Unified Theory of Particles－ Symmetry of Spinor and Source of Mass －SEIKO KONO河野 誠公 著

Introduction

Assume quark and lepton as matter elementary particle, among
three interaction with them and various (gauge) boson, describe
electromagnetic and weak interaction in electro-weak (EW) gauge
theory integrally and compile a theory to describe strong interaction in
QCD gauge theory, and which is called a standard theory (model). In
this theory, the mass that an elementary particle has been thought tohold it essentially is explained for quantity provided secondarily by
spontaneous symmetry breaking. (Higgs mechanism) However, only by
this Higgs mechanism, not only the mixing phenomenon of quark threegenerations and neutrino vibration cannot be explained, but also theregularity of each mass of quark, lepton three generations cannot befound at all. Furthermore, because the Einstein gravity field that is
another interaction field has difficulty in quantization, it cannot be
incorporated in a framework of standard theory as gauge theory. On theother hand, energy scale ~1015 GeV of the grand unified theory (GUT)to try unification of three interaction except the gravitational
interaction, and the Planck scale ~1019 GeV that the unification of thegravitational interaction is expected, it is thought that there is thehierarchy problem that how can maintain naturally in the big
hierarchy characteristics with those scales and energy scale ~102 GeV ofthe electro-weak theory. 1), 2), 11)
In this book, discuss the process to form each field that each matterelementary particle and boson leave the thermal equilibrium state thatis a vacuum and is released in the 4D (4-dimensions) outer spaceaccording to the space expansion in the space formative period, anddevelop “the unified theory of particles” to describe all known
elementary particles and four interaction integrally. By this unifiedtheory, show that it gives one solution or interpretation for most of

ii
unsolved problems that the standard theory has.
This book is constructed in 2 parts. In PartⅠ, review process that thestandard theory had been built and reconfirm each unsolved problem tohave. At first in Chapter 1 and 2, review about Lorentz transformation
led by the principle of relativity which is essential in describing theparticle equation of motion, the equation of motion in free field led by
global symmetry of space-time and local symmetry appearing in eachinteraction. In Chapter 3, look back toward the electro-weak (EW)
theory being the nucleus of the standard model. In Chapter 4 and 5,
review process to lead to a quark model and constitution of the hadrons,
and confirm problem of the quark confinement and problem of theorigin of hadrons mass. In Chapter 6, look back on the derivation of theEinstein gravity field and confirm ground that the quantization isdifficult. In Chapter 7 of the last of partⅠ, summarize to refer todocuments (1),(2),(11), etc. unsolved problems that the standard theory
has.
In PartⅡ, develop “unified theory of particles” based on “new particle
theory beyond standard model”15),16), “new cosmic formation” 12), “newquantum gravity theory and mass charge”21) and etc.17),19) which I
already published. At first, in Chapter 8, suppose a vacuum whichcauses whole space and has structure of D=2(n+1) dimensions spin
space expressed by combinations of n+1 unit helicity. Describe theformative period of space where 4D space-time which has three helicity
to a component in n+1 unit ones expands and all known elementary
particles are produced as some vacuum energy released in the outerspace. In Chapter 9, scalar field, vector field and metric tensor field arecreated in n = 0, 2, 4, and quark and lepton of the 1st, 2nd, 3rd
generations are created in n = 1, 3, 5(4) under the frame of 4D Lorentz
group concretely. In Chapter 10, each boson and matter field separates
and leaves the vacuum in the thermal equilibrium state according tothe space expansion and the change of each energy density. And discussa process forming those each field. In Chapter 11, give newinterpretation for the color enclosure with the gluon field composed of

