spin and the other one does not. In (6.12), the loop with
length P
mL

 is the main loop celled loop-1 and the
other loop is loop-2. The dual loop structure of electron
corresponds to two components. The dual loop structure
is not only for electron but also for other Dirac type fer-
mions, which will be discussed in later sections.
The basic parameters for electron are listed below.
Mass:
231 /)13(510998910.010)45(10938215.9 cMeVkgMe 
(6.14)
Compton wavelength:
mcMh eeC 12 1042631022.2)/(   
. (6.15)
Converting factor:
23
10501197.1/  ePe MMN . (6.16)
Loop parameters:
125.0
1x, 375.0 2x, 5.0 e
l, 1
e
L. (6.17)
At 125.0
1x, 375.0 2x, )(11)(21xSxS  , the
probability compensation is exact corresponding to elec-
tron as a stable particle. At other locations, the probabil-
ity compensation is not exact corresponding to unstable
particles.
Electron is unique. Its mass servers as basic unit used
for calculating other fermion’s mass. The general for-
mula to determine )( 12 xx  for fermion with mass
M
is:
 M
M
xx
xx
xx
xx e
e




)(4
1
)(4
)(4
)(4
12
12
12
12 . (6.18)
The reason for (6.18) is that, loop length
)(4 12 xxL 
is inversely proportional to mass.
According to (6.18), the values of 1
x and 2 x of the
fermion with mass
M
are:








 M
M xx
xe
8
25.0
2
25.0 12
1, (6.19a)
125.0xx  . (6.19b)
Along the x-axis, according to (2.19) and (2.20), the
region between two special points c
x and d
x is:
  3726973550.25001284 ,73026452499871562.0,
dc xx
(6.20)
Inside region


dc xx,, both
 1
1xS and

 1
2

xS ,
probability transportation for unitarity does not make
sense.
Rule 6.1: The special points c
x sets a mass upper
limit Max
M for standalone fermions:
2
/97323432.4
25.0
125.0 cGeVM
x
Me
c
Max 

. (6.21)
A fermion with mass heavier than Max
M cannot stand
alone. It must associate with an anti-fermion as compan-
ion to form a boson state.
Rule 6.2: A particle with 1
x and 2 x inside region


dc xx, belong to gauge bosons with spin .
Rule 6.3: The region


cc xx','

belongs to scalar bo-
sons with spin 0. Point c
x' is defined as:
5
10552843726973.1
73026452499871562.025.025.0'







ccxx . (6.22)
The meaning and the effectiveness of these rules will
be given in later sections.
This section serves as the introduction of electron for
SQS theory. It will be followed by later sections in much
more details.
Section 7: DS-Function on k-Plane as
Particles Spectrum
In Section 6, the)(xDS as a function of x is defined as:

 
1
2
122 5.0





 





i
i
i
i
x
xx
x
xx eexDS 
. (6.1)
Taking Fourier transformation to convert )(xDS into
)(kDSk on complex k-plane yields:
 



  .
4
1
1
2
1
2
1
2
1
5.0
4
5.0
2
22
keee
dxeeedxexDSkDS
j
kjiijk
k
ikx
j
xjxjikx
k
































(7.1)
Summation index i
x in (6.1) is replaced by index j in
(7.1) for simplicity. In (7.1), k is the wave-number on
complex k-plane. Normalize k with respect to

2 as:


2/kk

. (7.2)
In terms of k, the )(kDSk function (7.1) becomes:


 
.
4
15.022
2keeekDS
j
kjikjik
k









 


 (7.3)

 kDSk and

 kDSk are the DS-functions on the
complex k-plane and k-plane, respectively. Because
x
and i
x in (6.1) are normalized with respect to Planck
length P
L as numbers. In the Fourier transformation
process, xk and kxi are also numbers, so k and k

Z. Y. SHEN 1233
are normalized with respect to P
L/1 .
Definition 7.1: The real part r
k and imaginary part
i
k of k are related to particle’s complex mass as:
e
i
e
irM
M
i
M
M
kikk  . (7.4)
M
and i
Mare the real mass and the imaginary mass
of the particle, respectively. e
M is electron’s mass
serving as the basic mass unit.
Explanation: Definition 7.1 is based on the concept
that, k-plane serves as the spectrum of particles. Ac-
cording to (7.4), particle’s mass
M
and its decay time t
are:
er MkM, (7.5a)
i
eC
kc
t

. (7.5b)
In which, e
M and eC 
are the mass and Compton
wavelength of electron, respectively.
Formula (7.5a) is derived from real part of (7.4).
Formula (7.5b) is derived from imaginary part of (7.4)
as:
cthc
hf
hc
E
h
cM
M
M
keCieCieCieC
e
i
i





.
(7.6)
i
E andi
f are imaginary part of energy and frequency of
the particle, respectively.
Numerical calculations of
 kDSk found the follow-
ing results.
1)
 01 kDSk, 1k is a root of

 0kDSk.
According to (7.5a) and (7.5b):
e
MM , t. (7.7)
 0kDSk at 1k corresponds to electron as a
fermion.
2)
  0kDSk, 0k is a pole of

 kDSk.
According to (7.5a) and (7.5b):
0M, t. (7.8)
 0kDSk, 0k corresponds to photon as a
boson.
Rule 7.1: In general, the local minimum of

 kDSk
corresponds to a fermion, while the local maximum of
 kDSk corresponds to a boson. At the local minimum
or local maximum of
 kDSk, k with real value corres-
ponds to a stable particle, k with complex value corre-
sponds to an unstable particle.
Explanation: Rule 7.1 is the generalization of
 01 kDSk for electron as a fermion and
  0kDSk for photon as a boson.
Consider the factor
2
4
1k
e



in (7.3):
iririr kkikkkik
keeee



2)()( 222
2
4
1
4
1
4
1
 (7.9)
In (7.9), for 4
r
k and
rikk

,
 22
10
4
122 
 
ir kk
e


,
which drastically suppresses the magnitudes of local
minimum and local maximum of
 kDSk and makes
numerical calculation difficult. In (7.3), the

-function

 k

does not contribute except for
0

k. Let’s disregard the factor )4/(
2

 k
e and
drop the


k

term to define the simplified version
of

 kDSk as:








j
kjikji
keekDS 5.022

. (7.10)
In terms of the k value at local minimum or local
maximum of

 kDSk, the error caused by simplifica-
tion is evaluated in Appendix 3, which is negligible in
most cases.
The simplified

 kDSk of (7.10) is taken for technical
reasons. It does not mean ignoring the importance of the
factor )4/(
2

 k
e and the term 
k

in the original

 kDSk of (7.3). In fact, the factor )4/(
2

 k
e serves as
the suppression factor for the original
 kDSk of (7.3).
The suppression factor plays an important role in Section
15 for unifications. In addition, as the suppression factor
value decreases to extremely low level, the magnitudes
of the local minimums and local maximums are sup-
pressed too much and no longer distinguishable from the
background noise. This scenario may relate to the early
universe with extremely high temperatures. The term


k

in the original

 kDSk of (7.3) comes from the uni-
tarity term “1” in


xDS of (6.1). In Section 9,

 kDSk
is extended based on the extension of the

 k

term.
Then the extended version is Fourier transformed back to
the complex x
-plane, a number of new things show up,
which will be discussed in Section 9.

 kDSk serves as particles spectrum with fermions at
local minimum and bosons at local maximum. Particle’s
mass and decay time can be calculated from ir kikk 
according to (7.5). The summation index j in (7.10) must
be truncated at integer n. The rules for truncation are:
For odd n:
 




 2/)1(
2/)1(
)5.0(2
n
nj
kjikji
keekDS

, (7.11a)
For even n:
 


 2/
)12/(
)5.0(2
n
nj
kjikji
keekDS

,
or

Z. Y. SHEN 1234
 




 12/
2/
)5.0(2
n
nj
kjikji
keekDS

. (7.11b)
The numerical parameter n assigned to particles is
from the mass ratio:
r
e
k
M
M
n
p. (7.12)
For
 kDSk serving as spectrum, the number n for
truncation in (7.12) must be integer, if the n-parameter in
(7.12) is not an integer, multiplication is taken to convert
it into an integer for the truncation in (7.11).
In (7.12), n and p are the second and third numerical
parameters introduced after the first one of m introduced
in Section 6. For a particle, the set of three numerical
parameters m, n, p plays important roles for particles
models and parameters, which will be explained in later
sections.
As examples, (7.11) is used to calculate the parameters
of muon and taon. The results of 16-digit numerical cal-
culation are listed in Table 7.1. In which, the reason for
taking the values of numerical parameters m, n, p will be
given in later sections.
Table 7.1. The calculated parameters of muon and taon.
*The listed i
k value corresponds to particle’s lifetime. **The relative dis-
crepancy of mass is calculated with the medium value of 2010-PDG data.
For the truncated
 kDSk of (7.11), the locations of
local minimums and local maximums depend on the val-
ue of n, which must be given beforehand. In other words,
different n values give different mass values for different
particles. Fortunately, the n value of a particle can be
determined by other means. For instance, quarks’ n is
selected from a set of prime numbers and it is tightly
correlated to strong interactions. It can be determined
within a narrow range and in many cases uniquely. The
details will be given in later sections.
Look at the spectrum from another perspective,
 kDSk actually provides a dynamic spectrum for all
particles. As the value of n-parameter increases, the loca-
tions of local maximums and minimums change accord-
ingly corresponding to different particles. It is conceiv-
able that, for the full range of n-parameter,

 kDSk ser-
ves as the spectrum of all elementary particles. Whether
it includes composite particles or not, which is an inter-
esting open issue.
Using 16-digit numerical calculations found that, for a
given value of r
ksuch as
30777692307692.206

r
k
for muon, there are a series of local minimums located at
different values of i
k. Table 7.2 shows muon twenty
three i
k values over a narrow range from
15
1068348.3  
i
k to 15
1068387.3 
i
k.
there are 6 local minimums corresponding to 6 possible
decay times.
Table 7.3 shows )(kDSk profile as a function of i
k
over a broad range of i
k alone
30777692307692.206

r
k
line.
As shown in Table 7.3 muon i
kvalues from 0

i
k to
1

i
kdivided into three regions. In Region-1


18
100  i k, average values of )(kDSk as base line
keep constant. In Region-2


1418 1010  i
k, )(kDSk
base line is in the global minimum region. Region-2 is
the effective region of muon’s decay activities. In which,
15
10683739.3 
i
k
corresponds to muon’s mean life of
s
6
10197034.2   
.
In Range-3


110 14 
i
k, )(kDSk base line increases
monotonically.
Table 7.2. Muon decay data in narrow range*.
*The parameters m, n, p and r
k are the same as those listed in Table 7.1.

