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PROGRESS

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IN PHYSICS

2011 Volume 2

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APRIL 2011 VOLUME 2

CONTENTS

Daywitt W. C. The Lorentz Transformation as a Planck Vacuum Phenomenon in a Gali-lean Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Minasyan V. and Samoilov V. Charged Polaritons with Spin 1 . . . . . . . . . . . . . . . . . . . . . . . . . 7

Minasyan V. and Samoilov V. New FundamentalLight Particle and Breakdown ofStefan-Boltzmann’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Lehnert B. The Point Mass Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

Zhang T. X. Quark Annihilation and Lepton Formation versus Pair Production and Neu-trino Oscillation: The Fourth Generation of Leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Belyakov A. V. On the Independent Determination of the Ultimate Density of PhysicalVacuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27

Feinstein C. A. An Elegant Argument thatP,NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Daywitt W. C. The Compton Radius, the de Broglie Radius, the Planck Constant, andthe Bohr Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Shnoll S.E., Rubinstein I.A., Shapovalov S.N., Kolombet V.A., Kharakoz D.P.Histo-grams Constructed from the Data of 239-Pu Alpha-Activity Manifest a Tendencyfor Change in the Similar Way as at the Moments when the Sun, the Moon,Venus, Mars and Mercury Intersect the Celestial Equator . . . . . . . . . . . . . . . . . . . . . . . . 34

Khazan A. Electron Configuration, and Element No.155 of the Periodic Table of Elem-ents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Cahill R. T. Dynamical 3-Space: Cosmic Filaments, Sheets and Voids . . . . . . . . . . . . . . . . . 44

Ndikilar C. E. Black Holes in the Framework of the Metric Tensor Exterior to the Sunand Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52

Khazan A. Isotopes and the Electron Configuration of the Blocks in the Periodic Tableof Elements, upto the Last Element No.155. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55

Assis A.V. D. B. On the Cold Big Bang Cosmology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58

LETTERS

Minasyan V. and Samoilov V. Arthur Marshall Stoneham (1940–2011) . . . . . . . . . . . . . . . L1

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April, 2011 PROGRESS IN PHYSICS Volume 2

The Lorentz Transformation as a Planck Vacuum Phenomenon in a GalileanCoordinate System

William C. DaywittNational Institute for Standards and Technology (retired), Boulder, Colorado, USA

E-mail: [email protected]

In a seminal Masters’ dissertation [1] Pemper derived the relativistic electric and mag-netic fields of a uniformly moving charge from the response of some continuum tothe perturbation from the charge’s Coulomb field. The results seem to imply that theMaxwell equations and the Lorentz transformation are associated with some type ofvacuum state. Unbeknownst at the time, Pemper had discovered the Planck vacuum(PV) quasi-continuum [2] and its interaction with the free charge. The importance ofthis derivation, its obscurity in the literature, and its connection to the PV justifies thefollowing rework of that derivation.

1 Pemper Derivation

When a free, massless, bare charge e∗ travels in a straight lineat a uniform velocity v its bare Coulomb field e∗/r2 perturbs(polarizes) the PV [2]. If there were no PV, the bare fieldwould propagate as a frozen pattern with the same velocityand there would be no accompanying magnetic field. Thecorresponding force perturbing the PV is e2∗/r

2, where one ofthe charges e∗ in the product e2∗ belongs to the free chargeand the other to the individual Planck particles making up thedegenerate negative-energy PV.

This charge-vacuum interaction is described by Pemper[1] as a series (n = 1, 2, 3, . . .) of electric and magnetic fields(generated by the vacuum)

∇ × En = −1c∂Bn∂t

(1)

andBn+1 = ββ × En (2)

that respond in a iterative fashion to the bare charge’sCoulomb field, leading to the well-known relativistic elec-tric and magnetic fields that are traditionally ascribed to thecharge as a single entity. The serial electric and magneticfields are En and Bn and ββ = v/c. The curl equation in (1) isrecognized as the Faraday equation and the magnetic field in(2) is due to the free-charge field rotating the induced dipoleswithin the PV. The series of partial fields is not envisionedas a series in time — the PV response is assumed to happeninstantaneously at each field point.

The initial magnetic field in the series is B1 = ββ × E0,where the bare charge’s laboratory-observed Coulomb fieldis

E0 =err3=

ee∗

e∗rr3= α1/2

e∗rr3, (3)

where α is Planck’s constant. The serial electric fields areassumed to be radial; so the final electric field is radial with amagnitude equal to the sum

E = E0 + E1 + E2 + E3 + . . . , (4)

where the En are the magnitudes of the Ens and the final mag-netic field is ββ × E. Assuming that the En = En(r, θ), thecharge-PV feedback equations (1) and (2) reduce to

∂En∂θ=

rc∂Bn∂t

(5)

andBn+1 = βEn sin θ (6)

in the azimuthal direction about the z-axis.Calculating the first partial field E1 in the series begins

with (6)B1 = βE0 sin θ (7)

and leads to (Appendix A)

Ḃ1 =3cβ2E0 sin θ cos θ

r, (8)

where the overhead dot represents a partial differentiationwith respect to time. Then from (5)

dE1 =rḂ1

cdθ = 3β2E0 sin θ cos θ dθ, (9)

which integrates over the limits (0, θ) to

E1 =3β2E0 sin2 θ

2− λ1E0, (10)

where the reference field E1(θ = 0) = −λ1E0 with λ1 a con-stant to be determined.

The second iteration for the electric field begins with

B2 = βE1 sin θ =3β3E0 sin3 θ

2− λ1B1 (11)

and yields (Appendix A)

Ḃ2 =15cβ4E0 sin3 θ cos θ

2r− λ1Ḃ1 . (12)

William C. Daywitt. The Lorentz Transformation as a Planck Vacuum Phenomenon in a Galilean Coordinate System 3

Volume 2 PROGRESS IN PHYSICS April, 2011

Equation (5) then leads to

dE2 =rḂ2

cdθ =

(15β4E0 sin3 θ cos θ

2− λ1rḂ1

c

)dθ, (13)

which integrates to

E2 =15β4E0 sin4 θ

8− λ1

3β2E0 sin2 θ2

− λ2E0, (14)

where again E2(θ = 0) = −λ2E0 .The third iteration proceeds as before and results in (Ap-

pendix A)

Ḃ3 =3 · 5 · 7cβ6E0 sin5 θ cos θ

8r− λ1

3 · 5cβ4E0 sin3 θ cos θ2r

−λ23cβ2E0 sin θ cos θ

r(15)

and

E3 =3 · 5 · 7β6E0 sin6 θ

6 · 8 − λ13 · 5β4E0 sin4 θ

2 · 4

−λ23βE0 sin2 θ

2− λ3E0 (16)

for the third partial field.Inserting (10), (14), and (16) (plus the remaining infinity

of partial fields) into (4) gives

E = E0 +3β2E0 sin2 θ

2+

3 · 5β4E0 sin4 θ8

+3 · 5 · 7β6E0 sin6 θ

48+ . . .

−λ1(E0 +

3β2E0 sin2 θ2

+3 · 5β4E0 sin4 θ

8+ . . .

)

−λ2(E0 +

3βE0 sin2 θ2

+ . . .

)− λ3(E0 + . . .) + . . .

= E0

(1 +

3β2 sin2 θ2

+3 · 5β4 sin4 θ

2 · 4

+3 · 5 · 7β6 sin6 θ

2 · 4 · 6 + . . .)

(1 − λ) , (17)

where

λ ≡∞∑

n=1

λn (18)

is a constant. The sum after the final equal sign in (17) isrecognized as the function (1 − β2 sin2 θ)−3/2; so E can beexpressed as

E =(1 − λ)E0

(1 − β2 sin2 θ)3/2. (19)

Finally, the constant λ can be evaluated from Gauss’ lawand the conservation of bare charge e∗:∫

D · dS = 4πe∗ −→∫

E · dS = 4πe, (20)

where D = (e∗/e)E is used to arrive at the second integral.Inserting (19) into (20) and integrating yields

λ = β2, (21)

which, inserted back into (19), gives the relativistic electricfield of a uniformly moving charge. That this field is the sameas that derived from the Lorentz transformed Coulomb fieldis shown in Appendix B.

2 Conclusions and Comments

The calculations of the previous section suggest that theLorentz transformation owes its existence to interactions be-tween free-space particles and the negative-energy PV. Freespace is defined here as “the classical void + the zero-pointelectromagnetic vacuum” [3].

The fact that the bare charge is massless makes the Pem-per derivation significantly less involved and more straight-forward than the related case for the massive point charge(Dirac electron). Nevertheless, the uniform motion of theDirac electron too exhibits electron-PV effects. When a barecharge is injected into free space (presumably from the PV) itvery quickly (∼ 10−30 sec) develops a mass from being drivenby the random fields of the electromagnetic vacuum. The cor-responding electron-PV connection is easily recognized in theLorentz-covariant Dirac equation [4, p. 90], [5]:(

ic~γµ∂µ − mc2)ψ = 0 −→

(ie2∗γ

µ∂µ − mc2)ψ = 0, (22)

where the PV relation c~ = e2∗ is used to arrive at the equa-tion on the right. A nonrelativistic expression for the electronmass is given by Puthoff [3, 6]

m =23

〈ṙ2

〉1/2c

m∗, (23)

where ṙ represents the random excursions of the zero-point-driven bare charge about its center of (random) motion at r =0 and m∗ is the Planck mass.