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9-4 section. In Chapter 12, take up a problem in the origin of nucleon
spin and show that this problem is solved by the hierarchy structure ofquark three generations composed of 9-3 section. In Chapter 13,
perform the quantization by the operational calculus for metric tensor
field to be comprised of two connections, spin and affine connections
composed of 9-5 section. Because the Lagrangian density of thisquantum metric tensor field is given with the sum of Lagrangiandensity of two connection gauge fields, it is not necessary to treat
differential term of the metric tensor in the Feynman diagram directly,
and can treat it like the effective field theory that is renormalizable. In
Chapter 14, perform the BRS conversion that is quantization by thepath integral calculus for this metric tensor field. Then the relation ofgauge phase of two connection gauge fields is led. A new charge “massunit-charge” which gives the hierarchy structure of the mass of three
generations matter particles can be constructed by composition withthe phase difference of two connections and the helicity which matterdoublet exchanges in the interaction. In Chapter 15, lead mass
expression of quark and lepton doublet of three generations using this“mass unit-charge” and identify each mass. In Chapter 16, apply toquark three generations mixing and neutrino vibration phenomena.
Each generation mixing is available to explain integrally as aphenomenon to exchange a “mass unit-charge” which each quark or
neutrino of three generations holds, and show that each mixing angle isidentified by the exchange probability amplitude defined using a massunit-charge. In Chapter 17, apply it to the mass of hadrons which are
the composite particles of quarks. Using “the mass charge” that a quark(u, d, s) constitutingπ, K meson and nucleon, Λ particle holds, lead
each mass expression and show that the mass of each hadron can be
identified with high precision.
Finally, in Chapter 18 and 19, summarize “unified theory of particles”
developed with partⅡ and unsolved problems of standard theory
resolved by this unified theory. In Chapter 18, summarize unified
expression of three generations matter particle and boson field using

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the spinor expression at first, and summarize unified expression of fourinteraction expressed by Yukawa coupling of the direction restoring
supersymmetry. In 18-4 section, each quantum number to characterizeeach elementary particle and each interaction is expressed using spinor
symmetry (symmetry of helicity). In Chapter 19, summarize solutions
or interpretation provided by this unified theory for unsolved problems
summarized in Chapter 7 which the standard theory has. In 19-2
section, show problems to still remain as non-solution and problems toproduce by this unified theory newly. I hope that it is helpful for thegrowth of the future theory of elementary particles and cosmology.

January, 2020 (in Japanese)
December, 2020 (translated into English)
Seiko Kono
(河野 誠公)v
Contents

PartⅠ Standard Theory (Model) and Unsolved Problems
Chapter 1
1-1
1-2
1-3

Symmetry of 4D Space-Time and Lorentz Group ・・ 1
Principle of Relativity and Lorentz Transformation
Creation Operator of Lorentz Group
Lorentz Invariant and Helicity

Chapter 2 Energy-Momentum Conservation and Wave Equation

・・・・・・ 16

2-1
2-2
2-3
2-4
2-5

Klein-Gordon Equation and Dirac Equation
Angular Momentum Conservation and Spin Operator
Maxwell Equation of Electro Magnetic Field
Electro Magnetic Interaction and Gauge Symmetry
Generator of Local Conversion Group and Electric Charge

Chapter 33-1
3-2
3-3

Weak Interaction and Electro-Weak (EW) Theory ・・ 28
Weak Interaction and Current
Gauge Theory of Weak Interaction
Electro-Weak (EW) Theory and Higgs Mechanism

Chapter 44-1
4-2
4-3
4-4

Strong Interaction and Color Enclosure ・・・・・ 38
Experiment of Electron-Positron Annihilation
Experiment of Electron-Nucleon Inelastic Scattering
Quark Model and Structure of Hadrons
Quark Confinement and Anti-Dielectricity of QCD

Chapter 55-1
5-2
5-3

Problem of Mass Origin of Hadrons
Chiral Symmetry Breaking by QCD・・・・・・・ 49

Mass Formula of Meson based-on Chiral Perturbation Theory

Mass Calculation of Hadrons by Lattice QCD

vi
Chapter 6
6-1
6-2
6-3
6-4
Chapter 7
7-1
7-2
7-3
7-4
7-5

Einstein Gravity Field and Quantization Problem ・・ 63
From Special Relativity to General Relativity
Gauss Theory of Curved-Surface and Riemannian Geometry