Z. Y. SHEN 1235
Table 7.3. )( kDSk over broad range for muon*.
*The parameters m, n, p and r
kare the same as those listed in Table 7.1.
**2010-PDG listed muon’s mean life

 s
6
10000021.0197034.2  
.
Table 7.4 listed some samples of local minimums dis-
tribution at 11 locations, which are used to estimate the
average value of the separation between two adjacent
local minimums.
Table 7.4. Samples of local minimums of )(kDSk for
muon at 30777692307692.206
r
k*.
*The parameters m, n, p and r
kare the same as those listed in Table 7.1.
A distinctive feature of these theoretical results is that,
along a constkr straight line, )(kDSk has a series
of local minimums corresponding to a series of possible
decay times for a particle such as muon. Does it make
sense? From the theoretical viewpoint, it does. According
to the first fundamental postulation, SQS is a statistic
theory in the first place. A series of )(kDSk local mi-
nimums corresponding to a serious of possible decay
times should be expected. On the practical side, muon’s
mean life having a definitive value s
6
10197034.2   
is for large numbers of muons as a group. For an indi-
vidual muon,

is the statistical average value of many
possible decay times, it by no means must decay exactly
at


t.
As shown in Table 7.4, the 121 local minimums are
taken as samples from 18
101 
i
k to 12 101  
i
k
with 21
10as variation step. It shows that, local mini-
mums behavior randomly. The average separation be-
tween two adjacent minimums is calculated from these
samples as
20
20
20
10389.1
10641.1
10909.3 




 i k,
which roughly kept constant over a broad region. These
data is used to estimate the total number of local mini-
mums in Region-2 between18
1, 101 

i
k and
14
2, 101

i
k as:
5
1,2,10558.2 



i
ii
k
kk
N. (7.13)
Region-2 with decay time from st 7 100933.8  
to st 3 100933.8   is the effective region of muon’s
decay activity. There are 5
10558.2N local mini-
mums in this region, each one corresponds to a possible
decay time. The locations of local minimums determine
the values of possible decay times. Besides Region-2,
there are local minimums in Region-1 and in part of Re-
gion-3, which will be discussed later.
By counting all local minimums of )(kDSk, in prin-
ciple, the theoretical mean life

of muon can be calcu-
lated by extensive number crunching. But it requires a
tailor made program. In the meantime, let’s take a rough
estimate.
According to (7.5b), the separation tof two adja-
cent possible decay times and corresponding decay
time’s density (number of possible decays per unit time)
N

are:
ck
k
ck
t
i
ieC
i
eC
2










. (7.14a)
ieC
ik
ck
t
N 



2
1. (7.14b)
In the i
k domain, the local minimums have roughly
even distribution as shown in Table 7.4. In the time do-
main, because of the inverse relation 2
/1i kt  of
(7.14a), the local minimum of )(kDSk in the i
k do-
main corresponds to the temporal response as the local
maximum in the time domain. As shown by (7.14a), the
local maximums in time domain are unevenly distributed
caused by the 2
i
k factor in denominator of t.

Z. Y. SHEN 1236
The effective Region-2 is divided into four sub-re-
gions:
Region-2a: 17181010i
k with center at 18
105  
i
k;
Region-2b: 1617 1010 i
k with center at 17
105 
i
k;
Region-2c: 1516 1010   i
k with center at 16
105  
i
k;
Region-2d: 1415 1010   i
k with center at 15
105 
i
k.
The values of i
k, t, t, N and Nt at center
of each sub-regions calculated according to (7.14) and
Table 7.4 are listed in Table 7.5.
Table 7.5. Parameters in the center of four sub-regions.
The values at the center of each sub-region are treated
as the average values for that sub-region. Take

d
ajjj NN/as the probability for muon decay in
),,,( dcbajj sub-region, muon’s mean life is roughly
estimated as:
s
N
tN
td
aj j
j
d
aj j6
10838.1 

 


. (7.15)
The value of t

is 83.7% of muon’s measured
mean life s
6
10197034.2  
, which is in the ballpark.
Since only the activity in Region-2 is counted, the 16.3%
discrepancy is understandable. The ballpark agreement
shows that, the spectrum does contain the information of
mean lifetime in the muon’s case and Region-2 is the
effective region.
The rough estimation is based on the assumption that,
Region-2 is the effective region for muon’ decay activity.
The effects of other two regions are not taken into ac-
count, which need justification.
The local minimums are not restricted in Region-2,
they extended to Region-1 from 19
10~ 
i
k to 23 10~ 
i
k.
According to (7.14b), the decay time density N

is
proportional to 2
i
k, in Region-1, N value decreases
rapidly as i
k value decreasing. For instance, the N

value at the boundary of Region-1 and Region-2,
18
10 
i
k, is roughly less than 7 10  of the N value
at the center of Region-2 where muon’s mean life is
close by. In other words, muon rarely decays in Region-1
with extremely low probability.
Prediction 7.1: The probability of muon decay time
longer than
st3 100933.8  
corresponding to
18
101 
i
kis less than 7
10 of the
probability of muon decay at
st 6 10197034.2   
corresponding to
15
10684.3  
i
k.
Explanation: According (7.14b), N is proportional
to 2
i
k. The ratio of decay probabilities at 18 101  
i
k
and at 15 10684.3  
i
k is estimated as:
78
2
15
18
1010368.7
10684.3
101 










. (7.16)
For Region-1 with 18
101 
i
k, the ratio of decay
possibilities is much less than 8
10368.7  , which can be
estimated the same way. So the rough estimation of
t


disregarding Region-1 is justified.
The local minimums are also extended in Region-3
with rapidly increasing density. However, it does not
mean that muon decays more frequently in Region-3. In
fact, muon decays rarely in Region-3, which needs ex-
planation. )(kDSk serves as spectrum with fermion at
local minimum. In the spectrum, the tendency for muon
as a fermion to reach minimum value of )(kDSk actu-
ally is in two senses, locally and globally. The former
was considered, now it’s the time to consider the latter.
In Region-1 and Region-2, as shown in Table 7.3, the
base line of )(kDSk is almost flat with minor variations.
The vast numbers of local minimums with different den-
sities compete for the possible decay time. The base line
of )(kDSk increases monotonically in Region-3 and the
bottom values of local minimums increase with it. In
most part of Region-3, the bottom values of local mini-
mums are higher than the base line level in Region-1 and
Region-2. The turning point is probably at the vicinity of
13
101  
i
kcorresponding to st 8
100933.8   . muons
have very low probability for decay times shorter than
st 8 100933.8   despite the fact that the values of
N

are many orders of magnitudes larger than those in
Region-2. The abrupt drop of decay probability in Re-
gion-3 is caused by the local minimums disqualified in
the global sense, because their bottom values are higher
than the base line in Region-1 and Region-2. So the
rough estimation of t


disregarding Region-3 for
muon is also justified.
Moreover, according to (7.14a), the time separation
t

is proportional to the inverse of 2
i
k. In Region-3, as
i
k increases, t

decreases rapidly. At certain point, the
extremely crowded local maximums in time domain are
overlapped and no longer distinguishable.
Prediction 7.2: Muon has zero probability to decay at
times shorter than st 13
min 102 
.
Explanation: The disappearance of local maximums
in Region-3 happens at the point that, separation t

becomes shorter than the width of the response in time
domain for muon. At that point, individual response in
time domain is no longer distinguishable. Muon’s decay
is caused by weak interaction mediated by gauge bosons

W or 0
Z
with mean lifetime of s
ZW 25 ,102 
 
. The

Z. Y. SHEN 1237
muon’s decaying process must complete before its me-
diators’ decay, which roughly determines the width of
individual response in time domain. The criterion is
ZW
t,

. According to (7.5b) and (7.14a), the min
tfor
muon is:
s
kckc
t
k
tc
ckc
t
i
ZWeC
i
eC
ieC
eC
i
eC 13
,
min 10075.2 













(7.17)
In which 20
10909.3   i k is the medium average
value cited from Table 7.4.
As another example, electron’s )(kDSk profile over
broad range is sown in Table 7.6. The reason for taking
such values of numerical parameters m, n, p for electron
will be given in later sections.
Table 7.6. )( kDSk over broad range for electron at 1

r
k
with m = 2, n = 1, p = 1.
The distinctive features of electron’s )(kDSk profile
over broad range are the disappearance of Region-2 and
Region-1 becoming global minimum region with only
one local minimum of 0)0(
k
DS at 0
i
k corre-
sponding to t. It is consistent with the fact that
electron is stable. This is an important check point to
verify that, the rough estimation of mean life based on
the global minimum concept for muon is correct. It also
increases the credibility of )( kDSk serving as particles
spectrum with information of decay times and in some
way related to mean life. But nuon and electron are just
two examples, which are by no means sufficient to draw
a conclusion. The real correlation between )( kDSk as
spectrum and particle’s mean life is still an open issue.
More works along this line are needed.
As illustrated in this section, )(kDSk as a member of
the S-equation family has rich physics meanings. In gen-
eral, )(kDSk serving as particle mass spectrum is con-
ditional. It subjects to a prior knowledge of numerical
parameters. Even though, it does provide useful informa-
tion. More importantly, )(kDSk serves as the base for
an extended version, which reveals more physics signifi-
cance. Details will be given in later sections.
In this section, muon and taon are used as examples
for )(kDSk serving as particles spectrum on the com-
plex k-plane. More details of muon and taon will be giv-
en in later sections.
Section 8: Electron Torus Model and
Trajectories
As mentioned in Section 6, electron has two-loop struc-
ture. Loop-1 is the primary loop with loop length1 L.
Loop-2 perpendicular to loop-1 is the secondary loop
with loop length2
L. Loop-2 center rotates around loop-1
circumference to forms a torus surface. According to
SQS theory, all Dirac type fermions’ models are based
on torus. Torus is a genus-1 topological manifold with
one center hole and four tiny holes 1
h,  h,

h, 2

h
corresponding to four branch points on Riemann surfaces
described in Section 3.
To begin with, torus as a topological manifold has
neither definitive shape nor determined dimensions. The
four tiny holes 1
h,  h,

h, 2

h without fixed loca-
tion can move around on torus surface. To represent a
particle such as electron, the torus model must have de-
finitive shape and determined dimensions, and the loca-
tion of four tiny holes must be fixed as well. To deter-
mine these geometrical parameters, additional informa-
tion is needed, which comes from SQS theory first prin-
ciple.
Fig. 8.1 shows the torus serving as electron model.
There are three circles on x-y cross section shown in Fig.
8.1a. The two solid line circles represent the inner and
outer edges of torus, and the dot-dashed line circle
represents loop-1 and the trace of rotating loop-2 center.
In Figure 8.1b, the right and left circles shown torus two
cross sections are cut from line GO1 and line HO1on
x-y plane, respectively.
According to SQS theory, a set of three numerical pa-
rameters, m, n, p is assigned to each fermion defined as:
1
2
L
L
m
n, (8.1a)
e
M
M
n
p (8.1b)

Z. Y. SHEN 1238
In which,
M
and e
M are the mass of the fermion
and electron, respectively.
For electron, its original m, n, p parameters are se-
lected as:
2m, 1n, 1p. (8.2)
Substituting (8.2) into (8.1) yields:
2
1
1
2
L
L, (8.3a)
1
1

e
M
M. (8.3b)
The torus surface is divided into two halves as shown
in Figure 8.1b. The outer half has positive curvature and
the inner half has negative curvature. According to S-
equation of (3.20), unitarity requires:


0112
)(  


N
Nj
xjx
exS

. (8.4)
In (8.4), the original summation index i x is replaced
by
j
for simplicity. The lower and upper summation
limits are truncated at
N
j
 for numerical calcula-
tion. A sufficient large
N
is selected for

 1

xS to
converge. As discussed in Section 6, the two points on
real r
x-axis of Figure 3.4 representing electron are:
125.0
1x, and 375.0 2x. (8.5)
Substituting (8.5) into (8.4) and solving for )(x

yields:


 200378771029244.3)125.0( 1 x; (8.6a)


 543918645924982.2)375.0( 2 x. (8.6b)
)(1 x 
and )(2
x 
serve as the messengers to transfer
information from S-equation to torus model.