The massive point charge perturbs the PV with the two-fold force [5]

e2∗r2− mc

2

r, (24)

where the first and second terms are the polarization and cur-vature∗ forces respectively. It is the interaction of this com-posite force with the PV that is responsible for the Dirac equa-tion as evidenced by the e2∗ and mc

2 in (22) and (24). Thus∗Using the PV relations G = e2∗/m

2∗ and e

2∗ = r∗m∗c

2 in the curvatureforce leads to mc2/r = mm∗G/rr∗ and shows the direct gravitational interac-tion between the electron mass and the Planck particle masses within the PV.

4 William C. Daywitt. The Lorentz Transformation as a Planck Vacuum Phenomenon in a Galilean Coordinate System


both the Pemper derivation and the Dirac equation argue com-pellingly for the existence of the Planck vacuum state and itsplace in the physical scheme of things. It is noted in pass-ing that the force in (24) vanishes at the electron’s Comptonradius rc = e2∗/mc

2.

Appendix A: Galilean Coordinate System

The laboratory system in which the charge propagates is con-sidered to be a Galilean reference system. In that system(x, y, z) represents the radius vector from the system originto any field point (considered in the calculations to be fixed).The position of the charge traveling at a constant rate v alongthe positive z-axis is (0, 0, vt); so at time t = 0 the chargecrosses the origin. Since the field point is fixed, the vector inthe x-y plane

b = b b̂ ≡ x + y (A1)is constant. The radius vector from the position of the chargeto the field point is then

r = (x, y, z − vt) . (A2)

Combining (A1) and (A2) gives

r =[b2 + (z − vt)2

]1/2(A3)

for the magnitude of that vector.If θ is the angle between the radius r and the positive z-

axis, it is easy to show from (A1)—(A3) that

r sin θ = b (A4)

andr cos θ = z − vt (A5)

and from (A3)—(A5) that

ṙ = −v cos θ (A6)

andrθ̇ = v sin θ, (A7)

where the overhead dot represents a partial derivative withrespect to time.

From (7) the initial magnetic field in the charge-PV inter-action is

B1 = βE0 sin θ = β ·er2· b

r=

βeb[b2 + (z − vt)2]3/2 (A8)

whose time differential leads to

Ḃ1 =3cβ2E0 sin θ cos θ

r(A9)

in a straightforward manner.From (11) in the text

B2 = βE1 sin θ =3β3eb3

2[b2 + (z − vt)2]5/2 − λ1B1, (A10)

which leads to

Ḃ2 =15cβ4Eo sin3 θ cos θ

2r− λ1Ḃ1 . (A11)

From B3 = βE2 sin θ,

B3 =15β5E0 sin5 θ

8− λ1

3β3E0 sin3 θ2

− λ2βE0 sin θ

=15β5eb5

8[b2 + (z − vt)2]7/2 − λ1 3β

3eb3

2[b2 + (z − vt)2]5/2

−λ2βeb[

b2 + (z − vt)2]3/2 (A12)and

Ḃ3 =3 · 5 · 7cβ6E0 sin5 θ cos θ

8r− λ1

3 · 5cβ4E0 sin3 θ cos θ2r

−λ23cβ2E0 sin θ cos θ

r. (A13)

Appendix B: Lorentz Transformed Fields

The Lorentz transformation coefficients aµν in the coordinatetransformation [7, pp. 380–381]

x′µ = aµνxµ =

1 0 0 00 1 0 00 0 γ iβγ0 0 −iβγ γ

xyz

ict

=

xy

γ(z − vt)iγ(ct − βz)

(B1)lead to the Lorentz transformed fields

F′µν = aµσaντFστ, (B2)

where the F′µν, etc., are the electromagnetic field tensors. Theprimed and unprimed parameters refer respectively to thecharge-at-rest and laboratory systems, where the charge sys-tem travels along the z-axis of the laboratory system with aconstant velocity v.

Using the static Coulomb field in the charge system andtransforming it to the laboratory system with the inverse of(B2) leads to the magnitude

E =γe

[b2 + (z − vt)2

]1/2[b2 + γ(z − vt)2]3/2 (B3)

for the electric field, where γ = 1/(1− β2)1/2. (B3) reduces to(19) in the following way:

E =γe

[b2 + (z − vt)2

]1/2γ3

[b2 + (z − vt)2 − β2b2]3/2

William C. Daywitt. The Lorentz Transformation as a Planck Vacuum Phenomenon in a Galilean Coordinate System 5


=e/

[b2 + (z − vt)2

]γ2

[1 − β2b2/[b2 + (z − vt)2]]3/2=

(1 − β2) E0(1 − β2 sin2 θ

)3/2 . (B4)Submitted on January 5, 2011 / Accepted on January 6, 2011

References1. Pemper R.R. A classical foundation for electrodynamics. Master Dis-

sertation, Univ. of Texas, El Paso, 1977. Barnes T.G. Physics of theFuture – A Classical Unification of Physics, Institute for Creation Re-search, California, 1983.

2. Daywitt W.C. The Planck vacuum. Progress in Physics, 2009, v. 1, 20–26.

3. Daywitt W.C. The Source of the Quantum Vacuum. Progress inPhysics, 2009, v. 1, 27–32.

4. Gingrich D.M. Practical Quantum Electrodynamics, CRC – The Taylor& Francis Group, Boca Raton, 2006.

5. Daywitt W.C. The Dirac Electron in the Planck Vacuum Theory.Progress in Physics, 2010, v. 4, 69–71.

6. Puthoff H.E. Gravity as a zero-point-fluctuation force. Physical ReviewA, 1989, v. 39, no. 5, 2333–2342.

7. Jackson J.D. Classical Electrodynamics. John Wiley & Sons, 1st ed.,2nd printing, New York, 1962.

6 William C. Daywitt. The Lorentz Transformation as a Planck Vacuum Phenomenon in a Galilean Coordinate System


Charged Polaritons with Spin 1

Vahan Minasyan and Valentin SamoilovScientific Center of Applied Research, JINR, Joliot-Curie 6, Dubna, 141980, Russia

E-mails: [email protected]; [email protected]

We present a new model for metal which is based on the stimulated vibration of in-dependent charged Fermi-ions, representing as independent harmonic oscillators withnatural frequencies, under action of longitudinal and transverse elastic waves. Due toapplication of the elastic wave-particle principle and ion-wave dualities, we predict theexistence of two types of charged Polaritons with spin 1 which are induced by longitu-dinal and transverse elastic fields. As result of presented theory, at small wavenumbers,these charged polaritons represent charged phonons.

1 Introduction

In our recent paper [1], we proposed a new model for dielec-tric materials consisting of neutral Fermi atoms. By the stim-ulated vibration of independent charged Fermi-atoms, repre-senting as independent harmonic oscillators with natural fre-quencies by actions of the longitudinal and transverse elasticwaves, due to application of the principle of elastic wave-particle duality, we predicted the lattice of a solid consistsof two types of Sound Boson-Particles with spin 1, with fi-nite masses around 500 times smaller than the atom mass.Namely, we had shown that these lattice Sound-Particles ex-cite the longitudinal and transverse phonons with spin 1. Inthis context, we proposed new model for solids representingas dielectric substance which is different from the well-knownmodels of Einstein [2] and Debye [3] because: 1), we suggestthat the atoms are the Fermi particles which are absent in theEinstein and Debye models; 2), we consider the stimulatedoscillation of atoms by action of longitudinal and transverselattice waves which in turn consist of the Sound Particles.

Thus, the elastic lattice waves stimulate the vibration ofthe fermion-atoms with one natural wavelength, we suggestedthat ions have two independent natural frequencies by underaction of a longitudinal and a transverse wave. Introduction ofthe application of the principle of elastic wave-particle dualityas well as the model of hard spheres we found an appearanceof a cut off in the spectrum energy of phonons which havespin 1 [1].

In this letter, we treat the thermodynamic property ofmetal under action of the ultrasonic waves. We propose anew model for metal where the charged Fermi-ions vibratewith natural frequencies Ωl and Ωt, by under action of lon-gitudinal and transverse elastic waves. Thus, we consider amodel for metal as independent charged Fermi-ions of latticeand gas of free electrons or free Frölich-Schafroth chargedbosons (singlet electron pairs) [4]. Each charged ion is cou-pled with a point of lattice knot by spring, creating an iondipole [5,6]. The lattice knots define the equilibrium posi-tions of all ions which vibrate with natural frequencies Ωl andΩt, under action of longitudinal and transverse elastic fields

which in turn leads to creation of the transverse electromag-netic fields moving with speeds cl and ct. These transverseelectromagnetic waves describe the ions by the principle ofion-wave duality [7]. Using the representation of the elec-tromagnetic field structure of one ion with ion-wave dualityin analogous manner, as it was presented in a homogenousmedium for an electromagnetic wave [8], we obtain that theneutral phonons cannot be excited in such substances as met-als, they may be induced only in dielectric material [1]. Inthis respect, we find the charged polaritons with spin 1 whichare always excited in a metal, and at small wavenumbers, theyrepresent as charged phonons.

