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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) Copyright © 2013 SciRes. JMP Stochastic Quantum Space Theory on Particle Physics and Cosmology —A New Version of Unified Field Theory Zhi-Yuan Shen Email: zyshen@comcast.net 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 1R. (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 25.0x 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 Copyright © 2013 SciRes. JMP 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 25.0x. It extends the region of 1xS and shrinks the region of 1xS . 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 Copyright © 2013 SciRes. JMP 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 pare 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 Copyright © 2013 SciRes. JMP 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,, 3zyxS. 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,, 3zyxS , 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,, 3zyxS 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 Copyright © 2013 SciRes. JMP 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,, 3zyxS , 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 Copyright © 2013 SciRes. JMP 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 1xS at 1 x trans- ports along a set of complex lines 1 and 2 to 2 x where probability having deficient 1 2xS. 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 2xS 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 Copyright © 2013 SciRes. JMP 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 13V 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 Copyright © 2013 SciRes. JMP 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)( 1x , the repulsive force pushes the event associated with its probability towards 2 x. When it ar- rived 2 xx where 1)(2x , 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 2n 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 1n 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 10dm, 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 25.0x 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 25.0x. -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 , 0xDS has two roots: 125.0 1x, 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 2x 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 Copyright © 2013 SciRes. JMP 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 1x, 375.0 2x, 5.0 e l, 1 e L. (6.17) At 125.0 1x, 375.0 2x, )(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 1xS 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 Copyright © 2013 SciRes. JMP 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, 1k is a root of 0kDSk. According to (7.5a) and (7.5b): e MM , t. (7.7) 0kDSk at 1k corresponds to electron as a fermion. 2) 0kDSk, 0k is a pole of kDSk. According to (7.5a) and (7.5b): 0M, t. (7.8) 0kDSk, 0k 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 10as 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 10558.2N 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 tof 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 Copyright © 2013 SciRes. JMP 1236 The effective Region-2 is divided into four sub-re- gions: Region-2a: 17181010i 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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: 2m, 1n, 1p. (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 1x, and 375.0 2x. (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 1d 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 1d, 5.0 2a. 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 1d, solving the two equations of (8.26) for 2 'a yields: 222154918171173.0'2a. (8.27) AT and PS are two independent equations with one un- known 2 'a. Both equations are satisfied simultaneously with the same solution 222154918171173.0'2a. 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 2a, 5.0 2 b, 222154918171173.0'2a, 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' 2b. (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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 10to 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 1246 ratio 1/1// npMM e are determined by a set of three numerical parameters, 2m, 1n, 1p. 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, 2k (12/ kk ) in kDSk is a root of 0kDSk represents electron as a fermion. In kEDSk, the1'j, 2k 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 10and 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), 0xSS . The val- ues of xSS fluctuate around 1715 10~10 and oc- casionally equal to zero, 0xSS , which are the roots of 0 xSS . As shown in Section 6, 0xDS is a real equation on the real x-axis. It has a root at 125.0 1x on the Z. Y. SHEN Copyright © 2013 SciRes. JMP 1247 x-axis corresponding to electron. On the other hand, 0xSS is a complex equation and 125.0 1x is not its root. Instead, a root of 0xSS is found by nu- merical calculations at: W i exx 11' , (9.6a) 78213151240811255.0'1x, (9.6b) 1384598708641.28 W . (9.6c) 1384598708641.28 W is electron original Weinberg angle of (8.45a) before f-modification. 78213151240811255.0'1x is slightly less than 125.0 1x. According to (6.19a), 78213151240811255.0'1x 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) 3/1F, 3/2F, 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 1x 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, 0xSS 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 Copyright © 2013 SciRes. JMP 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: 18m, 4 1 29n, 6048p; (10.2a) Taon: 42m, 120n, 417270p. (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 1x, 86904760.25060453 2x; (10.4a) Taon: 20526280.24996405 1x, 79473720.25003594 2x. (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)( 1x , 40639671394911815.3)( 2x ; (10.5a) Taon: 268531416714823.3)( 1x , 14241414262265.3)( 2x . (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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 0RYX. (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 1'2da 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 Copyright © 2013 SciRes. JMP 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: 0m, 0n, 0p, (11.6a) Fermion with left handedness: 0 m, 0n, 0p, (11.6b) Anti-fermion with right handedness: 0m, 0 n, 0p, (11.6c) Anti-fermion with left handedness: 0 m, 0n, 0p. (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, 0m 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 Copyright © 2013 SciRes. JMP 1252 Z. Y. SHEN Copyright © 2013 SciRes. JMP 1253 Z. Y. SHEN Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 0000001.04918172.0'2a. 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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 Copyright © 2013 SciRes. JMP 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</ |