)(1 x
corresponds to negative curvature on the inner half of
torus; and


)( 2 x corresponds to positive curvature
on the outer half of torus.
The distance between two loops’ centers is d, which
is the radius of loop-1. For electron, loop-1 circumfer-
ence equals to one Planck wavelength, 1
1


PP LL

,
which corresponds to


2/12/
1 Ld . For conven-
ience, let’s set 1d as the reference length for other
lengths on the torus models, and consider its real value
later.
According to (8.3a) and 1

d, the radius of loop-2 for
electron is determined as:
5.0
2
1
1
2
2 dd
L
L
a. (8.7)
The two dimensions of torus as electron model are de-
termined as 1d, 5.0
2a. The next step is to fix the
locations for four tiny holes 1
h,  h,

h, 2

h shown
in Fig. 3.5b.
Figure 8.1. Electron torus model: (a) x-y cross section; (b)
Right is cross section along line GO1, left is cross section
along line HO1.
In fact, the electron torus model is shared with its an-
ti-particle, the positron. For the four tiny holes 1
h,  h,

h, 2

h, two of them belong to electron and the other
two belong to positron. The values of )(2
x

and
)( 1 x

determine the locations of two characteristic
points
A
,B for electron.




543918645924982.2)( 2 x of (8.6b) corre-
sponds to the torus outer half with positive curvature like
a sphere. On the GO1 cross section at the right of Fig
8.1b, the location of point 2
A at 22 ,ZzXx

 with
origin at cross section center 2 O is determined by
)( 2 x

according to the following formulas:

0
)(2
cossin
22
2
2
2
2
2
2



xZ
dttbta




, 22 ab

, (8.8a)
2
2
2
cos a
X
 
, (8.8b)
01
2
2
2
2
2
2

















b
Z
a
X, 22 ab. (8.8c)
As shown in Fig. 8.1b, point 2
A and two loops’ cen-
ters 1
O, 2
O form a triangle 212OOA. The three inner
angles 2

, 2

, 2

of triangle 212 OOA are deter-
mined by:

Z. Y. SHEN 1239
2
2
2
tan X
Z
 
, 22 180


 , (8.9a)
2
2
2
tan Xd
Z

 
, (8.9b)
222



 . (8.9c)
On x-y plane shown by Fig. 8.1a, the location of point
G at 33,YyXx  with origin at 1
O is determined
by angle 3

from )(2
x

according to following for-
mulas:
0
)(sin)(
)(
232
32 


xad
ad




, (8.10a)
2
2
2
3
2
3)( adYX  . (8.10b)
The three inner angles 3

, 3

,3

of triangle
21OGO are determined by:
dX
Y


3
3
3
tan 
, 33 180


 , (8.11a)
3
3
3
tan X
Y
 
, (8.11b)
333
  . (8.11c)


 44200373.87710292)(1 x of (8.6a) corresponds
to the inner half of torus with negative curvature like a
saddle surface with sinusoidal variation. The parameters
of saddle surface are determined by )( 1
x

according to
following formulas:


0
2
2cos21
1
2
0
2







x
dttAm, (8.12a)

12
cossin
2
0
2
1
2
1





dttbta
A
A
m
, 11 ab . (8.12b)
m
A and
A
are the amplitudes of saddle sinusoidal
variation on circles with radius 1 and radius 1 a, re-
spectively. The locations of points 1
B and point D are
determined by the following simultaneous equations:


0
22
1
2 RaAR, 22 1aadR 

, (8.13a)
01
2
2
1
2
2
2

















b
b
a
Aa , 11ab , 22 ab . (8.13b)
Equation (8.13a) represents a circle with radius R
centered at 1
O. The location of point D at
1
),(ayARx  with origin at 1
Ois determined.
Equation (8.13b) represents the circle with radius 2 a
centered at 2 'O on the HO1 cross section in Fig. 8.1b.
The location of point 1 B at 12 ),(byAax

 with
origin at 2
'O is determined.
In the HO1 cross section in Fig. 8.1b, the three inner
angles 1

, 1

, 1

of triangle 21 'DOB and angle 1

are determined by:
Aa
b


2
1
1
tan 
, (8.14a)
A
b1
1
tan  
, (8.14b)
111 180






, (8.14c)
A
R
b

1
1
tan 
. (8.14d)
On the x-y plane shown by Fig. 8.1a, The three inner
angles 0

, 0

, 0

of triangle 2
'DEO and angle 0

are determined by:
A
a1
0
tan  
, 00 180
  , (8.15a)
Aa
a


2
1
0
tan 
, (8.15b)
000
  , (8.15c)
A
R
a

1
0
tan 
. (8.15d)
According to the torus model and two characteristic
points A, B determined by )( 1
x

and )( 2
x 
from the
S-equation, electron parameters calculated with the above
formulas are listed in Table 8.1.
Table 8.1. Parameters for electron torus model*.
*All data are from 16-digit numerical calculations, only 8-digit after the
decimal point is presented. **The reduced numerical parameters are the
original numerical parameters divided by m.
In Table 8.1, notice that:

24442009.24
32 

, (8.16a)

45987086.28
23222

, (8.16b)

257547.55
10 

, (8.16c)

88107155.29
01 

. (8.16d)

Z. Y. SHEN 1240
Let’s consider the meanings of (8.16a). 2

is the an-
gle at the center of loop-2 between line GO2 and line
22AO as shown in Fig.8.1b, which serves as the initial
phase angles of cyclic movements along loop-2. 3

is
the angle at the center of loop-1 between the x-axis and
line GO1on x-y cross section shown in Fig. 8.1a, which
serves as the initial phase angles of cyclic movements
along loop-1. 32
  means that the two cyclic move-
ments around loop-2 and loop-1 are synchronized in
phase. (8.16b) indicates that the phase angles’ differences
of 23
  and 22


 both equal to

45987086.28
2

,
which is close to the Weinberg angle W

. This is the
first hint that, the characteristic points such as point
A
and the triangle 212 OOA have something to do with par-
ticle’s interaction parameters. (8.16c) and (8.16d) indi-
cate that, the some types of synchronizations as (8.16a)
and (8.16b) hold between angles 0

and 1

as well as
between 1

and 0

in the inner half of torus shown by
Fig. 8.1a and Fig. 8.1b on left side.
These types of synchronizations are interpreted as the
geometrical foundation of electron’s stability. It is the
first conclusion drawn from electron’s torus model.
The torus model represents electron, it must have all
electron parameters expressed in geometrical terms. This
is the job a model supposed to do. But the torus has only
two geometrical parameters dand 2 a to determine its
shape and size, which are by no means sufficient to rep-
resent all parameters. )( 1
x

and )(2
x

come to help.
They serve as the messengers to transfer information
from S-equation to torus model to define the locations of
characteristic points and the triangles associated with
them. In this way, the torus model with defined charac-
teristic points and triangles is capable to represent all
parameters of electron. The details will be given later.
For the standard model, particle is represented by a
point. A point carries no information except its location
and movement. That is why twenty some parameters are
handpicked and put in for standard model. For SQS the-
ory, parameters are derived from the first principle and
represented by geometrical model. In which, two mes-
sengers )( 1
x

, )(2
x

, the characteristic points and
triangles play pivotal roles.
The torus model provides a curved surface to support
the trajectory of electron’s internal movement. Electron
internal movement includes three types: (1) cyclic
movement along loop-1; (2) cyclic movement along
loop-2; (3) sinusoidal oscillation along trajectory. Fig.
8.2a and Fig. 8.2b show the projections of electron’s tra-
jectory on x-y plane and x-z plane, respectively. On x-y
plane shown in Fig. 8.2a, the top trajectory is for electron,
and the bottom trajectory is for positron. Because these
two trajectories are symmetrical, to explain the one for
electron is sufficient to understand the other.
Figure 8.2. Electron and positron trajectories on torus mo-
del: (a) Projection on x-y plane; (b) Projection on x-z plane.
The trajectory is a closed loop. It can start anywhere
on the loop as long as it comes back to close the loop.
Let’s look at trajectory starting at point A on torus out-
er half bottom surface represented by the short dashed
curve shown in Fig. 8.2(a). It passes through the torus
outer edge and goes to the upper surface shown by solid
curve. It passes the top center line getting into the inner
half and reaches point B on torus inner half top surface
to complete its first half journey. The second half journey
starts from point
B
. At the torus inner edge, it goes back
to the bottom surface shown by dashed curve. It passes
through the bottom center line and comes back to point
A to complete a full cycle. The trajectory repeats its
journey again and again. The x-z plane projection of the
trajectory is shown in Fig. 8.2(b).
The trajectory shown in Fig. 8.2 is a rough sketch. Its
exact shape is determined by two geodesics on the torus
surface. One from point A to point
B
; the other from
point
B
back to point A to close the trajectory loop.
The characteristic points A and
B
not only carry the

Z. Y. SHEN 1241
parameters information to define the triangles but also
serve as the terminals for the two geodesics to form the
trajectory.
Notice that, in Figure 8.1 and Figure 8.2(a), the three
points
A
, 1
O,
B
are not aligned. The difference be-
tween two angles 0

and 3

is:
 01312691.13942.2444200257547.55
30 

.(8.17)

 is the angle deviated from 
180 representing a
perfect alignment of three points
A
, 1
O,
B
. It is impor-
tant to point out that,
A
and
B
are not fixed points.
Instead, they define two circles, circle-A and circle-B,
with radius AO1 and BO1, respectively. The trajectory
may start at a point on circle-A halfway through a point
at circle-B and comes back to point A. The trajectory is
legitimate as long as it kept the same angle of BAO1
:
 98687309.166180
1
 BAO. (8. 18)
There are many trajectories on torus surface with the
same angle BAO1
 given by (8.18), all of them contain
the same information carried by )(1
x

and )( 2
x

.
These trajectories spread over torus entire surface. As
shown in later sections, trajectories are discrete in nature
and the number of trajectories is countable, which form a
set of discrete trajectories on torus surface. At a given
time, electron is represented by a trajectory. As time
passing by, it jumps to other trajectories. The scenario is
dynamic and stochastic. Physically, jumping trajectories
on the same torus surface corresponds to emitting and
absorbing a virtual photon by the electron.
For the x-y projection shown in Figure 8.2a, the tra-
jectory on the bottom for positron goes through two cha-
racteristic points 'A and 'B with anti-clockwise direc-
tion. As shown in Figure 8.2b, the x-z projections of two
trajectories are coincided with opposite directions: an-
ti-clockwise for electron and clockwise for positron.
In essence, the S-equation determine the value of
)( 1 x 
and )(2
x 
from 1
x and 2 x; )( 2 x 
and )( 1
x 
determine the location of characteristic points A and
B
on torus model; Points
A
,
B
and two geodesics be-
tween them define a trajectory on torus surface; Rotating
points
A
and
B
defines circle-A and circle-B along
with a set of discrete trajectories on torus model.
The sinusoidal oscillation along trajectory path is rep-
resented by a term in two ad hoc equations. Figure 8.3
shows two orthogonal differential vectors


d and

da 2 ':
2
2
2
2
2'
'
'a
ad
da
d



, (8.19a)


cos'2 ad  . (8.19b)
The oscillation on trajectory is represented by a sinu-
soidal term:



sin'2 a, (8.20a)






