2 New model for metal

The Einstein model of a solid considers the solid as gas ofN atoms in a box with volume V . Each atom is coupledwith a point of the lattice knot. The lattice knots define thedynamical equilibrium position of each atom which vibrateswith natural frequency Ω0. The vibration of atom occurs nearequilibrium position corresponding to the minimum of po-tential energy (harmonic approximation of close neighbors).We presented the model of ion-dipoles [5,6] which representsions coupled with points of lattice knots. It differs from theEinstein model of solids where the neutral independent atomsare considered in lattice knots, these ions are vibrating withnatural frequencies Ωl and Ωt forming ion-dipoles by underaction longitudinal and transverse ultrasonic lattice fields.

Usually, matters are simplified assuming the transfer ofheat from one part of the body to another occurs very slowly.This is a reason to suggest that the heat exchange during timesof the order of the period of oscillatory motions in the bodyis negligible, therefore, we can regard any part of the body asthermally insulated, and there occur adiabatic deformations.Since all deformations are supposed to be small, the motionsconsidered in the theory of elasticity are small elastic oscilla-tions. In this respect, the equation of motion for elastic con-tinuum medium [9] represents as

%~̈u = c2t ∇2~u + (c2l − c2t ) grad div ~u, (1)

Vahan Minasyan and Valentin Samoilov. Charged Polaritons with Spin 1 7


where ~u = ~u(~r, t) is the vectorial displacement of any parti-cle in the solid; cl and ct are, respectively, the velocities of alongitudinal and a transverse ultrasonic wave.

We shall begin by discussing a plane longitudinal elasticwave with condition curl ~u = 0 and a plane transverse elasticwave with condition div ~u = 0 in an infinite isotropic medium.In this respect, the vector displacement ~u is the sum of thevector displacements of a longitudinal ul and of a transverseultrasonic wave ut:

~u = ~ul + ~ut. (2)

In turn, the equations of motion for a longitudinal and a trans-verse elastic wave take the form of the wave-equations:

∇2~ul − 1c2l

d2~uldt2

= 0, (3)

∇2~ut − 1c2t

d2~uldt2

= 0. (4)

It is well known, in quantum mechanics, a matter waveis determined by electromagnetic wave-particle duality or deBroglie wave of matter [7]. We argue that in analogous man-ner, we may apply the elastic wave-particle duality. This rea-soning allows us to present a model of elastic field as theBose-gas consisting of the Sound Bose-particles with spin 1having non-zero rest masses which are interacting with eachother. In this respect, we may express the vector displace-ments of a longitudinal ul and of a transverse ultrasonic waveut via the second quantization vector wave functions of SoundBosons as

~ul = Cl

(φ(~r, t) + φ+(~r, t)

)(5)

and

~ut = Ct

(ψ(~r, t) + ψ+(~r, t)

), (6)

where Cl and Ct are unknown constant normalization coeffi-cients; ~φ(~r, t) and ~φ+(~r, t) are, respectively, the second quan-tization wave vector functions for one Sound-Particle, corre-sponding to the longitudinal elastic wave, at coordinate ~r andtime t; ~ψ(~r, t) and ~ψ+(~r, t) are, respectively, the second quan-tization wave vector functions for one Sound-Particle, corre-sponding to the transverse elastic wave, at coordinate ~r andtime t:

~φ(~r, t) =1√V

∑

~k,σ

~a~k,σei(~k~r+kclt) (7)

~φ+(~r, t) =1√V

∑

~k,σ

~a+~k,σe−i(~k~r+kclt) (8)

and~ψ(~r, t) =

1√V

∑

~k,σ

~b~k,σei(~k~r+kct t) (9)

~ψ+(~r, t) =1√V

∑

~k,σ

~b+~k,σe−i(~k~r+kct t), (10)

where ~a+~k,σ

and ~a~k,σ are, respectively, the Bose vector-oper-ators of creation and annihilation for one free longitudinalSound Particle with spin 1, described by a vector ~k whose di-rection gives the direction of motion of the longitudinal wave;~b+~k,σ

and ~b~k,σ are, respectively, the Bose vector-operators ofcreation and annihilation for one free transverse Sound Parti-cle with spin 1, described by a vector ~k whose direction givesthe direction of motion of the transverse wave.

In this respect, the vector-operators~a+~k,σ

, ~a~k,σ and~b+~k,σ

, ~b~k,σsatisfy the Bose commutation relations as:

[â~k,σ, â

+~k′ ,σ′

]= δ~k, ~k′ · δσ,σ′

[â~k,σ, â~k′ ,σ′ ] = 0

[â+~k,σ, â+~k′ ,σ′

] = 0

and[b̂~k,σ, b̂

+~k′ ,σ′

]= δ~k, ~k′ · δσ,σ′

[b̂~k,σ, b̂~k′ ,σ′ ] = 0

[b̂+~k,σ, b̂+~k′ ,σ′

] = 0.

Thus, as we see the vector displacements of a longitu-dinal ul and of a transverse ultrasonic wave ut satisfy thewave-equations of (3) and (4) because they have the follow-ing forms due to application of (5) and (6):

~ul =Cl√

V

∑

~k,σ

(~a~k,σe

i(~k~r+kclt) + ~a+~k,σe−i(~k~r+kclt)

)(11)

and

~ut =Ct√

V

∑

~k,σ

(~b~k,σe

i(~k~r+kct t) + ~b+~k,σe−i(~k~r+kct t)

). (12)

In this context, we may emphasize that the Bose vectoroperators ~a+

~k,σ, ~a~k,σ and ~b

+~k, σ and ~b~k, σ communicate with

each other because the vector displacements of a longitudinal~ul and a transverse ultrasonic wave ~ut are independent, and inturn, satisfy the condition of a scalar multiplication ~ul · ~ut = 0.

8 Vahan Minasyan and Valentin Samoilov. Charged Polaritons with Spin 1


Consequently, the Hamiltonian operator Ĥ of the system,consisting of the vibrating Fermi-ions with mass M, is repre-sented in the following form:

Ĥ = Ĥl + Ĥt, (13)

where

Ĥl =MNV

∫ (d~uldt

)2dV +

NMΩ2lV

∫(~ul)2dV (14)

and

Ĥt =MNV

∫ (d~utdt

)2dV +

NMΩ2tV

∫(~ut)2dV, (15)

where Ωl and Ωt are, respectively, the natural frequencies ofthe atom through action of the longitudinal and transverseelastic waves.

To find the Hamiltonian operator Ĥ of the system, we usethe formalism of Dirac [10]:

d~uldt

=iclCl√

V

∑

~k,σ

k(~a~k,σe

ikclt − ~a+−~k,σe−ikclt

)ei~k~r (16)

and

d~utdt

=ictCt√

V

∑

~k,σ

k(~b~k,σe

ikct t − ~b+−~k,σe−ikct t

)ei~k~r, (17)

which by substituting into (14) and (15), using (11) and (12),gives the reduced form of the Hamiltonian operators Ĥl andĤt:

Ĥl =∑

~k,σ

(2MNC2l c

2l k

2

V+

2MNC2l Ω2l

V

)~a+~k,σ~a~k,σ−

−∑

~k,σ

(2MNC2l c

2l k

2

V− 2MNC

2l Ω

2l

V

)(a~k,σ~a−~k,σ+a

+

−~k~a+~k,σ

)(18)

and

Ĥt =∑

~k,σ

(2MNC2t c

2t k

2

V+

2MNC2t Ω2t

V

)~b+~k,σ

~b~k,σ−

−∑

~k,σ

(2MNC2t c

2t k

2

V−2MNC

2t Ω

2t

V

)(b~k,σ~b−~k,σ+b

+

−~k~b+~k,σ

), (19)

where the normalization coefficients Cl and Ct are defined bythe first term of right side of (18) and (19) which representthe kinetic energies of longitudinal Sound Particles ~

2k22ml

and

transverse Sound Particles ~2k2

2mtwith masses ml and mt, re-

spectively. Therefore we suggest to find Cl and Ct:

2MNC2l c2l k

2

V=~2k2

2ml(20)

and

2MNC2t c2t k

2

V=~2k2

2mt, (21)

which in turn determine

Cl =~

2cl√

mlρ(22)

and

Ct =~

2ct√

mtρ, (23)

where ρ = MNV is the density of solid.As we had shown in [1], at absolute zero T = 0, the Fermi

ions fill the Fermi sphere in momentum space. Thus, there aretwo type Fermi atoms by the value of its spin z-componentµ = ± 12 with the boundary wave number k f of the Fermi,which, in turn, is determined by a condition:

V2π2

∫ k f0

k2dk =N2,

where N is the total number of Fermi-ions in the solid. Thisreasoning together with the model of hard spheres claims theimportant condition to introduce the boundary wave number

k f =(

3π2NV

) 13

coinciding with kl and kt. Then, there is an

important condition k f = kl = kt which determines a relation-ship between natural oscillator frequencies

k f =Ωl

cl=

Ωt

ct. (24)

3 Charged Polaritons

In papers [5, 6], we demostrated the so-called transformationof longitudinal and transverse elastic waves into transverseelectromagnetic fields with vectors of the electric waves ~Eland ~Et, corresponding to the ion displacements ~ul and ~ut, re-spectively. In turn, the equations of motion are presented inthe following forms [5, 6]:

Md2~uldt2

+ MΩ2l ~ul = −e~El (25)and

Md2~utdt2

+ MΩ2t ~ut = −e~Et. (26)

The vector of the electric waves ~El and ~Et are defined by sub-stitution of the meaning of ~ul and ~ut from (11) and (12), re-spectively, into (25) and (26):

Vahan Minasyan and Valentin Samoilov. Charged Polaritons with Spin 1 9


~El(~r, t) =Cl

e√

V

∑

~k,σ

γ~k,l

(~a~k,σe

i(~k~r+kclt) + ~a+~k,σe−i(~k~r+kclt)

)(27)

and

~Et(~r, t) =Ct

e√

V

∑

~k,σ

γ~k,t

(~b~k,σe

i(~k~r+kct t) + ~b+~k,σe−i(~k~r+kct t)

), (28)

where

γ~k,l = M(k2c2l −Ω2l

)(29)

and

γ~k,t = M(k2c2t −Ω2t

). (30)

On the other hand, by action of the longitudinal and trans-verse ultrasonic waves on the charged ion [5, 6], these ultra-sonic waves are transformed into transverse electromagneticfields with electric wave vectors ~El and ~Et which in turn de-scribe the de Broglie wave of charged ions expressed via elec-tric ~El(~r, t) and ~Et(~r, t) fields of one ion-wave particle in ho-mogeneous medium. In fact, these electric ~El(~r, t) and ~Et(~r, t)fields satisfy the Maxwell’s equations in dielectric medium:

curl ~Hl − εlcd ~Eldt

= 0 (31)

curl ~El +1c

d ~Hldt

= 0 (32)

div ~El = 0 (33)

div ~Hl = 0 (34)

and

curl ~Ht − εtcd ~Etdt

= 0 (35)

curl ~Et +1c

d ~Htdt

= 0 (36)

div ~Et = 0 (37)

div ~Ht = 0 (38)

with √εl =

ccl

(39)

and

√εl =

ccl, (40)

where ~Hl = ~Hl(~r, t) and ~Ht = ~Ht(~r, t) are, respectively, the lo-cal magnetic fields, corresponding to longitudinal and trans-verse ultrasonic waves, depending on space coordinate ~r andtime t; εl and εt are, respectively, the dielectric constants fortransverse electric fields ~El(~r, t) and ~Et(~r, t) corresponding tolongitudinal and transverse ultrasonic waves; c is the velocityof electromagnetic wave in vacuum; µ = 1 is the magneticsusceptibility.

When using Eqs. (31–40) and results of letter [8], we maypresent the transverse electric fields ~El(~r, t) and ~Et(~r, t) by thequantization forms:

~El(~r, t) =Al√

V

∑

~k

(~c~ke

i(~k~r+kclt) + ~c+~k e−i(~k~r+kclt)

)(41)

and

~Et(~r, t) =At√

V

∑

~k,0

(~d~ke

i(~k~r+kct t) + ~d+~k e−i(~k~r+kct t)

), (42)

where Al and At are the unknown constants which are foundas below; ~c+

~k, ~d+

~kand ~c~k, ~d~k are, respectively, the Bose vector-

operators of creation and annihilation of electric fields of oneion-wave particle with wave vector~k which are directed alongof the wave normal ~s or ~k = k~s. These Bose vector-operators~El(~r, t) and ~Et(~r, t) are directed to the direction of the unitvectors ~l and ~t which are perpendicular to the wave normal ~s;N̂ is the operator total number of charged ions.

In this context, we indicate that the vector-operators ~c+~k,σ

,

~c~k,σ and ~d+~k,σ

, ~d~k,σ satisfy the Bose commutation relations as:[ĉ~k,σ, ĉ

+~k′ ,σ′

]= δ~k, ~k′ · δσ,σ′

[ĉ~k,σ, ĉ~k′ ,σ′ ] = 0

[ĉ+~k,σ, ĉ+~k′ ,σ′

] = 0

and[d̂~k,σ, d̂

+~k′ ,σ′

]= δ~k, ~k′ · δσ,σ′

[d̂~k,σ, d̂~k′ ,σ′ ] = 0

[d̂+~k,σ, d̂+~k′ ,σ′

] = 0.

Comparing (41) with (27) and (42) with (28), we get

~a~k,σ =eAl

Clγ~k,l~c~k,σ (43)



and

~b~k,σ =eAt

Ctγ~k,t~d~k,σ. (44)

Now, substituting ~a~k,σ and ~b~k,σ into (18) and (19), we ob-tain the reduced form of the Hamiltonian operators Ĥl andĤl which are expressed via terms of the electric fields of theion-wave particle:

Ĥl =∑~k,σ

e2A2lγ2~k,l

[(2MNc2l k

2

V +2MNΩ2l

V

)~c+~k,σ

c~k,σ−

−(

MNc2l k2

V− MNΩ

2l

V

)(~c−~k,σ~c~k,σ + ~c

+~k,σ~c+−~k,σ

)] (45)

and

Ĥt =∑~k,σ

e2A2tγ2~k,l

[(2MNc2t k

2

V +2MNΩ2t

V

)~d+~k,σ

d~k,σ−

−(

MNc2t k2

V− MNΩ

2t

V

)(~d−~k,σ ~d~k,σ + ~d

+~k,σ~d+−~k,σ

)].

(46)

To evaluate the energy levels of the operators Ĥl (45) andĤt (46) within the diagonal form, we use a transformation ofthe vector-Bose-operators:

~c~k,σ =~l~k,σ + L~k~l

+

−~k,σ√1 − L2

~k

(47)

and

~d~k,σ =~t~k,σ + M~k~t

+

−~k,σ√1 − M2

~k

, (48)

where L~k and M~k are, respectively, the real symmetrical func-tions of a wave vector ~k.

Consequently,

Ĥl =∑

k


Thus, we predicted the existence of a new type of chargedquasiparticles in nature. On the other hand, we note that thequantization of elastic fields is fulfilled for the new model ofmetals. In analogous manner, as it was presented in [1], wemay show that the acoustic field operator does not commutewith its momentum density.

Submitted on January 6, 2011 / Accepted on January 10, 2011

References1. Minasyan V. N., Samoilov V. N. Sound-Particles and Phonons with

Spin 1. Progress in Physics, 2011, v. 1, 81–86.

2. Einstein A. Die Plancksche Theorie der Strahlung und die Theorie derspezifischen Warme. Annalen der Physik, 1907, v. 22, 180–190.

3. Debye P. Zur Theorie der spezifischen Waerme. Annalen der Physik,1912, v. 39, 789–839.

4. Minasyan V. N., Samoilov V. N. Formation of Singlet Fermion Pairs inthe Dilute Gas of Boson-Fermion Mixture. Progress in Physics, 2010,v. 4, 3–9.

5. Minasyan V. N., Samoilov V. N. The Intensity of the Light Diffractionby Supersonic Longitudinal Waves in Solid, Progress in Physics, 2010,v. 2, 60–63.

6. Minasyan V. N., Samoilov V. N. Dispersion of Own Frequency of Ion-dipole by Supersonic Transverse Wave in Solid, Progress in Physics,2010, v. 4, 10–12.

7. de Broglie L. Researches on the quantum theory. Annalen der Physik,1925, v. 3, 22–32.

8. Minasyan V. N., Samoilov V. N. New resonance-polariton Bose-quasiparticles enhances optical transmission into nanoholes in metalfilms. Physics Letters A, 2011, v. 375, 698–711.

9. Landau L. D., Lifshiz E. M. Theory of Elasticity. Theoretical Physics,1987, v. 11, 124–127.

10. Dirac P. A. M. The Principles of Quantum Mechanics. Clarendon press,Oxford, (1958).



New Fundamental Light Particle and Breakdown of Stefan-Boltzmann’s LawVahan Minasyan and Valentin Samoilov

Scientific Center of Applied Research, JINR, Dubna, 141980, RussiaE-mails: [email protected]; [email protected]

Recently, we predicted the existence of fundamental particles in Nature, neutral LightParticles with spin 1 and rest mass m = 1.8×10−4me, in addition to electrons, neutronsand protons. We call these particles Light Bosons because they create electromagneticfield which represents Planck’s gas of massless photons together with a gas of Light Par-ticles in the condensate. Such reasoning leads to a breakdown of Stefan–Boltzmann’slaw at low temperature. On the other hand, the existence of new fundamental neutralLight Particles leads to correction of such physical concepts as Bose-Einstein conden-sation of photons, polaritons and exciton polaritons.