Einstein Gravity Field
Quantization Problem of Einstein Gravity Field
Standard Theory and Unsolved Problems

・・・・ 78

Mixing Phenomena and Mass Irregularity of Matter Particles

Renormalization Group and Grand Unified Theory (GUT)
Quantization of Gravity Theory and Planck Mass
Mystery of Cosmology and Relation with Particle Theory
Unsolved Problems ( Summary )Part Ⅱ Unified Theory of Particles
Chapter 8 Creation of 4D Space-Time and Release ofVacuum Energy
・・・・・・・・・・・・・・・・ 88
8-1 Appearance of Time-axis and Expansion Space
8-2 Two Vacuums and Dark Energy
8-3 Release of Vacuum Energy（Inflation Period）
8-4 Beginning of Radiation Dominated Era
Chapter 9 Particles Creation and Spinor Expression ・・・・ 101
9-1 4D Lorentz Group and Symmetry of Spinor
9-2 Definition of Vacuum and Creation of Scalar Field(Higgs Field)
9-3 Creation of Three Generations Matter Particles(Quark, Lepton)
9-4 Creation of Vector Boson (Photon, W-Boson, Gluon)
9-5 Creation of Metric Tensor Field

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Chapter 10 Cosmic Expansion and Formation of Each Elementary
Particle Field・・・・・・・・ 116
10-1 Change Ratio of Energy Density of Each Boson Field
10-2 Change Ratio of Energy Density of Matter Field
Chapter 11 Formation of QCD Field and Color Enclosure ・・ 122
11-1 4D Space-Time Constitution and Color Enclosure11-2 Quantum Color Dynamics (QCD) and Color Enclosure
Chapter 12 Origin of Nucleon Spin and Hierarchy Structureof Three Generations Quark
・・・・・・・・ 129
12-1 Ratio of Quark Spin in Nucleon Spin12-2 Interpretation based on Spinor Expression of Quark
Chapter 13 Quantization of Metric Tensor Field and Renormalizable・・・・・・・・・ 133
13-1 Canonical Quantization of Metric Tensor Field
13-2 Derivation of Einstein-Gravity Field by Classic Approx.
13-3 Renormalization of Quantum Metric Tensor Field
13-4 Origin of Coupling Constant（Gravitational Constant）
Chapter 14
14-1
14-2
14-3

Construction of “Mass Charge”
・・ ・・・・・ 144
Quantization of Metric Tensor Field by BRS-Conversion
Gauge Phase Difference of Spin and Affine Connections
Construction of Mass (Unit-) Charge

Chapter 15 Mass of Three Generations Matter Particles(Quark, Lepton)
・・・・・・ 154
15-1 Mass Expression of Three Generations Matter Doublet
15-2 Constitution of “Mass Charge” of 3G Matter Doublet
15-3 Comparing with Mass Database of Each Particle
15-4 Identifying Coupling Constantviii
Chapter 16 Three Generations Mixing of Quark, Neutrino andMixing Angle・・・・・・ 173
16-1 Expression of 3G Mixing Matrix using “Mass-Charge”
16-2 Quark Three Generations Mixing and Mixing Angle
16-3 Neutrino Vibration and Identifying Mixing Angle
Chapter 17 Identifying Hadrons Mass・・・・・・・・・・ 183
17-1 Mass Expression of π, K-Meson and Identifying Mass
17-2 Mass Expression of Nucleon, Λ-Particle and Identifying
MassChapter 1818-1
18-2
18-3
18-4

Unified Theory of Elementary Particles ・・・・ 190
Unified Expression of Three Generations Matter Field
Unified Expression of Boson Field
Interaction of Matter Field and Boson Field
Unified Expression of Quantum Number using Helicity

Chapter 19 Resolutions for Unsolved Problems (Summary) ・ 203
19-1 Resolution and Interpretation based on Unified Theory19-2 Unsolved Problems and Problems to arise Newly
References