1
2
22
2
L
L
M
M
m
n
n
p
m
p
e

. (8.20b)
Figure 8.3. Differential vectors on torus model.
In which,
M
and e
Mare the mass of the particle and
electron, respectively. For electron1// npMM e,
2/1// 12  mnLL and 1


, (8.20a) becomes:
)sin(')sin('22


aa

. (8.20c)
As shown by (8.20), the sinusoidal oscillation term
)sin(' 2

a is related to mass, it is called the “mass
term”. Adding the mass term of (8.20c) to the numerator
on right side of (8.19a) yields:
2
2
2
2
2
2'
sin''
'a
aad
da
d




, (8.21a)
or

  daadd sin'' 2
2
2
2 . (8.21b)
According to Figure 8.3 and (8.21b), the combined
differential vector length is:
   daadaddadl 2
2
2
2
22
2
22
2sin'''' . (8.22)
Take the integral of (8.22) from 0 to

2:



 


2
0
2
2
2
2
22
2
2
0
sin''' daadadlL . (8.23)
According to (8.21b) and (8.19b), the differential an-
gle along the

-direction is:






 d
ad
aad
d
aad
dcos'
sin''sin''
2
2
2
2
2
2
2
2
2




. (8.24)
Take the integral of

d from 0 to

2:
 







2
02
2
2
2
2 2
0cos'
sin'' d
ad
aad
d. (8.25)
Definition 8.1: Define the Angle Tilt (AT) equation
and the Phase Sync (PS) equation as:

Z. Y. SHEN 1242
1) AT-equation:





2
02
2
2
2
2
22
20
'2
sin'''
2
1
a
d
daada
d; (8.26a)
2) PS-equation:








2
02
2
2
2
2
01
cos'
sin''
2
1d
ad
aad . (8.26b)
In (8.26a), the factor 2 in the denominator of second
term comes from Section 3:
2
)120cos(
1
)cos(
1


 
. (3.12)

120 
is the separation angle of three lines on the
complex plane shown in Figure 3.2.
For 1d, solving the two equations of (8.26) for 2 'a
yields:
222154918171173.0'2a. (8.27)
AT and PS are two independent equations with one un-
known 2 'a. Both equations are satisfied simultaneously
with the same solution
222154918171173.0'2a.
It indicates that they are self consistent and mean
something.
22
'aa  means that, the torus original circular cross
section is distorted. To keep loop lengths ratio
mnLL // 12  unchanged, the original cross section pa-
rameters, 2
a and 22 ab must be changed accord-
ingly, which makes the torus cross section elliptical.
Definition 8.2: The Modification Factors (MF) of the
f-modification are defined as:
2
2
'
a
a
fa, (8.28a)
2
2
'
b
b
fb. (8.28b)
For electron, 5.0
2a, 5.0 2

b,
222154918171173.0'2a, and 2 'b
is determined by:
dttbtadttbta  
 2
0
22
2
22
2
2
0
22
2
22
2)(cos)(sin)(cos)'()(sin)'( , (8.29a)
197555081164600.0' 2b. (8.29b)
Explanation: In essence, the f-modification is intro-
duced to satisfy (8.26) and to keep loop-2 length2
L un-
changed as shown by (8.29a). It is important to keep loop
length ratio mnLL // 12  unchanged, because it is re-
lated to interactions.
According to Definition 8.2, the modification factors
of electron are calculated as:
4443039836342346.0
'
2
2 a
a
fa, (8.30a)
39510162329200.1
'
2
2 b
b
fb. (8.30b)
After the f-modification, the geometrical parameters
are changed accordingly. The rules are to keep the initial
phase angles unchanged as the originals:
22
'



, (8.31a)
33
'



, (8.31b)
11
'



, (8.31c)
00
'   . (8.31d)
The other geometrical parameters of the modified to-
rus model change accordingly. The rules are: (1) To keep
the initial phase angles given by (8.31) unchanged; (2)
The torus cross section becomes elliptical with 2
'a and
2
'b given by (8.27) and (8.29b), respectively. The rest is
from geometry.
The modified point 2
'A and triangle 212 'OOA re-
lated angles are determined by:
222 180'180'


  , (8.32a)
2
2
2'
'
'tan Xd
Z

 
, 222'''



 , (8.32b)

2
2 2
2
2
2
2'tan'/1'/1
1
'
 ba
X , (8.32c)
222 'tan''

XZ

. (8.32d)
The modified point 'G and triangle 21
'OOG related
angles are determined by:
dX
Y


3
3
3'
'
'tan 
, 33'180'


 , (8.33a)
33
'  , 333 '''   , (8.33b)

3
2
2
3'tan1
'
'
 

ad X, (8.33c)
333 'tan''

XY

. (8.33d)
The modified point 1
'B at 11',' ZyXx with origin
at 2
'O and triangle 21 ''' ODB related angles are deter-
mined by:
12
1
1''
'
'tan Xa
Z



, (8.34a)
11
'



, 111''180'
  , (8.34b)
1
1
1'
'
'tan Xd
Z

  , (8.34c)

1
2
2
2
2
2
1'tan'/1'/1
1
'
 ba
X , (8.34d)
111 'tan''

XZ

. (8.34e)
The modified point 'D at 00',' YyXx  with ori-
gin at 1
O and triangle 2 ''' OED related angles are de-
termined by:

Z. Y. SHEN 1243
0
0
0'
'
'tanXd
Y

 
, (8.35a)
0
0
0'
'
'tan XR
Y

 
, 00 '180'
  , (8.35b)
000 '''
  , (8.35c)
00
'  , (8.35d)

0
2
0'tan1
'
 
R X, (8.35e)
000 'tan''

XY . (8.35f)
The modified data for electron are listed in Table 8.2.
In which the effective parameters after f-modification are
marked with the ‘ sign.
Table 8.2. Modified parameters for electron torus model*.
*All data are from 16-digit numerical calculations, only 8-digit after the
decimal point is presented. **The reduced numerical parameters are the
original numerical parameters divided by m.
After modification, despite the change of
428.4794845''2

 
W
from original 628.4598708
2

 
W, as shown in Table
8.2, three out of four synchronizations still hold with one
slightly off:

24442009.24''32  
, (8.36a)

47948454.28''''' 23222  
, (8.36b)

257547.55'' 10  
, (8.36c)

44799569.30'06432177.30' 0
0
1

. (8.36d)
It indicates that electron stability is persistent and ro-
bust.
To understand the meaning of f-modification, in the
AT-equation, let’s set the mass term 0sin'2


a to
see its effect, (8.26a) and (8.28) become:









2
02
2
022
2
2
2
22
20
'2
1
1
'22'2
''
2
1
aa
d
d
d
d
a
d
dada
d
,
22 5.05.0' ada



, 22 5.0' bb 

; 1

baff. (8.37)
1


ba ff means no f-modification. It clearly shows
that the effect of f-modification is caused by the added
mass term of

sin'2 a, which represents the mass ef-
fect.
In the standard model, particle acquires mass through
symmetry broken. Likewise, in SQS theory, the mass
term of

sin'2 a breaks the 3-fold symmetry with

120 
on the complex plane. This analogue plus the
simultaneous satisfaction of two independent equations
with the same solution 2
'a give some legitimacy to
AT-equation and PS-equation despite their ad hoc nature.
Let’s look at the geometrical meaning of the f-modi-
fication. As shown in Section 3, the angle separates three
lines on complex plane is:

120
3
2


. (3.8d)
The f-modification causes the angle having a slight tilt
from

to '

:
a
f
a
ad


















4443159836342346.0
)120cos(
'
'
arctan180cos
cos
'cos 2
2
2
2




(8.38a)
 224600855504.119
'
'
arctan180'
2
2
2
2









 a
ad

,(8.38b)

785399144495.0'  
. (8.38c)


is the tilting angle deviated from 
120 
. It indi-
cates that, original 
120 
3-fold symmetry is slightly
broken by tilting angle


for electron having mass.
After f-modification, AT-equation and PS -equation are
satisfied simultaneously. It indicates that, the two cyclic
movements of two loops and the sinusoidal oscillation
along the trajectory are synchronized perfectly for elec-
tron as a stable particle.
Numerical calculations found that, AT-equation of
(8.26a) has only one root 2
'a given by (8.27) with the
a
f value given by (8.30a). On the other hand,
PS-equation of (8.26b) has a series of roots. Start from
4443039836342346.0

a
f, varying its value with 16
10 
steps calculate the values of )( a
f

as a function of a
f:










2
02
2
2
2
2
1
cos'
sin''
2
1
)( d
ad
aad
fa,

Z. Y. SHEN 1244
aa fafa 5.0'22  . (8.39)
A sample of numerical calculated results are listed in
Table 8.3.
In Table 8.3,
 0
0





means phase precisely
synchronized, and
 0
0



means off sync.
The results of Table 8.3 are interpreted as that, electron’s
torus model is dynamic and stochastic in nature. It
changes its loop-2 tilting angle constantly corresponding
to different a
f, b
f and 2
'a, 2 'b values representing
different torus surfaces. Electron’s trajectory changes
accordingly. The tilting angle changes discretely, so does
the trajectory, which means that trajectories are quan-
tized. At a given time, electron is represented by a tra-
jectory on a torus surface. As time passing by, it jumps to
other trajectories on another torus surface. It is a stochas-
tic scenario of jumping trajectories on different torus
surfaces. Physically, it corresponds to interactions such
as emitting and absorbing a photon. As mentioned pre-
viously, jumping trajectories on the same torus surface
corresponds to emitting and absorbing a virtual photon
by electron.
Table 8.3. Some roots of equation (8.26b).
*Note: )( a
f


 and )4443039836342346.0( 00  a
f


.
As shown in Table 8.3, PS-equation has 23 roots in re-
gion 15
1050  
a
f 
corresponding root density of:
15
15 106.4
105
23 

 D. (8.40)
The root density D roughly kept constant in the effec-
tive region
 1,
a
f. For orders of magnitude estimation,
the total number of roots for PS-equation in region

 1,
a
f
is:



1315
1984.0 10544.7106.49836.01  DfN a. (8.41)
As mentioned previously, there is a set of discrete tra-
jectories on the same surface of a torus surface. Now on
top of it, there is another set of discrete trajectories on
13
10544.7 N different torus surfaces caused by
f-modifications. At a given time, the real trajectory is the
one randomly chosen from these two sets of discrete tra-
jectories. In other words, electron trajectories are dy-
namic and stochastic in nature, which spread like clouds
around the torus surfaces. The term “electron clouds”
was used to describe electron’s behavior around a nu-
cleon according to quantum mechanics wave function.
Here the clouds appear in a deeper level, which should
not be a surprise.
As shown in Figure 3.4 of Section 3, the loop on the
complex plane connecting 1
x and 2 x has many dif-
ferent paths with the same loop length. That scenario is
consistent with the different trajectories with the same
length on different torus surfaces and different locations.
It shows the consistency of the theory.
In Table 8.3, the step ofa
f

variations and step of 2
'a

and 2
'b 
variations are in the order of 16
10to 15 10
Planck length corresponding to 51 10 to 50
10  meters.
The step of torus surface variations is extremely tiny. As
the torus’ loop-2 tilts, the electron’s trajectory jumps
from one torus surface to the other. In fact, this dynamic
picture is expected from quantum theory. The three types
of movement for electron described in this section all are
deterministic in nature. Without trajectory jumping, the
deterministic movements are contradictory to the uncer-
tainty principle. Moreover, the Gaussian Probability
Postulation of SQS theory is stochastic in the first place.
The trajectory jumping is ultimately originated from the
Gaussian probability assigned to discrete points in space.
The


0





fluctuating data listed in Table 8.3 is
an indication of the stochastic nature of SQS theory, even
though the PS-equation of (8.26b) is not derived from the
first principle.
Figure 8.4 shows the right side of Figure 8.1b in de-
tails. Points
A
,
F
,2
O define a right triangle 2
AFO ,
which contains two additional right triangles:
A
F
K
and
2
FKO . The triangle 2 AFO is indentified as the Gla-
show-Weinberg-Salam triangle, GWS-triangle for short.
In the 1


c unit system, the sides of GWS-triangle
are related to electroweak coupling parameters:
eFK

,gAF

,'
2gFO,22 2'ggAO  . (8.42)
e, and
g
, 'g are electric charge and two weak cou-
pling constants, respectively. The following formulas are
from geometry:
22 '
'
sin gg
g
g
e
W
 
, (8.43a)
'
cos g
e
W

. (8.43b)
Combining (8.43a) and (8.43b) yields:
22'
cossin gg
e
WW 
 
. (8.44)

Z. Y. SHEN 1245
Formula (8.44) is used extensively in later sections as
the criterion to construct the model for other fermions.
Figure 8.4. Glashow-Weinberg-Salam triangle.
According to 16-digit numerical calculation, the
original and effective Weinberg angles of electron are:
Original:
64113828.4598708
o2

 
We , (8.45a)
Effective:
3780828.4794845'2

 
WeM . (8.45b)
One of SQS theory final goals is that, all parameters of
an elementary particle should be derived from its model.
To identify the GWS-triangle with Weinberg angle in the
torus model is a step toward the final goal. Some other
characteristic triangles will be introduced in later sec-
tions.
From Einstein’s unified field theory viewpoint, every-
thing including all elementary particles and interactions
are originated from geometry. For SQS theory, the model
plays that role. Torus as a genus-1 topological manifold
has one center hole, its shape and size are arbitrary to
begin with. In order for the torus model to represent a
particle with its parameters, additional steps must be
taken. Take electron as an example. As the first step, the
shape and dimensions of torus are determined by loop-2
to loop-1 length ratio of 2/1// 12 mnLL and
dL

2
1. The second step is to fix the locations of cha-
racteristic points
A
and B on torus surface by utiliz-
ing the curvature information carried by )( 2
x

and
)(1 x

from the S-equation. In this way, the triangles
such as the GWS-triangle are determined and the pa-
rameters are determined as well. The process shows ma-
thematics at work. The mathematics at work viewpoint
will be enhanced further in later sections.
Recall in Section 3, the four tiny holes 1
h,  h,

h,
2

hserved as four branch points 1, ,

, 2

on the
Riemann surface. Moreover, the way Penrose built the
torus is to glue a pair of slits on two sheets of Riemann
surface together [2]. In fact, there are infinite sheets of
Riemann surface corresponding to a general form of
(3.22):


2
0
in
ez  , 









2
3
2
1
ni
ez ,










2
3
4
2
2ni
ez ,



 3,2,1,0n. (8.46)
These sheets can be combined into pairs to build many
genus-1 torus surfaces, which serve as the topological
base of many torus surfaces with slightly different pa-
rameters 2
'a and 2 'b derived from PS-equation as
discussed earlier. After all, there are enormous numbers
of torus surfaces provided by (8.46) for trajectory to
jump on. This argument gives more credit to the ad hoc
PS-equation.
Moreover, the torus with four tiny holes shown in Fig.
8.5a is topologically equivalent to a pair of trousers with
a large hole in their waistband shown in Figure 8.5b. The
four tiny holes on torus with their edge extended out-
wards form four tubes as the four ports. According to
[11], if the loops around trousers shrink to points, the
trousers with four ports degenerate to a Feynman dia-
gram with one closed loop and four branch lines shown
in Figure 8.5c. Feynman diagram is correlated to interac-
tions. Therefore, the triangles such as GWS-triangle de-
fined by characteristic points carry interactions informa-
tion are natural.
Figure 8.5(a). Torus with four tiny holes; (b) Four tiny
holes’ edge extended into four tubes; (c) De generated into a
Feynman diagram with one loop.
In summary, electron’s torus model is built on three
bases:
1) Loop lengths ratio 2/1// 12  mnLL and masses

Z. Y. SHEN 1246
ratio 1/1//  npMM e are determined by a set of three
numerical parameters, 2m, 1n, 1p.
2) The 3-dimensional Gaussian probability’s 0

, 1

,
2

plus  are identified as four branch points on the
Riemann surface, which are topologically equivalent to
four tiny holes on torus.
3) The four tiny holes on torus correspond to charac-
teristic points A, Band '
A
, '
B
. Their locations are
fixed according to the information carried by )( 1
x

and
)( 2 x

, which are the solutions of the 1-dimensional
S-equation.
In the three bases, No.2 and No.3 are originated from
SQS theory first fundamental postulation, the Gaussian
Probability Postulation. No.1 is a set of three numerical
parameters. It is related to the second fundamental pos-
tulation of SQS theory, which will be introduced in later
section. These are the only things needed to build the
model for a particle such as electron to carry all its pa-
rameters. It shows the power and the simplicity of the
first principle of SQS theory.
The electron torus model introduced in this section
serves as the basic building block. It is not the final ver-
sion. The details will be given in Section 12.
Section 9. Complex x
-Plane and Fine
Structure Constant

kDSk of (7.1) is the Fourier transformation of

 xDS
of (6.1):


  .
4
15.0
4
2
keeekDS
j
kjiijk
k
k









 



 (7.1)
 kDSk serves as particles spectrum. The local mini-
mums of

kDSk correspond to fermions and the local
maximums of

kDSk correspond to bosons. In this sec-
tion,
 kDSk is extended as  kEDSk. Then

 kEDSk is
Fourier transformed back to the complex x
-plane and
compared with
 xDS to find some physics implica-
tions.
Definition 9.1: Define the
 kEDSk function as the
extension of
 kDSk function


   
 












'
4
5.0
45.0'4'4
4
1
22
j
k
j
kjiijk
k
kjkjkeeeekEDS



(9.1)
Explanation: In the

 kEDSk, the original term

 k

in
 kDSk of (7.1) is extended by the second summation
terms with two sets of

-functions. The first term with
0' j in the second summation, )()'4( 0' kjkj
  
is the original delta function )(k

in
 kDSk, and all the
other terms in the second summation are newly added
delta functions. The extension adds a series of additional
local maximums for

 kEDSk representing bosons.
Look at (9.1) closely, the added

-functions also affect
fermions in (7.1). For instance,

2k (12/



kk )
in

 kDSk is a root of
 0kDSk represents electron
as a fermion. In

 kEDSk, the1'j,

2k term
  )0())5.0'(4( 2,1' 
 kj
jk
causes

   1'1' 12 j
k
j
kkEDSkEDS

.
It represents a boson.
Using Fourier transform to transfer
 kEDSk back to
the complexx
-plane:
 




dkekEDSxEDS ikx
kx
. (9.2)
Substituting (9.1) into (9.2) yields the

 xEDSx
- func-
tion on the complex x
-plane:
 


 

 




 
'
5.0'45.0'4'4'45.0 2
2 22
2
1
)(
j
xjijxjij
j
xjxj
xeeeeeexEDS


.
(9.3)
In the

 xEDSx
, the first summation is

 1

xDS as
expected; the second summation includes the unitarity
term:


1
0'
'4'4 2

 j xjij ee 
.
The other terms in the second summation correspond
to bosons representing interactions, which are originated
from delta functions added in
 kEDSk.
Numerical calculations found that:
In general on x
-plane:

 )5.0( xEDSxEDS xx


 . (9.4a)
On the real x-axis:

 xSxEDSx

1
. (9.4b)
Errors of approximations are around 15
10and 5
10 for
(9.4a) and (9.4b), respectively.
Definition 9.2: Define the SS-function and SS- equa-
tion on the complex x
-plane as:








 

  






















j
xjijxjij
j
xjxj
m
xjixjxjij
j
xjxj
xx
eeeeee
eeeeee
xEDSxEDSxSS
,
2
1
2
1
5.0
5.05.045.045.0445.05.05.0
5.045.04445.0
2 2 22
2
2
22



(9.5a)





 05.0


xEDSxEDSxSS xx  . (9.5b)
According to (9.4a) and (9.5a),
 0xSS . The val-
ues of


xSS fluctuate around 1715 10~10  and oc-
casionally equal to zero,
 0xSS , which are the roots
of

 0

xSS .
As shown in Section 6,
 0xDS is a real equation
on the real x-axis. It has a root at 125.0
1x on the

Z. Y. SHEN 1247
x-axis corresponding to electron. On the other hand,
 0xSS is a complex equation and 125.0
1x is not
its root. Instead, a root of
 0xSS is found by nu-
merical calculations at:
W
i
exx

11'
, (9.6a)
78213151240811255.0'1x, (9.6b)

1384598708641.28
W
 . (9.6c)

1384598708641.28
W

is electron original Weinberg
angle of (8.45a) before f-modification.
78213151240811255.0'1x
is slightly less than 125.0
1x. According to (6.19a),
78213151240811255.0'1x correlates to the mass e
M'
slightly less than e
M:
)'25.0(8
1 '
1
xM
M
e
e

. (9.7)
As shown in Appendix-4, charged particle mass sub-
jects to electromagnetic modification. According to
(A4.5) and (9.7):

306879927026474.0
'25.08
1
1
'
1




xM
MM
M
M
e
EMe
e
e

.
(9.8)
In which

is “fine structure constant” of electron.
Solving (9.8) for

yields:

50359990834.137
'25.08
1
1
1
1
1










x

. (9.9)
According to references [3,4], 2010-PDG (p.126) pro-
vides the experimental data:
)51(035999084.137
1 

. (9.10)
The relative deviation of SQS theoretical value and
2010-PDG data medium value is
12
10013.4   
. (9.11)

xSS is also used for calculating the 1 
values
for electron quantum states with fractional charges. Ac-
cording to (8.44) with assumption of constgg  22 ',
the Weinberg angle FW ,

for particles with fractional
charges are determined by:
F
WW
FWFW 



cossin
cossin ,, . (9.12)
3/1F, 3/2F, are for fractional charges, e/3, 2e/3,
respectively. Formula (9.12) and

1384598708641.28
W

are used to calculate the values of FW,

.
The definition of fine structure constant

is:
hc
e
0
2
2 

. (9.13)
According to (9.13),

is proportion to 2
e. For the
electron states with fractional charges 3/e, 3/2e,
(9.8) and (9.9) are changed accordingly as.