1 Introduction

First, the quantization scheme for the local electromagneticfield in vacuum was presented by Planck in his black bodyradiation studies [1]. In this context, the classical Maxwellequations lead to appearance of the so-called ultraviolet catas-trophe; to remove this problem, Planck proposed the modelof the electromagnetic field as an ideal Bose gas of masslessphotons with spin one. However, Dirac [2] showed the Planckphoton-gas could be obtained through a quantization schemefor the local electromagnetic field, presenting a theoreticaldescription of the quantization of the local electromagneticfield in vacuum by use of a model Bose-gas of local planeelectromagnetic waves propagating by speed c in vacuum.

In a different way, in regard to Plank and Dirac’s mod-els, we consider the structure of the electromagnetic field [3]as a non-ideal gas consisting of N neutral Light Bose Par-ticles with spin 1 and finite mass m, confined in a box ofvolume V . The form of potential interaction between LightParticles is defined by introduction of the principle of wave-particle duality of de Broglie [4] and principle of gauge in-variance. In this respect, a non-ideal Bose-gas consistingof Light Particles with spin 1 and non-zero rest mass is de-scribed by Planck’s gas of massless photons together with agas consisting of Light Particles in the condensate. In thiscontext, we defined the Light Particle by the model of hardsphere particles [5]. Such definition of Light Particles leadsto cutting off the spectrum of the electromagnetic wave bythe boundary wave number k0 = mc~ or boundary frequencyωγ = 1018 Hz of gamma radiation at the value of the restmass of the Light Particle m = 1.8 × 10−4me. On the otherhand, the existence of the boundary wave number k0 = mc~for the electromagnetic field in vacuum is connected with thecharacteristic length of the interaction between two neighbor-ing Light Bosons in the coordinate space with the minimaldistance d = 1k0 =

~mc = 2×10

−9m. This reasoning determinesthe density of Light Bosons NV as

NV =

34πd3 = 0.3×10

26m−3.It is well known that Stefan-Boltzmann’s law [6] for ther-

mal radiation, presented by Planck’s formula [1], determines

the average energy density UV as

UV=

2V

∑0≤k


L2~k =~2k22m +

mc22 − ~kc

~2k22m +

mc22 + ~kc

. (5)

Our calculation shows that at absolute zero the value of~i+~k~i~k = 0, and therefore the average energy density of black

radiation UV reduces to

UV=

mc2N0,T=0V

=mc2N

V− m

4c5B(2, 3)4π2~3

≈ mc2N

V, (6)

where B(2, 3) =∫ 1

0 x(1 − x)2dx = 0.1 is the beta function.

Thus, the average energy density of black radiation UV isa constant at absolute zero. In fact, there is a breakdown ofStefan-Boltzmann’s law for thermal radiation.

In conclusion, it should be also noted that Light Bosonsin vacuum create photons, while Light Bosons in a homoge-neous medium generate the so-called polaritons. This factimplies that photons and polaritons are quasiparticles, there-fore, Bose-Einstein condensation of photons [7], polaritons[8] and exciton polaritons [9] has no physical sense.

Acknowledgements

This work is dedicated to the memory of the Great BritishPhysicist Prof. Marshall Stoneham, F.R.S., (London Centrefor Nanotechnology, and Department of Physics and Astron-omy University College London, UK), who helped us withEnglish. We are very grateful to him.


References1. Planck M. On the Law of Distribution of Energy in the Normal Spec-

trum. Annalen der Physik, 1901, v. 4, 553–563.2. Dirac P. A. M. The Principles of Quantum Mechanics. Clarendon Press,

Oxford, 1958.3. Minasyan V. N., Samoilov V. N. New resonance-polariton Bose-

quasiparticles enhances optical transmission into nanoholes in metalfilms. Physics Letters A, 2011, v. 375, 698–711.

4. de Broglie L. Researches on the quantum theory. Annalen der Physik,1925, v. 3, 22–32.

5. Huang K. Statistical Mechanics. John Wiley, New York, 1963.6. Stefan J. Uber die Beziehung zwischen der Warmestrahlung

und der Temperatur. In: Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wis-senschaften, Bd. 79 (Wien 1879), 391–428.

7. Klaers J., Schmitt J., Vewinger F., Weitz M. Bose-Einstein condensa-tion of photons in an optical microcavity. Nature, 2010, v. 468, 545–548.

8. Balili R., Hartwell V., Snoke D., Pfeiffer L., West K. Bose-Einsteincondensation of microcavity polaritons in a trap. Science, 2007, v. 316,1007–1010.

9. Kasprzak J. et al. Bose-Einstein condensation of exciton polaritons. Na-ture, 2006, v. 443, 409–414.

14 Vahan Minasyan and Valentin Samoilov. New Fundamental Light Particle and Breakdown of Stefan-Boltzmann’s Law


The Point Mass Concept

Bo LehnertAlfvén Laboratory, Royal Institute of Technology, SE–10044 Stockholm, Sweden. E-mail: [email protected]

A point-mass concept has been elaborated from the equations of the gravitationalfield. One application of these deductions results in a black hole configuration of theSchwarzschild type, having no electric charge and no angular momentum. The criticalmass of a gravitational collapse with respect to the nuclear binding energy is found to bein the range of 0.4 to 90 solar masses. A second application is connected with the spec-ulation about an extended symmetric law of gravitation, based on the options of positiveand negative mass for a particle at given positive energy. This would make masses ofequal polarity attract each other, while masses of opposite polarity repel each other.Matter and antimatter are further proposed to be associated with the states of positiveand negative mass. Under fully symmetric conditions this could provide a mechanismfor the separation of antimatter from matter at an early stage of the universe.

1 Introduction

In connection with an earlier elaborated revised quantum-electrodynamic theory, a revised renormalisation procedurehas been developed to solve the problem of infinite self-energy of the point-charge-like electron [1, 2]. In the presentinvestigation an analogous procedure is applied to the basicequations of gravitation, to formulate a corresponding pointmass concept. Two applications result from such a treatment.The first concerns the special Schwarzschild case of a blackhole with its critical limit of gravitational collapse. The sec-ond application is represented by the speculation about an ex-tended form of the gravitation law, in which full symmetry isobtained by including both positive and negative mass con-cepts. This further leads to the question whether such con-cepts could have their correspondence in matter and antimat-ter, and in their mutual separation.

2 The conventional law of gravitation

In this investigation the analysis is limited to the steady caseof spherical symmetry, in a corresponding frame where r isthe only independent variable.

2.1 Basic equations

Following Bergmann [3], a steady gravitational field strength

g = −∇φ (1)

is considered which originates from the potential φ (r). Thesource of the field strength is a mass density ρ related to g by

−div g = 4πGρ = ∇2φ = 1r2

ddr

(r2

dφdr

), (2)

where G = 6.6726× 10−11 m3kg−1s−2 is the constant of gravi-tation in SI units. The associated force density becomes

f = ρ g . (3)

In the conventional interpretation there only exists a posi-tive mass density ρ> 0. This makes in a way the gravitationalfield asymmetric, as compared to the electrostatic field whichincludes both polarities of electric charge density.

A complete form of the potential φ would consist of a se-ries of both positive and negative powers of r, but the presentanalysis will be restricted and simplified by studying eachpower separately, in the form

φ (r) = φ0

(rr0

)α. (4)

Here φ0 is a constant, r0 represents a characteristic dimensionand α is a positive or negative integer. Equation (2) yields

4πGρ =φ0rα0α (α + 1) rα−2 > 0 . (5)

When limiting the investigations by the condition ρ> 0, thecases α= 0 and α=−1 have to be excluded, leaving the re-gimes of positive α= (1, 2, . . .) and negative α= (−2,−3, . . .)to be considered for positive values of φ0.

2.2 Point mass formation

For reasons to become clear from the deductions which fol-low, we now study a spherical configuration in which themass density ρ is zero within an inner hollow region 06 r6 ri,and where ρ> 0 in the outer region r> ri. From relation (5)the total integrated mass P (r) inside the radius r then be-comes

P (r) =

r∫

0

ρ 4πr2dr =1Gφ0rα0α(rα+1− rα+1i

)> 0 (6)

with a resulting local field strength g = (g, 0, 0) given by

g (r) = −G P (r)r2

= −φ0rα0

α

r2(rα+1− rα+1i

)< 0 (7)

Bo Lehnert. The Point Mass Concept 15


and a local force density f = ( f , 0, 0) where

f (r) = − 14πG

φ0rα0

2

α2 (α + 1) rα−4(rα+1− rα+1i

)< 0 . (8)

Here all (P, g, f ) refer to the range r> ri, and φ0 > 0.A distinction is further made between the two regimes of

positive and negative α:

• When α= (1, 2, . . .) of a convergent potential (4), thishollow configuration has an integrated mass (6) whichincreases monotonically with r, from zero at r = ri tolarge values. This behaviour is the same for a vanishingri and does not lead to a point-like mass at small ri.