・・・・・・・・・・・・・・・・・・・・・・ 212

Appendix A
Appendix B
Appendix C
Appendix D
Appendix E
Appendix F

Spinor algebra and Super-symmetry algebra ・・ 214
Gravity Gauge Theory ・・・・・・・・・・・・ 219
Spinor expression of the 3rd gen. Matter field ・・ 221
Database of Quark three gen. Mixing Matrix ・・ 224
Estimate of Vacuum Energy Densityρv ・・・・ 225
Comparison results with Anisotropic Observation of・・・・ 229
Microwave Background Radiation

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PartⅠ Standard Theory (Model) and Unsolved Problems
Chapter 1 Symmetry of 4D Space-Time and Lorentz Group
１－１．Principle of Relativity and Lorentz Transformation 3), 4), 21)
All natural rules are the same in every inertial frame of reference
when they obey the theory of relativity that is a law learned byexperience. In other words, the equation representing the natural rule
is invariant in coordinate transforming from one inertial frame ofreference to other inertial ones.
The interaction between the material particles was expressed as afunction of the coordinate of the particle which interacted. It was
implied that the interaction carried out instantaneously. However, weknow empirically that there is no interaction which carries out
instantaneously and a certain limited time is necessary. It is called
propagation velocity of interaction that a distance between two
interacting particles is divided by its finite time. This propagationvelocity must be called the maximal velocity of the interaction and be
the same in all inertial frames of reference, according to the theory ofrelativity. This propagation velocity is a universal constant and isregarded as the speed that light is transmitted through the vacuum
namely light velocity. 3)
In the Galileo’s principle of relativity which follows the principle ofrelativity, the interaction occurred instantaneously. In the Einstein’sprinciple of relativity to distinguish from it, finiteness of the light
velocity which is propagation speed of the interaction is combined withthe principle of relativity.
In the classical mechanics, space is relative, however, time is absolute.
However, when obey to the composition law of the velocity vector, thevelocity of the compositing motion becomes the vector sum of eachvelocity of the motion to constitute it. As this law should be applied to2propagation velocity of the interaction, the propagation velocity will bedifferent in the different inertial frame of reference. This completely
contradicts the Einstein’s principle of relativity. By the measurement of
the light velocity by Michelson (1881), it was confirmed that the light
velocity did not depend on its spread direction, and the Einstein’s
principle of relativity was confirmed. 3) It was concluded that time is not
absolute and there is inherent time in an inertial frame of reference.
（Inherent Time and Lorentz Transformation）3)
In inertial frame of reference, consider to observe a clock moving any
activity for static frame. The ‘moving clock’ advances only to distance√dx2+dy2+dz2 during minute time interval dt measured by a clock inthe static frame. As this clock is static state in the coordinate (x’, y’, z’, t’)of ‘moving clock’, it is dx’ = dy’ = dz’ = 0. By invariance of element ds ofthe length,ds2 = c2 dt2－dx2－dy2－dz2 = c2 dt’ 2(1-1)∴ dt’ = ds / c = 1/c √c2 dt2－dx2－dy2－dz2Here, assume velocity of ‘moving clock’υ,υ2 = (dx2 + dy2 + dz2 ) / dt2(1-2)

When substitute it for eq.(1-1),
dt’ = ds / c = dt√1－υ2 / c2 = dt√1－β2 , β≡υ/ c(1-3)

Thus, the length at the time that ‘moving clock’ ticks away is as follows,
while ‘clock in static state’ ticks only t2－t1.
t2t’2－t’1 = ∫ dt √1－β2t1(1-4)

This interval t’2－t’1 which ‘moving clock’ ticks is the inherent time ofthis frame.