1
2
'25.08
1
1x F  
, (9.14)

1
1
21
'25.08
1
1








 x F 
. (9.15)
The SQS theoretical values of 1

, W

and FW ,

for electron states with different charges from 16-digit
numerical calculations are listed in Table 9.1.
Table 9.1.

, W

, FW,

for electron with different char-
ges.
*Note:

is the relative deviation from 2010-PDG medium value of
035999084.137
1 

.
In fact, the electron fractional charge states did show
up in the quantum Hall effect experiments.
The

effect on mass is originated from electro-
magnetic interaction. It is consistent with the fact
that

 xDS does not include interactions and


xSS
does. It also explains why 125.0
1x on the real x-axis
does not require mass correction with

and
78213151240811255.0'1

x
with phase angle 
1384598708641.28
W

on the
complex x -plane does.
The values listed in Table 9.1 are not unique. In fact,


0xSS has a series of roots corresponding to a series
of different

values. The multi-value behavior re-
flects the fact that

is a running constant and the sto-
chastic nature of SQS theory. The details will be dis-
cussed in later sections.
The

 xEDSx
 function introduced in this section is
not only used to define


xSS function but also has oth-
er important applications, which will be given in Section
15.
Section 10. Muon and Taon Torus Model
and Parameters
Muon and taon belong to the second and third genera-
tions of lepton family. Their torus model is similar to
electron torus model except that the x-z cross section is
elliptical for the original version. Instead of one radius
2
a for the circular cross section of electron torus model,
the elliptical cross section has two radii 2
a and 2 b. To

Z. Y. SHEN 1248
determine the parameter 2
b requires an additional equ-
ation. The option taken in this section is to keep the
original (before f-modification) Weinberg angle the same
for all three charged leptons:
WeOWO


. (10.1a)
WeO

is the original Weinberg angle for electron,
WO

is the original Weinberg angle for muon or taon.
According to (10.1a) and (8.45a), the original angle
WeOWO




2 for muon and taon is determined:

1384598708641.28
2 WeOWO

. (10.1b)
The original numerical parameters m, n, p for
muon and taon are selected as:
Muon:
18m, 4
1
29n, 6048p; (10.2a)
Taon:
42m, 120n, 417270p. (10.2b)
The reasons for selecting such values of m, n, p
will be given in later sections.
The values of 1 x and 2 x for muon and taon are
calculated according to (6.19):
p
n
M
M
xe
8
25.0
8
25.0
1
,
(10.3a)
12 5.0xx  . (10.3b)
In (10.3a), )8/()8/(pnMMe is according to
npMM e//  of (8.1b). Substitute the values of p and n
given by (10.2) into (10.3) yields:
Muon:
13095240.24939546
1x,
86904760.25060453
2x; (10.4a)
Taon:
20526280.24996405
1x,
79473720.25003594
2x. (10.4b)
Substituting 1
x,2 x of (10.4) into the S-equation
(3.20) and solving for )( 1
x 
and )( 2
x

yield:
Muon:
424961436156775.3)( 1x

,
40639671394911815.3)( 2x

; (10.5a)
Taon:
268531416714823.3)( 1x

,
14241414262265.3)( 2x

. (10.5b)
Most formulas of electron torus model to determine
characteristic point
A
, point B locations and other
geometrical parameters in Section 8 are valid for muon
and taon except some differences caused by the cross
section change from circular to elliptical.
The formula to calculate loop length ratio
mnLL//12

is:
m
n
L
L
d
dttbta  

1
2
2
0
2
2
2
2
2
)cos()sin(


.
(10.6)
For the torus outer half, formulas (8.8b), (8.9a)
through (8.9c), (8.10a) through (8.10b), (8.11a) through
(8.11c) are also valid for muon and taon. The changes are
(8.8a) and (8.8c), in which 22 ab  is replaced by
22 ab

.
For the torus inner half, formulas (8.12a), (8.13a),
(8.14a) through (8.14d), (8.15a) through (8.15d) are valid
for muon and taon. The changes are: in (8.12b), 11 ab

is replace by 11 ab; in (8.13b), 11 ab  and 22ab

are replaced by 11 ab

and 22 ab .
For the f-modification, (8.26a) and (8.26b) are for
electron. For other fermions including muon and taon,
they are generalized as:
AT-equation:






2
02
2
2
2
2
22
20
'2
)sin('''
2
1
a
d
daada
d
; (10.7a)
PS-equation:








2
02
2
2
2
2
01
cos'
)sin(''
2
1d
ad
aad ; (10.7b)
Mass term’s

:
e
M
M
L
L
n
p
m
n
m
p





















1
2
22
2

. (10.7c)
The

cos in the denominator of PS-equation does
not change, because it is originated from geometry rela-
tion of (8.19b) and has nothing to do with mass.
The rest of formulas for the f-modification, (8.28a),
(8.28b), (8.29a), (8.31a) through (8.31d), (8.32a) through
(8.32d), (8.33a) through (8.33d), (8.34a) through (8.34e),
(8.35a) through (8.35f), angle tilt formulas (8.38a)
through (8.38c) and (8.39) all are valid for muon and
tuaon without change. The GWS-triangle and related
formulas (8.40), (8.41) and (8.42) are also valid for muon
and tuaon.
Table 10.1 and 10.2 list the calculated parameters for
muon and taon, respectively. In these tables, the parame-
ters with the ‘mark are effective, i.e. after the f-modi-
fication and the parameters without the mark are original,
i.e. before the f-modification.
The synchronization related angles in Table 10.1
are:
 63018464.3'15808314.52' 32 

, (10.8a)
 91156239.18''''6163361.29' 23222
 , (10.8b)

Z. Y. SHEN 1249
 38022705.34'40080232.9' 10 

, (10.8c)
 45108237.9'833.0215193' 01 

. (10.8d)
The synchronization of two loops cyclic movements
for electron described in Section 8 no longer holds for
muon. It indicates that, muon is not a stable particle. In
fact, muon has a mean life of
s 10)000021.0197034.2(6  
(2010-PDG data).
Table 10.1. The calculated parameters of muon torus mo-
del*.
Table 10.2. The calculated parameters of taon torus mo-
del*.
*All data are from 16-digit numerical calculations, only 8-digit after the
decimal point is presented. **The reduced numerical parameters are the
original numerical parameters divided by m.
The synchronization related angles in Table 10.2 are:
 02150092.1'89177471.51' 32 

, (10.9a)
68251145.21''''18776233.29' 23222 

, (10.9b)
 86247726.24'3.27752516' 10 

, (10.9c)
35382497.3'424.2338133' 01 

. (10.9d)
The synchronization of two loops cyclic movements
for electron described in Section 8 no longer holds for
taon. It indicates that, taon is not a stable particle. In fact,
taon has a mean life of
s 10)001.0906.2(13  
(2010-PDG data).
The parameters listed in Table 10.1 and Table 10.2 for
muon and taon are calculated according to the formulas
in Section 8 for electron with modifications introduced in
this section, in which some of them are optional and
subject to verification. If some of them are replaced by
other options, related parameters should be changed ac-
cordingly.
The characteristic points, the trajectory, the circle-A,
circle-B, the tilt angle


breaking 
180 
3-fold
symmetry, the jumping trajectories, the torus model with
four tiny holes equivalent to trousers with a large hole in
the waistband and 4 ports degenerated to Feynman dia-
gram, these and related issues discussed in Section 8 for
electron are also valid for muon and taon.
The torus models for muon and taon introduced in this
section serve as the basic building blocks, which are not
the final version. The final version of models will be in-
troduced in Section 12.
Section 11. Quarks Model and Parameters
Quarks torus model has elliptical x-z cross section. The
formulas for muon and taon in Section 10 are valid for
quarks with exception that formula (10.1) is replaced by
following formulas for quarks with fractional charges.
For up-type quarks:
0
3
2
cossin
cossin ,2,2 
WeOWeO
uOuO



,

2139796740885.16
,2  uO

; (11.1a)
For down-type quarks:
0
3
1
cossin
cossin ,2,2 
WeOWeO
dOdO



,

03841092834194.8
,2 dO

. (11.1b)
In which, uO,2

and dO,2

are original values of the
angle 221 OAO as shown in Figure 8.1 before the
f-modification for up type and down type quarks, respec-
tively. Formulas (11.1) is based on an assumption:
constgg 22 ', which is optional.
There is another difference. The top quark is different

Z. Y. SHEN 1250
from the other quarks. Because its mass exceeds the up-
per limit set by (6.21), top quark’s model is spindle type
torus with covered center hole as shown in Figure 11.1.
The inner half of spindle shape torus also has positive
curvature, which is consistent with top quark’s



)(1 x.
This difference makes top quark’s inner half two trian-
gles with different definitions and different physics
meanings.
Figure 11.1. Spindle type torus model for top quarks.
As shown in Figure 11.1, the location of points D
and B are determined by
 )(1 x the same way as
points G and A determined by


)( 2 x.
On x-y cross section:
0
)(sin10
0 xR
R




, daR  2, (11.2a)
0
22
0
2
0RYX. (11.2b)
On HO1cross section:

0
)(2
cossin
11
2
2
2
2
1
1


xZ
dttbta




, 22 ab , (11.3a)
2
1
1
cos a
X
 
, (11.3b)
01
2
2
1
2
2
1

















b
Z
a
X, 22 ab . (11.3c)
In Figure 11.1a, the triangle 21'ODO related angles
are:
0
0
0
tan X
Y
 
, (11.4a)
0
0
0
tan Xd
Y

 
, (11.4b)
00 180
 . (11.4c)
000





. (11.4d)
In Figure 11.1b, the triangle 211 'OOB related angles
are:
1
1
1
tan X
Z
 