• When α= (−2,−3, . . .) of a divergent potential (4), thehollow configuration leads to a point-mass-like geome-try at small ri. This is similar to a point-charge-like ge-ometry earlier treated in a model of the electron [1, 2],and will be considered in the following analysis.

2.3 The renormalised point mass

In the range α6−2 expressions (6)–(8) are preferably castinto a form with γ=−α> 2 where

P (r) =1G

(φ0r

γ0

)γ(r−γ+1i − r−γ+1

)> 0 , (9)

g (r) = −(φ0r

γ0

) γr2

(r−γ+1i − r−γ+1

)< 0 , (10)

f (r) =− 14πG

(φ0r

γ0

)2γ2 (γ−1) r−γ−4

(r−γ+1i −r−γ+1

)< 0. (11)

Here an erroneous result would be obtained if the terms in-cluding ri are dropped and the hollow configuration is aban-doned. Due to eqs. (9)–(11) this would namely result in anegative mass P, a positive field strength g, and a repulsivelocal gravitational force density f .

The radius ri of the hollow inner region is now made toapproach zero. The total integrated mass of eq. (9) is thenconcentrated to an infinitesimally small layer. Applying a re-vised renormalisation procedure in analogy with an earlierscheme [1,2], we “shrink” the combined parameters φ0r

γ0 and

rγ−1i in such a way that

φ0rγ0 = cφr · ε rγ−1i = ci · ε 0 < ε � 1 , (12)

where ε is a smallness parameter and cφr and ci are positiveconstants. A further introduction of

P0 ≡ 1Gγ cφr

ci(13)

results in P (r) = 0 for r6 ri and

P (r) = P0

[1 −

( rir

)γ−1]r > ri . (14)

In the limit ε→ 0 and ri→ 0 there is then a point mass P0 atthe origin. This mass generates a field strength

g (r) = −G P0r2

(15)

at the distance r according to equations (10), (12) and (13).With another point mass P1 at the distance r, there is a mutualattraction force

F01 = P1g (r) = −G P0P1r2 , (16)which is identical with the gravitation law for two pointmasses.

To further elucidate the result of eqs. (12)-(16) it is firstobserved that, in the conventional renormalisation procedure,the divergent behaviour of an infinite self-energy is outbal-anced by adding extra infinite ad-hoc counter-terms to theLagrangian, to obtain a finite difference between two “infini-ties”. Even if such a procedure has been successful, however,it does not appear to be quite acceptable from the logical andphysical points of view. The present revised procedure rep-resented by expressions (12) implies on the other hand thatthe “infinity” of the divergent potential φ0 at a shrinking ra-dius ri is instead outbalanced by the “zeros” of the inherentshrinking counter-factors cφr · ε and ci · ε.3 A black hole of Schwarzschild type

A star which collapses into a black hole under the compres-sive action of its own gravitational field is a subject of everincreasing interest. In its most generalized form the physicsof the black hole includes both gravitational and electromag-netic fields as well as problems of General Relativity, to ac-count for its mass, net electric charge, and its intrinsic angularmomentum. The associated theoretical analysis and relatedastronomical observations have been extensively describedin a review by Misner, Thorne and Wheeler [4] among oth-ers. Here the analysis of the previous section will be appliedto the far more simplified special case by Schwarzschild, inwhich there is no electric charge and no angular momentum.Thereby it has also to be noticed that no black hole in theuniverse has a substantial electric charge [4].

3.1 The inward directed gravitational pressure

From eq. (11) is seen that the inward directed local force den-sity is zero for r6 ri, increases with r to a maximum within athin shell, and finally drops to zero at large r. The integratedinward directed gravitational pressure on this shell thus be-comes

p =

∞∫

0

f dr = − G8π

(γ − 1)2(γ + 1) (γ + 3)

P20r4i

(17)

in the limit of small ε and ri.When this pressure becomes comparable to the relevant

energy density of the compressed matter, a correspondinggravitational collapse is expected to occur.

16 Bo Lehnert. The Point Mass Concept


3.2 Gravitational collapse of the nuclear binding forces

Here we consider the limit at which matter is compressed intoa body of densely packed nucleons, and when the pressure ofeq. (17) tends to exceed the energy density of the nucleonbinding energy. The radius of a nucleus is [5]

rN = 1.5 × 10−15A1/2 [m], (18)where A is the mass number. A densely packed sphere of Nnuclei has the volume

VN =43πNr3N =

43πr3eq , (19)

where req is the equivalent radius of the sphere. The totalbinding energy of a nucleus is further conceived as the workrequired to completely dissociate it into its component nucle-ons. This energy is about 8 MeV per nucleon [5, 6]. With Anucleons per nucleus, the total binding energy of a body of Nnuclei thus becomes

WN = NAwN , (20)

where wN = 8 MeV = 1.28× 10−12 J. The equivalent bindingenergy density of the body is then

pN =WNVN

=34

AwNπ r3N

= 0.907 × 1032A−1/2 [J ×m−3]. (21)

The shell-like region of gravitational pressure has a forcedensity (11) which reaches its maximum at the radius

rm = ri

(2γ + 3γ + 4

)1/(γ−1)(22)

being only a little larger than ri. This implies that the radiusreq of eq. (19) is roughly equal to ri and

ri � rN N1/3. (23)

With N nuclei of the mass Amp and mp as the proton mass,the total mass becomes

P0 = NAmp , (24)

which yields

ri � rN

(P0

Amp

)1/3= 1.26 × 10−6A1/6P1/30 [m]. (25)

This result finally combines with eq. (17) to an equivalentgravitational pressure

p = −1.1 × 1012 (γ − 1)2

(γ + 1) (γ + 3)A−2/3P2/30 [J ×m−3]. (26)

For a gravitational collapse defined by −p> pN the point massP0 then has to exceed the critical limit

P0c � 7.5 × 1029[(γ + 1) (γ + 3)

(γ − 1)2]3/2

A1/4 [kg]. (27)

For γ> 2 and 16 A6 250, the critical mass would then befound in the range of about 0.46 P0c 6 90 solar masses ofabout 1.98 × 1030 kg.

4 Speculations about a generalized law of gravitation

The Coulomb law of interaction between electrically chargedbodies is symmetric in the sense that it includes both polari-ties of charge and attractive as well as repulsive forces. Theclassical Newtonian law of gravitation includes on the otherhand only one polarity of mass and only attractive forces. Infact, this asymmetry does not come out as a necessity fromthe basic equations (1)–(3) of a curl-free gravitational fieldstrength. The question could therefore be raised whether amore general and symmetric law of gravitation could be de-duced from the same equations, and whether this could havea relevant physical interpretation.

4.1 Mass polarity

In relativistic mechanics the momentum p of a particle withthe velocity u and rest mass m0 becomes [7]

p = m0u[1 −

(uc

)2]−1/2. (28)

With the energy E of the particle, the Lorentz invariance fur-ther leads to the relation

p2 − E2

c2= −m20c2, (29)

where p2 = p2 and u2 = u2. Equations (28) and (29) yield

E2 = m20c4[1 −

(uc

)2]−1≡ m2c4, (30)

leading in principle to two roots

E = ±mc2. (31)

In this investigation the discussion is limited to a positive en-ergy E, resulting in positive and negative gravitational masses

m = ± Ec2

E > 0 . (32)

This interpretation differs from that of the negative energystates of positrons proposed in the “hole” theory by Dirac [8]corresponding to the plus sign in eq. (31) and where both Eand m are negative.

4.2 An extended law of gravitation

With the possibility of negative gravitational masses in mind,we now return to the potential φ of equations (1) and (4)where the amplitude factor φ0 can now adopt both positiveand negative values, as defined by the notation φ0+ > 0 andφ0− < 0, and where corresponding subscripts are introducedfor (P, g, f , P0) of eqs. (9)–(11) and (13). Then P+ > 0, P− < 0,P0+ > 0, P0− < 0, g+ < 0, g− > 0, but f+ < 0 and f− < 0 alwaysrepresent an attraction force due to the quadratic dependenceon φ0 in eq. (11). With P1+ or P1− as an additional point mass



at the distance r from P0+ or P0−, there is then an extendedform of the law (16), as represented by the forces

P1+g+ = −G P0+P1+r2 = P1−g− = −GP0−P1−

r2, (33)

P1+g− = GP0−P1+

r2= P1−g+ = G

P0−P1+r2

. (34)

These relations are symmetric in the gravitational force inter-actions, where masses of equal polarity attract each other, andmasses of opposite polarity repel each other. It would implythat the interactions in a universe consisting entirely of neg-ative masses would become the same as those in a universeconsisting entirely of positive masses. In this way specificmass polarity could, in fact, become a matter of definition.

4.3 A possible rôle of antimatter

At this point the further question may be raised whether thestates of positive and negative mass could be associated withthose of matter and antimatter, respectively. A number ofpoints become related to such a proposal.

The first point concerns an experimental test of the re-pulsive behaviour due to eq. (34). If an electrically neutralbeam of anti-matter, such as of antihydrogen atoms, could beformed in a horizontal direction, such a beam would be de-flected upwards if consisting of negative mass. However, thedeflection is expected to be small and difficult to measure.