3Then, suppose a coordinate transformation between two frames tomove at uniform velocity each other. Transform from frame K(x, y, z, t)
to frame K’ (x’, y’, z’, t’). The K’ frame moves in constant velocityυinforward direction of x-axis on the basis of K. This problem is easily
solved in the classical mechanics using the Galilei transformation;x’ = x +υt , y’ = y , z’ = z , t’ = t(1-5)

However, the relativistic transformation formula is expressed as arotation of the 4D-coordinate frame (x, y, z, t). Suppose the rotation of τ－x plane using a parameterτ= i ct. The y- and z-axis do not change by
this rotation. When takeθas rotation-angle, the relations between oldand new coordinates are expressed as follows;x = x’ cosθ－τ’sinθ, τ= x’ sinθ + τ’cosθ(1-6)

The coordinate transformation results in a problem to seek this rotation
angleθ.θdepends only on relative velocityυbetween frames K and K’.
Suppose movement of the origin of K’ in K frame. As it becomes x’ = 0,eq.(1-6) is
x = －τ’sinθx
τ= τ’cosθ∴ τ = － tanθ
(1-7)
As x /τ ~ x / t is velocityυof K’ for K frame clearly,
xix=tanθ= － τ =ct∴ sinθ=iβ√1－β2iυ= iβc, cosθ=(1-8)

1√1－β2(1-9)

When replace it withτ= i ct,τ’ = i ct’ and substitute eq.(1-9) for eq.(1-6),x=x’ +υt’√1－β2, y = y’, z = z’,

t=t’ +βx’/ c√1－β2(1-10)

4This eq.(1-10) is the formula of Lorentz-transformation.
The inherent length is obtained by eq.(1-10), in the same way theinherent time is given by eq.(1-4). A length of stick in static state isexpressed in ℓ0 = △x = x2－x1, and a length of stick of a frame K’ isexpressed in ℓ = △x’ = x’2－x’1.
ℓ0 = △x = x2－x1 ==x’2 +υt’x’1 +υt’－2√1－β√1－β2x’2－x’ 1√1－β2= ℓ /√1－β2∴ ℓ = ℓ0 √1－β2(1-11)

A length of stick becomes maximum in a static frame, and it shrinks√1－β2 times in a frame moving in velocityυ. (Lorentz-contraction)
Then, time and space are expressed to emphasize 4-dimensions (4D)
space-time as follows;
x0 = ct , x1 = x , x2 = y , x3 = z(1-12)
Element of the length ds that is invariant is expressed below for
Lorentz transformation when express the minute displacement of eachcoordinate direction in dxμ.
ds2 = ημνdxμdxν
(1-13)
here, η0 0 = +1, ηi i =－1, ημν= 0 (μ≠ν)
When the same subscript appears up and down, add it from 0 to 3.
The Lorentz-transformation of eq.(1-10) is expressed by using a matrix
as follows;
ct’x’y’z’=k －kβ－kβ k000000
10000
1

ctxyz

(1-14)

5here, k = 1 / √1－β2 , β = υ/ c
WhenΛμν represents component (μ,ν) of matrix, eq.(1-14) becomesx’μ = Λμν xν

(1-15)

Here, give below a 4D velocity vector;
μuμ ≡ c dx
ds(1-16)

When a particle is in static state, it is dx = dy = dz = 0. Using eq.(1-13),
0u0 = c dx = c √dx0 dx0 / ds2 = c √1 /η0 0 = cdsi
ui = c dx = 0ds

at i = 1, 2, 3(1-17)

When a particle moves in velocityυto x-direction, use the Lorentz
transformation eq.(1-14) in which the coordinate moves in velocity－υ,0u0 = c dx = c・k = c / √1－β2 , u2 = u3 = 0ds

u1 = cdx1ds(1-18)= c (－k・(－υ/ c )) = k・υ= υ/ √1－β2

4D-momentum pμ is obtained when multiply mass m by 4D-velocity
vector. This particle energy E is equal to 0-component p0 times c.
(1-19)

E = c p0 = mc u0 = mc2 / √1－β2

When particle velocityυis small enough for light velocity c (υ≪c),
2
E～～ mc ( 1 +1β2 ) = mc2 +21mυ22(1-20)

Kinetic energy of the Newton dynamics is provided in the 2nd term.