, (11.5a)
dX
Z


1
1
1
tan 
, (11.5b)
11180
  , (11.5c)
111





. (11.5d)
The generalized AT- and PS-equations of (10.7) are
applicable to all quarks except the top quark. The top
quark’s model must have 1'2 da to qualify as the
spindle type torus. The f-modification reduces 1
2

da
to 15.0'2 da , that is not valid for spindle type torus.
The effectiveness of f-modification for top quarks is lim-
ited to the 1'2da part, which does not includes the
root for the AT-equation.
Before going further, one question must be answered:
How many quarks are there?
Postulation 11.1: Quarks with the same flavor and
different colors are different elementary particles. There
are eighteen quarks in three generations.
Explanation: Elementary particles are distinguished
from each other according to their different intrinsic pa-
rameters. Quarks with the same flavor and different col-
ors have at least two different intrinsic parameters: one is
color and the other is mass. To recognize them as differ-
ent elementary particles is inevitable and legitimate.
According to Postulation 11.1, there are eighteen dif-
ferent quarks instead of six, in which six flavors each has
three colors as shown in Table 11.1. Postulation 11.1 has
important impacts beyond quarks, which will be shown
in later sections.
Postulation 11.2: Prime Numbers Postulation. Pri-
me numbers are intrinsically correlated to elementary
particles’ parameters as well as cosmic space structure
and cosmic evolution.
Explanation: Prime Numbers Postulation serves as

Z. Y. SHEN 1251
the second fundamental postulation with importance next
to the first fundamental postulation of Gaussian probabil-
ity. It provides a principle. The details are given by cor-
responding rules.
Definition 11.1: A pair of two consecutive odd prime
numbers with average value equal to even number is de-
fined as an even pair. A pair of two consecutive odd
prime numbers with average value equal to odd number
is defined as an odd pair.
The numerical m-parameters of 18 quarks are selected
by the following rule.
Rule 11.1: The eighteen least odd prime numbers in-
cluding 1 are assigned as the m-parameters of eighteen
quarks as shown in Table 11.1. The m-parameters of
eighteen quarks are paired of up-type and down-type for
each color. All nine pairs are even pairs.
Table 11.1. 18 Prime numbers assigned to 18 quarks m-
parameters*.
*The m-parameters listed are their magnitude; the signs are defined by
(11.6).
Conclusion11.1: There are only three generations of
quarks.
Proof: As shown in Table 11.1, for the nine pairs of
quarks in three generations, their m-parameters: 1 & 3, 5
& 7, 11 & 13, 17 & 19, 23 & 29, 31 & 37, 41 & 43, 47 &
53, 59 & 61 all are even pairs. The next prime numbers
pair of 67 & 71 is not an even pair, which violates Rule
11.1. The fourth generation quarks are prohibited based
on the Prime Numbers Postulation and the prime num-
bers table. QED
In fact, no quarks beyond three generations have found
in experiments.
The numerical parameters n and p of quarks are se-
lected in the following rules.
Rule 11.2: The quarks’ n-parameters are selected from
prime numbers. The values of quarks n-parameter are
closely related to strong interactions among them, which
will be discussed in Section 13.
Rule 11.3: For a quark, the p-parameter is determined
by e
MMnp // , in which,
M
and e
M are the mass
of the quark and the mass of electron, respectively. The
ratio mp /2 equals to an integer.
The reasons for such rules will be explained later.
Definition 11.2: The signs of numerical parameters m,
n, p for fermions and anti-fermions with different hand-
edness are defined as:
Fermion with right handedness:
0m, 0n, 0p, (11.6a)
Fermion with left handedness:
0

m, 0n, 0p, (11.6b)
Anti-fermion with right handedness:
0m, 0

n, 0p, (11.6c)
Anti-fermion with left handedness:
0

m, 0n, 0p. (11.6d)
Explanation: According to definition 11.2, for all four
cases, the ratios np / for mass are always positive as
they should be. Loop ratios are different: 0/ mn for
fermions and 0/

mn for anti-fermions, which serve
as the mathematical distinction for fermions and an-
ti-fermions. For all fermions, 0m represents right
handedness, and 0

m represents left handedness.
The verifications and applications of Definition 11.2
will be given later.
The geometry parameters of quarks calculated by us-
ing above formulas and rules are listed in Table 11.2. In
which, for up, down, strange, charm, bottom quarks, the
parameters with the ‘mark are effective, i.e. after the
f-modification, and the parameters without the ‘ mark are
original, i.e. before the f-modification. All parameters for
top quarks listed in Table 11.2 are original.
Table 11.2. Calculated parameters for 18 quarks*.

Z. Y. SHEN 1252

Z. Y. SHEN 1253

Z. Y. SHEN 1254
*All data are from 16-digit numerical calculations, only 8-digit after the
decimal point is presented;**Except n/m and p/n, all other parameters in
quarks summary are average value of three colors.
The mass values for six quarks as the average values
of three colors for each flavor listed in Table11.2 are all
within 2010-PDG data error ranges. The PDG data are
not from direct measurements; they are extracted from
experimental data of baryons made of quarks. So the
agreements are indirect.
The three inner angles of the triangle 21 ''' ODB for six
quarks are listed in Table 11.3, which is averaged over
three colors for each flavor cited from the summary table
of Table 11.2.
Table 11.3. Three inner angles 1

, 1

, 1

of triangle 21 ''' ODB.
According to 2010-PDG (pp. 146-151)experimental
data, in the Cabibbo-Kobayashi-Maskawa (CKM)
triagles the three inner angles of the unitarity triangle are:



2.4
4.4
0.89

 
, (11.7a)



879.0
904.0
15.21

 
, (11.7b)



25
22
73 

 
. (11.7c)
Other five CKM-triangle all are elongated.
Comparing Table 11.3 to 2010-PDG data shows close
similarities:
1) The 21 ''' ODB triangle of up quark is very close to
the unitarity triangle given by (11.7). In fact, the SQS
theoretical values of two angles 1  and 1  are within
PDG data error ranges. The relative deviation of

922.0900584
1

from 2010-PDG medium value

15.21 
is -2
104.3  at its error range’s upper edge.
2) The experimental data show that, except for the un-
itarity triangle of (11.7), five other CKM-triangles are
elongated. In Table 11.3, except for 21 ''' ODB triangle of

Z. Y. SHEN 1255
the up quark, other four quarks’ 21 ''' ODB triangles are
elongated and the one for top quark is not valid.
3) Required by unitarity of probability, the side be-
tween angle

and angle

of CKM-triangle is nor-
malized to unity. The side ''2DO of triangle 21''' ODB
is normalized to unity for the other two sides represent-
ing probabilities.
According to SQS theory, there are fifteen 21 ''' ODB
triangles comparing to five CKM-triangles for five fla-
vored quarks except the top quark. This difference may
provide an important clue for the question regarding
CKM-triangle: Is the unitarity CKM-triangle really a tri-
angle? This is a serious question. If the answer is no, the
standard model must be revised. As shown by (11.7), two
angles

and

have large error ranges, and the sum
of three inner angles medium values equals to

15.183 instead of 
180 . From SQS theory standpoint,
the problem can be naturally resolved by recognized the
fact that, there are eighteen quarks with different flavors
as well as different colors. As a result, the unitarity
CKM-triangle isn’t a single triangle, it is a set of three
triangle corresponding to three different colored quarks
r
u, g
u, b
u. As listed in Table 11.2, three up quarks
r
u, g
u, b
u have

61137029.88
,1  ur

,
37440598.70
,1  ug

,
77604761.56
,1  ub

,
respectively. The large error range of 252273 

give by (11.7c) is the result of attempting to combine
three different triangles into one. The same argument is
applicable to angles

and

. So the large error
ranges of CKM-triangle data have a reasonable explana-
tion based on Postulation 11.1.
There are other reasons to identify triangles 21 ''' ODB
as the CKM-triangles. Quarks are represented by their
torus models and characteristic points carry information
from the S-equation to torus model. In principle, all pa-
rameters including the CKM-triangles should be derived
from the model. Moreover, if the angles are kept the
same, the triangles are similar. As one side is normalized,
the other two sides of the similar triangles also represent
the same information. In this way, the converting prob-
abilities among different quarks via weak interactions
indicated by the other two sides of the CKM-triangle
should be transferred to the 21 ''' ODB triangle as well.
For all these reasons, the 21 ''' ODB triangles are identi-
fied as the CKM-triangles. It is another step towards the
final goal: All physics parameters of an elementary parti-
cle are derived from its model.
The generalized AT- and PS-formulas of (10.7) are
used to calculate the angle tilt and phase sync data for
fifteen quarks listed in Table 11.4. Three top quarks are
excluded, because for them the f-modification is not fully
applicable. The data for three charged leptons are listed
for comparison.
Table 11. 4. Phase sync data for 15 quarks and 3 charged
leptons*.
*1.The data are from 16-digit numerical calculations. Only three effective
digits are listed. 2. The listed a f vary in -15
101  steps within range of
-15
10100  .
The features of these results are summarized as fol-
lows:
1) Electron, three up quarks and three down quarks
have perfect phase synch among two loops’ cyclic
movements and the sinusoidal oscillation of the mass
term indicated in Table 11.4 as “PS values at 0

AT ”
equal to zero. Their angle tilt equation (10.7a) and phase
sync equation (10.7b) are satisfied simultaneously. The
perfect synchronization is interpreted as electron, up
quarks and down quarks are stable fermions. In fact,
these three types of particles are stable and serve as the
building blocks of all atoms and molecules in the real
world.
2) The other particles listed in Table 11.4 namely
muon, taon, and strange, charm, bottom quarks are not
perfectly synced indicated by their “PS values at 0

AT ”
equal to nonzero values. According to the same reason, it
can be interpreted as they are not stable particles. In fact,
muon, taon, and all hadrons composited with strange,
charm, bottom quarks are unstable and subject to decay.
3) All fifteen quarks and three charged leptons have
fluctuation phase variations noted as the “PS value varia-
tion” in Table 11.4. It means that all these particles have
the trajectory jumping behavior similar to electron’s tra-
jectory jumping behavior described in Section 8.
Formulas of (8.38) are used to calculated the tilted an-
gle


deviated from 
120 
. The

 data along
with 2
a, 2 'a and a
f for three charged leptons and
fifteen quarks are listed in Table 11.5. Three top quarks