A model has earlier been elaborated for a particle with el-ementary charge, being symmetric in its applications to theelectron and the positron [1, 2]. The model includes an elec-tric charge q0, a rest mass m0, and an angular momentum s0of the particle. The corresponding relations between includedparameters are easily seen to be consistent with electron-positron pair formation in which q0 =−e, m0 = +E/c2 ands0 = +h/4π for the electron and q0 = +e, m0 =−E/c2 ands0 = +h/4π for the positron when the formation is due to aphoton of spin +h/2π. The energy of the photon is then atleast equal to 2E where the electron and the photon both havepositive energies E.

The energy of photons and their electromagnetic radia-tion field also have to be regarded as an equivalent mass dueto Einstein’s mass-energy relation. This raises the additionalquestion whether full symmetry also requires the photon tohave a positive or negative gravitational mass, as given by

mν = ±hνc2 . (35)

If equal proportions of matter and antimatter would havebeen formed at an early stage of the universe, the repulsivegravitational force between their positive and negative massescould provide a mechanism which expels antimatter frommatter and vice versa, also under fully symmetric conditions.Such a mechanism can become important even if the gravita-tional forces are much weaker than the electrostatic ones, be-

cause matter and antimatter are expected to appear as electri-cally quasi-neutral cosmical plasmas. The final result wouldcome out to be separate universes of matter and antimatter.

In a theory on the metagalaxy, Alfvén and Klein [9] haveearlier suggested that there should exist limited regions in ouruniverse which contain matter or antimatter, and being sepa-rated by thin boundary layers within which annihilation reac-tions take place. A simplified model of such layers has beenestablished in which the matter-antimatter “ambiplasma” isimmersed in a unidirectional magnetic field [10]. The sep-aration of the cells of matter from those of antimatter by aconfining magnetic field geometry in three spatial directionsis, however, a problem of at least the same complication asthat of a magnetically confined fusion reactor.

5 Conclusions

From the conventional equations of the gravitational field, thepoint-mass concept has in this investigation been elaboratedin terms of a revised renormalisation procedure. In a firstapplication a black hole configuration of the Schwarzschildtype has been studied, in which there is no electric charge andno angular momentum. A gravitational collapse in respect tothe nuclear binding energy is then found to occur at a criticalpoint mass in the range of about 0.4 to 90 solar masses. Thisresult becomes modified if the collapse is related to other re-strictions such as to the formation of “primordial black holes”growing by the accretion of radiation and matter [4], or tophenomena such as a strong centrifugal force.

A second application is represented by the speculationabout an extended law of gravitation, based on the optionsof positive and negative mass of a particle at a given posi-tive energy, and on the basic equations for a curl-free gravi-tational fieldstrength. This would lead to a fully symmetriclaw due to which masses of equal polarity attract each other,and masses of opposite polarity repel each other. A furtherproposal is made to associate matter and antimatter with thestates of positive and negative mass. Even under fully sym-metric conditions, this provides a mechanism for separatingantimatter from matter at an early stage of development of theuniverse.

After the completion of this work, the author has beeninformed of a hypothesis with negative mass by Choi [11],having some points in common with the present paper.


References1. Lehnert B. A revised electromagnetic theory with fundamental appli-

cations. Swedish Physics Archive. Edited by D. Rabounski, The Na-tional Library of Sweden, Stockholm, 2008; and Bogolyubov Institutefor Theoretical Physics. Edited by Z. I. Vakhnenko and A. Zagorodny,Kiev, 2008.

2. Lehnert B. Steady particle states of revised electromagnetics. Progressin Physics, 2006, v. 3, 43–50.

3. Bergmann P. G. Introduction to the theory of relativity. Prentice Hall,Inc., New York, 1942, Ch. X.

18 Bo Lehnert. The Point Mass Concept


4. Misner C. W., Thorne K. S., Wheeler J. A. Gravitation. W. H. Freemanand Co., San Francisco and Reading, 1973, Ch. 33.

5. Bethe H. A. Elementary nuclear theory. John Wiley & Sons, Inc., NewYork, Chapman and Hall, Ltd., London, 1947, pp. 8 and 17.

6. Fermi E. Nuclear physics. Revised edition. The University of ChicagoPress, 1950, p. 3.

7. Møller C. The theory of relativity. Oxford, Claredon Press, 1952,Ch. III.

8. Schiff L. I. Quantum mechanics. McGrawHill Book Comp., Inc., NewYork, Toronto, London, 1949, Ch. XII, Sec. 44.

9. Alfvén H., Klein O. Matter-antimatter annihilation and cosmology.Arkiv för Fysik, 1962, v. 23, 187–195.

10. Lehnert B. Problems of matter-antimatter boundary layers. Astro-physics and Space Science, 1977, v. 46, 61–71.

11. Choi Hyoyoung. Hypothesis of dark matter and dark energy with neg-ative mass. viXra.org/abs/0907.0015, 2010.



Quark Annihilation and Lepton Formation versus Pair Production and NeutrinoOscillation: The Fourth Generation of Leptons

T. X. Zhang

Department of Physics, Alabama A & M University, Normal, AlabamaE-mail: [email protected]

The emergence or formation of leptons from particles composed of quarks is still re-mained very poorly understood. In this paper, we propose that leptons are formed byquark-antiquark annihilations. There are two types of quark-antiquark annihilations.Type-I quark-antiquark annihilation annihilates only color charges, which is an incom-plete annihilation and forms structureless and colorless but electrically charged leptonssuch as electron, muon, and tau particles. Type-II quark-antiquark annihilation an-nihilates both electric and color charges, which is a complete annihilation and formsstructureless, colorless, and electrically neutral leptons such as electron, muon, and tauneutrinos. Analyzing these two types of annihilations between up and down quarks andantiquarks with an excited quantum state for each of them, we predict the fourth gener-ation of leptons named lambda particle and neutrino. On the contrary quark-antiquarkannihilation, a lepton particle or neutrino, when it collides, can be disintegrated intoa quark-antiquark pair. The disintegrated quark-antiquark pair, if it is excited and/orchanged in flavor during the collision, will annihilate into another type of lepton par-ticle or neutrino. This quark-antiquark annihilation and pair production scenario pro-vides unique understanding for the formation of leptons, predicts the fourth generationof leptons, and explains the oscillation of neutrinos without hurting the standard modelof particle physics. With this scenario, we can understand the recent OPERA measure-ment of a tau particle in a muon neutrino beam as well as the early measurements ofmuon particles in electron neutrino beams.

1 Introduction

Elementary particles can be categorized into hadrons and lep-tons in accord with whether they participate in the strong in-teraction or not. Hadrons participate in the strong interaction,while leptons do not. All hadrons are composites of quarks[1-3]. There are six types of quarks denoted as six differentflavors: up, down, charm, strange, top, and bottom, whichare usually grouped into three generations: {u, d}, {c, s}, {t, b}.Color charge is a fundamental property of quarks, which hasanalogies with the notion of electric charge of particles. Thereare three varieties of color charges: red, green, and blue. Anantiquark’s color is antired, antigreen, or antiblue. Quarksand antiquarks also hold electric charges but they are frac-tional, ±e/3 or ±2e/3, where e = 1.6 × 10−19 C is the chargeof proton.

There are also six types of leptons discovered so far,which are electron, muon, and tau particles and their cor-responding neutrinos. These six types of leptons are alsogrouped into three generations: {e−, νe}, {µ−, νµ}, {τ−, ντ}. Theantiparticles of the charged leptons have positive charges. Itis inappropriate to correspond the three generations of lep-tons to the three generations of quarks because all these threegenerations of leptons are formed or produced directly in as-sociation with only the first generation of quarks. We are stillunsure that how leptons form and whether the fourth genera-

tion of leptons exists or not [4-8].In this paper, we propose that leptons, including the fourth

generation, are formed by quark-antiquark annihilations.Electrically charged leptons are formed when the colorcharges of quarks and antiquarks with different flavors areannihilated, while neutrinos are formed when both the elec-tric and color charges of quarks and antiquarks with the sameflavor are annihilated. We also suggest that quarks and anti-quarks can be produced in pairs from disintegrations of lep-tons. This quark-antiquark annihilation and pair productionmodel predicts the fourth generation of leptons and explainsthe measurements of neutrino oscillations.

2 Quark Annihilation and Lepton Formation

Quark-antiquark annihilation is widely interested in particlephysics [9-13]. A quark and an antiquark may annihilateto form a lepton. There are two possible types of quark-antiquark annihilations. Type-I quark-antiquark annihilationonly annihilates their color charges. It is an incomplete anni-hilation usually occurred between different flavor quark andantiquark and forms structureless and colorless but electri-cally charged leptons such as e−, e+ , µ−, µ+, τ−, and τ+.Type-II quark-antiquark annihilation annihilates both electricand color charges. It is a complete annihilation usually oc-curred between same flavor quark and antiquark and forms

20 T. X. Zhang. Quark Annihilation and Lepton Formation versus Pair Production and Neutrino Oscillation


Fig. 1: Formation of the four generations of leptons by annihilationsof up and down quarks and antiquarks with an excited quantum state.

structureless, colorless, and electrically neutral leptons suchas νe, ν̄e, νµ, ν̄µ, ντ, and ν̄τ.