Z. Y. SHEN 1256
are excluded, because the f-modification is not fully ap-
plicable.
It is interesting to find out that, for the fifteen quarks
despite of their more than three orders of magnitude mass
differences, the values of  00001.053990.0 

are
within 5
10  degree, which corresponds to the values of
   120 within the same range. This is possible
because despite their very different mass and 2
a values,
the f-modification is capable to bring back the 2
'a val-
ues within a very narrow range of
0000001.04918172.0'2a.
These results are related to the )3(SU group symmetry
associated with quark’s flavors and colors, which will be
discussed in Section 22 and Section 24.
Table 11.5. Calculated a
f, 2 'a,

 data for 3 charged
leptons and 15 quarks*.
*The data for leptons are based on trefoil type model in Section 12.
The results shown in Table 11.4 and Table 11.5 indi-
cate that, even though the AT- and PS-equations are ad
hoc equations, they catch the essence of these particles.
Postulation 11.1 is important for SQS theory. To rec-
ognize quarks of same flavor with different colors as
different particles plays pivotal roles in many areas.
There are at least two facts to support Postulation 11.1.
As mentioned previously, the large error ranges of

and

for the unitarity triangle shown in (11.7) can be
explained naturally by three up quarks with different
colors as three particles instead of one. It serves as evi-
dence. The other evidence is quarks mass values. As
shown in the PDG data book, most of the weighted av-
erage curves for quarks’ mass have more than one peaks
corresponding to a flavored quark made of mul-
ti-components with different mass values. According to
Postulation 11.1, the multi-peak behavior corresponds to
quarks with the same flavor and different colors having
different masses. Moreover, compared to the 2008-PDA
data, the 2010-PDA data show more evidences of mul-
ti-peak behavior for quarks mass curves. This argument
is also supported by other evidence. In the PDG data
book, most weighted average mass curves for hadrons
made of quarks (anti-quarks) with different flavors show
similar multi-peak behavior as they should be. Quarks
with different flavors having different mass values are
recognized as different elementary particles, with the
same reason, so are quarks with different colors having
different mass values.
Experiments found that, a hadron is composed of
point-like constituents named “partons”. There are three
valence partons identified as three quarks, u, u, d as the
constituents of proton. According to Postulation 11.1,
proton is composed of nine quarks: r
u, g u, b u for an
u quark, r
u, g
u, b u for another u quark, r d, g
d, b
d
for the d quark. The question is: How the nine quarks
show up in a proton? There are two possible options.
Option-1: There are three smaller point-like constitu-
ents inside a valence parton simultaneously. If this is the
case, a flavored quark’s mass equals to the sum of three
constituents quarks. It is contradictory to fact that, as
shown by quark multi-peak weighted average mass curve,
a flavored quark’s mass equals to the average of con-
stituents’ mass. So this option is ruled out.
Option-2: For a quark with the same flavor and dif-
ferent colors such as r
u, g u, b
ueach one takes turns
to show up. At a given time, only one out of three shows
up. A flavored quark’s mass equals to the average of its
three constituent colored quarks’ mass. It fits the mul-
ti-peak weighted average mass curve well. This option is
accepted. But it raises a question: Does each colored
quark show up with different time intervals? If the an-
swer is yes, then the flavored quark’s mass equals to the
weighted average of three constituents mass. In this way,
the average mass for favored quark and the theoretical
value 
922.0900584
1

listed in Table 11.3 should be
re-calculated to include the weighting factors. The results
with weighting factors proportional to the reciprocal of
three colors’ mass values are as follows.
Weighted up quark mass value:
2
/3276313.2 cMeVMu, (11.8a)

Z. Y. SHEN 1257
Weighted up quark 1

value:

93059933.21
1

. (11.8b)
Both results are within 2010PDG data error ranges.
The importance of Postulation 11.2 and Rule 11.1 has
been shown by Conclusion 11.1. In fact, Postulation 11.2
as the second fundamental Postulation of SQS theory has
many important impacts far beyond quarks, which will
be given in later sections.
Section 12. Trefoil Type Model for Charged
Leptons
In this section, a broad view is taking to look at leptons.
Based on Prime Numbers Postulation and intrinsic rela-
tion between leptons and quarks, a new type of model
with torus as building blocks is introduced for charged
leptons.
In Section 11, nine even pairs of prime numbers are
assigned as the m-parameters for nine pairs of up type
and down type quarks as listed in Table 11.1.
Postulation 12.1: The original (before reduction)
m-parameter of a lepton is an even number equal to the
average value of the m-parameters of associated up type
quark and down type quark.
Explanation: In fact, this is the unstated reason in
Section 8 and Section 10 to select 2, 18, and 42 for the
original m-parameters of electron, muon and taon, re-
spectively.
22/)31(2/)( drure mmm , (12.1a)
182/)1719(2/)(  srcrmmm

, (12.1b)
422/)4143(2/)( brtr mmm

. (12.1c)
According to Postulation 12.1, the results for six lep-
tons are listed in Table 12.1. The m-parameters of eight-
een quarks are also listed for reference.
Table 12.1. The Leptons and quarks with assigned
m-parameters*.
*The m-parameters are their magnitude; signs are defined by (11.6).
Conclusion 12.1: There are only three generations of
quarks and leptons. The fourth generation is prohibited.
Proof: In the “End” column of Table 12.1, the average
of two m-parameters, 67 & 71, is an odd number:
692)7167(



. According to Postulation 12.1, the
fourth generation leptons are prohibited. According to
Conclusion 11.1, the fourth generation quarks are pro-
hibited. QED
Conclusion 12.1 is the extension of Conclusion 11.1
based on the Prime Numbers Postulation and the intrinsic
relation between quarks and leptons.
On the experiment side, according to 2010-PDG data,
the number of light neutrino types from direct measure-
ment of invisible Z width is 05.092.2 . The number
from ee colliders is 0082.09840.2 . Both results
show no trace of fourth generation neutrino existence.
These experimental data support Conclusion 12.1.
Notice that, there are vacant cells marked with “?” in
Table 12.1. The question is: Are there any undiscovered
leptons? In the three generations, there are twelve lepton
vacancies, in which six are e,

,

type, and the other
six are e

,


, 

type. If these vacancies correspond
to undiscovered leptons, the six e,

,

type would be
charged leptons with mass ranging from a few 2
/cMeV
to a few thousands 2
/cMeV . That is impossible, because
charged particles in such mass range should be discov-
ered already. The neutrinos e

, 

,


are intrinsi-
cally associated with their companions leptons, e,

,

respectively. If there are no undiscovered charged lep-
tons, so are no undiscovered neutrinos associated with
them.
To fill the vacancies with undiscovered leptons isn’t
the only way. The other way is that, these vacancies
serve as a hint for new structure of existing leptons.
The first generation fermions are divided into four
categories including two types of leptons e and e

, and
two flavors of quarks each with three colors, r
u, g
u,
b
u and r
d, g
d, b
d. The second and third generations
have the same structure. Should leptons also have colors?
This is the initial thought inspired by the vacancies in
Table 12.1.
The basic idea is that, leptons’ new model has three
branches. Each branch separately is a torus model. The
three branches combine to form the new model.
Leptons’ torus model has spin 2/. The new model
made of three torus should also have spin 2/. There
are two options to deal with the spin problem.
Option-1. Let two branches have spin 2/, and one
branches has spin 2/

. The sum of three branches spin
is 2/)2/(2/2/ 




. But this option makes the
new model lost three-fold circular symmetry. More seri-
ously, the opposite spin in one branch abruptly reverses
loop-1 movement direction, which violates the require-
ment for smooth trajectory. It is not acceptable.
Option-2. Let each branch has spin 6/. It can be
done by selecting the reduced m-parameter 3/1

m for
each branch. According to SQS theory, the lepton’s spin

Z. Y. SHEN 1258
equals to 2/m. For the new model as a whole entity,
the reduced m-parameter add up to
13/13/13/1 

m
corresponding to the spin 2/. This option is accepted
Next step is to find out how the three torus branches
and three trajectories are combined. According to Pen-
rose [12], there are two types of topological structures
with three branches. The trefoil-knot-type shown in Fig-
ure 12.1(a) is a single loop self-knotted to form a trefoil
structure. It fits the job to combine three trajectories on
three torus models into one trajectory on the trefoil type
model. The Borromean-ring-type structure shown in Fig.
12.1(b) is irrelevant to leptons model, because its three
loops do not combine into one.
In Figure 12.2, the three loop-1 circles shown by
dot-dashed lines touch each other tangentially from one
circle to the other circle with continuous first order de-
rivatives. In this way, loop-1 goes smoothly from one
branch to the other. The total length of combined loop-1
equals precisely the sum of three branches’ loop-1
lengths representing 2/6/6/6/ h  spin for
electron as a whole entity.
Figure 12.2 shows how the three branch trajectori-
escombined into a trefoil trajectory. As mentioned in
Section 8, on the electron torus surface, point-A and
point-B in Fig.8.2 actually represent two circles, circle-A
and circle-B. A trajectory may start at a point on circle-A
and halfway through at a point on circle- B to keep the
angle BAO1
:
 98687309.166180
1
 BAO. (8.18)
This rule is originated from the S-equation and strictly
related to)( 1
x

, )( 2
x

to determine curvatures of the
torus model. To construct the trefoil trajectory, (8.18) is
used to determine the location of point-B from the loca-
tion of point –A for each branch.
Figure 12.1. Three-branch patterns: (a) Trefoil-knot-type;
(b) Borromean rings type.
The other rules for the trefoil trajectory are:
1) The trefoil trajectory must go through points-A and
point-B of three branches to satisfy the requirements of
)(1 x

and )( 2
x

for each branch.
2) The trajectory is the geodesics between adjacent
point-A and point-B on trefoil type model surface.
3) The three branches of trefoil trajectory have the
same shape separated by 
120 for the 3-fold circular
symmetry.
In Figure 12.2, the trajectory on top surface is shown
by solid curve and on bottom surface is shown by dashed
curve. Electron’s trajectory goes anti-clockwise through
six characteristic points and back to close one cycle:
rrgbbrggbr ABBABBABBA  . (12.2)
Indeed, the trajectory is a trefoil type closed loop with
the correct topological structure and the 3-fold circular
symmetry.
The Weinberg angle 428.4794845'

W

is the same
for all three branches as well as for electron as a whole
entity. It needs explanation. As mentioned in Section 8,
Weinberg angle is a phase shift between loop-1 and
loop-2 periodic movements:
-222






W. (12.3)
For the trefoil trajectory, 2



W repeats three times
at three locations, r
A, g
A, b
A. The repetition means
the same phase shift kept no change along trajectory at
three locations. Therefore, the three angles should not be
added up toW

3. Look at it the other way, the combined
trajectory is the same one on the original genus-1 torus
surface, which is reconfigured to fit the genus-3 manifold.
The combined trajectory has one Weinberg angle
428.4794845''2

 
Wcorresponding to the charge of
e for electron.
Figure 12.2. The x-y plane cross section of electron trefoil
type model and trefoil trajectory projec tion on x-y plane.

Z. Y. SHEN 1259
The trajectory shown in Figure 12.2 is a samples se-
lected from two sets of discrete possible trajectories. The
jumping trajectories described in Section 8 for electron
torus model are also valid for the trefoil type model. As
long as the trajectories meet all rules, they are legitimate.
In other words, the “electron clouds” is also a visualized
description of electron behavior for the trefoil type model.
The same is true for the trajectories on trefoil type mod-
els of muon and taon.
Introducing the trefoil type model solves the vacancies
problem in Table 12.1. Table 12.2 shows the vacancies in
Table 12.1 are filled with leptons’ branches.
Table 12.2. The m-parameters of quarks and leptons with 3
branches*.
*The m-parameters listed are their magnitude; their signs are defined by
(11.6). **The number in parenthesis is the reduced m-parameter.
For three generations of charged leptons, the formulas
given by (12.1) of Postulation 12.1 are generalized for
the original m-parameters (before reduction) of trefoil
type model’s each branch and as a whole entity based on
the original m-parameters of corresponding up type quark
ji
quptype
m,
,and down type quark ji qdowntyp
m,
,.
For each branch:






23</