Mesons are quark-antiquark mixtures without annihilat-ing their charges. For instance, the meson pion π+ is a mix-ture of one up quark and one down antiquark. Meson’s colorcharges are not annihilated and thus participate in the stronginteraction. Leptons do not participate in the strong inter-action because their color charges are annihilated. Particlesformed from annihilations do not have structure such as γ-rays formed from particle-antiparticle annihilation. A baryonis a mixture of three quarks such as that a proton is composedof two up quarks and one down quark and that a neutron iscomposed of one up quark and two down quarks.

Recently, Zhang [14-15] considered the electric and colorcharges of quarks and antiquarks as two forms of imaginaryenergy in analogy with mass as a form of real energy and de-veloped a classical unification theory that unifies all naturalfundamental interactions with four natural fundamental ele-ments, which are radiation, mass, electric charge, and colorcharge. According to this consideration, the type-I quark-antiquark annihilation cancels only the color imaginary en-ergies of a quark and a different flavor antiquark, while thetype-II quark-antiquark annihilation cancels both the electricand color imaginary energies of a quark and a same flavorantiquark.

Figure 1 is a schematic diagram that shows formationsof four generations of leptons from annihilations of up anddown quarks and antiquarks with one excited quantum statefor each of them. The existence of quark excited states,though not yet directly discovered, has been investigated overthree decades [16-18]. That ρ+ is also a mixture of one up

Quarks u0 d0 u1 d1ū0 νe, ν̄e e− - τ−

d̄0 e+ νµ, ν̄µ µ+ -ū1 - µ− ντ, ν̄τ λ−

d̄1 τ+ - λ+ νλ, ν̄λ

Table 1: The up and down quarks and antiquarks in ground andexcited quantum states and four generations of leptons

quark and one down antiquark but has more mass than π+ andmany similar examples strongly support that quarks and anti-quarks have excited states. In Figure 1, the subscript ’0’ de-notes the ground state and ’1’ denotes the excited state. Thehigher excited states are not considered in this study. Thedashed arrow lines refer to type-I annihilations of quarks andantiquarks that form electrically charged leptons, while thesolid arrow lines refer to type-II annihilations of quarks andantiquarks that form colorless and electrically neutral leptons.These annihilations of quarks and antiquarks and formationsof leptons can also be represented in Table 1.

The first generation of leptons is formed by annihilationsbetween the ground state up, ground state antiup, ground statedown, and ground state antidown quarks (see the red arrowlines of Figure 1). The up quark u0 and the antiup quark ū0completely annihilate into an electron neutrino νe or an elec-tron antineutrino ν̄e. The antiup quark ū0 and the down quarkd0 incompletely annihilate into an electron e−. The up quarku0 and the antidown quark d̄0 incompletely annihilate into apositron e+.

The second generation of leptons are formed by annihila-tions between the ground state down, ground state antidown,excited up, and excited antiup quarks (see the blue arrow linesof Figure 1). The down quark d0 and the antidown quark d̄0completely annihilate into a muon neutrino νµ or an antimuonneutrino ν̄µ. The antiup quark ū1 and the down quark d0 in-completely annihilate into a negative muon µ−. The up quarku1 and the antidown quark d̄0 incompletely annihilate into apositive muon µ+.

The third generation of leptons are formed by annihila-tions between the ground state up, excited up, ground stateantiup, excited antiup, excited down, and excited antidownquarks (see the green lines of Figure 1). The up quark u1 andthe antiup quark ū1 completely annihilate into a tau neutrinoντ or a tau antineutrino ν̄τ. The antiup quark ū0 and the downquark d1 incompletely annihilate into a negative tau τ−. Theup quark u0 and the antidown quark d̄1 incompletely annihi-late into a positive tau τ+.

The fourth generation of leptons are formed by annihila-tions between excited up, excited antiup, excited down, andexcited antidown quarks (see the purple lines of Figure 1).The down quark d1 and the antidown quark d̄1 completely an-nihilate into a lambda neutrino νλ or a lambda antineutrino

T. X. Zhang. Quark Annihilation and Lepton Formation versus Pair Production and Neutrino Oscillation 21


Fig. 2: Formation of the first generation of leptons: (a) e− and ν̄ethrough beta decay and (b) e+ and νe through positron emission.

ν̄λ. The antiup quark ū1 and the down quark d1 incompletelyannihilate into a negative lambda λ−. The up quark u1 andthe antidown quark d̄1 incompletely annihilate into a positivelambda λ+.

3 Quark Pair Production and Lepton Disintegration

The first generation of leptons can be produced through thebeta decay of a neutron, n −→ p + e− + ν̄e (Figure 2a), andthe positron emission of a proton, energy + p −→ n + e+ + νe(Figure 2b).

In the beta decay, an excited down quark in the neutrondegenerates into a ground state down quark and an excited upand antiup quark pair, d1 −→ d0 + (u1ū1). The excited antiupquark further degenerates into a ground state up quark and aground state up and antiup quark pair, ū1 −→ ū0 + (u0ū0). Theground state antiup quark incompletely annihilates with theground state down quark into an electron, ū0 +d0 −→ e−. Theground state up and antiup quark pair completely annihilatesinto an electron antineutrino, u0 + ū0 −→ ν̄e.

In the positron emission, an excited up quark in the posi-tron after absorbing a certain amount of energy degeneratesinto a ground state up quark and produces an excited statedown and antidown quark pair, energy + u1 −→ u0 + (d1d̄1).The excited antidown quark further degenerates into a groundstate antidown quark and produces a ground state up and an-tiup quark pair, d̄1 −→ d̄0 + (u0ū0). The ground state upquark incompletely annihilates with the ground state anti-

Fig. 3: Production of other three electrically charged leptons froman energetic electron-positron collision. In the collision, electronand positron are first disintegrated into quark-antiquark pairs, whichare then excited and annihilated into other generations of electricallycharge leptons.

down quark to form a positron, u0 + d̄0 −→ e+. The groundstate up and antiup quark pair completely annihilates into anelectron neutrino, u0 + ū0 −→ νe.

The other three generations of electrically charged leptonscan be produced by an energetic electron-positron collision,

energy + e− + e+ −→µ− + µ+

τ− + τ+

λ− + λ+, (1)

as also shown in Figure 3. In the particle physics, it has beenexperimentally shown that the energetic electron-positroncollision can produce (µ−, µ+) and (τ−, τ+). But how theelectron-positron collisions produce µ and τ leptons is stillremained very poorly understood.

With the quark annihilation and pair production modelproposed in this paper, we can understand why an electron-positron can produce µ and τ particles. In addition, we predictthe existence of the fourth generation of leptons, λ particleand neutrino. The energetic electron-positron collision disin-tegrates the electron into a ground state antiup-downquark pair e− −→ (ū0d0) and the positron into a ground stateup-antidown quark pair e+ −→ (u0d̄0). During the collision,the quarks and antiquarks in the disintegrated electron andpositron quark-antiquark pairs absorb energy and become ex-cited. The excited quark-antiquark pairs incompletely anni-hilate into another generation of electrically charged leptons.

There are three possible excitation patterns, which lead tothree generations of leptons from the electron-positron colli-sion. If the antiup quark in the disintegrated electron quark-antiquark pair and the up quark in the disintegrated positronquark-antiquark pair are excited, then the annihilations pro-duce leptons µ− and µ+. If the down quark in the disinte-grated electron quark-antiquark pair and the antidown quarkin the disintegrated positron quark-antiquark pair are excited,



then the annihilations produce leptons τ− and τ+. If both theantiup and down quarks in the disintegrated electron quark-antiquark pair and both the up and antidown quarks in thedisintegrated positron quark-antiquark pair are excited, theannihilations produce the leptons λ− and λ+. An electron-positron collision in a different energy level produces a dif-ferent generation of electrically charged leptons. To producethe λ particles, a more energetic electron-positron collision isrequired than µ and τ lepton productions. On the other hand,the electron and positron, if they are not disintegrated intoquark-antiquark pairs during the collision, can directly anni-hilate into photons. The disintegrated electron and positronquark-antiquark pairs, if they are excited but not annihilated,can form the weak particles W− and W+.

A quark or antiquark can be excited when it absorbs en-ergy or captures a photon. An excited quark or antiquark candegenerate into its corresponding ground state quark or an-tiquark after it releases a photon and/or one or more quark-antiquark pairs. The decays of these three generations of elec-trically charged leptons (µ, τ, and λ particles) can producetheir corresponding neutrinos through degenerations and an-nihilations of quarks and antiquarks.

The currently discovered three generations of leptons in-cluding the fourth generation predicted in this paper areformed through the annihilations of the up and down quarksand antiquarks with an excited state. All these leptons are cor-responding to or associated with the first generation of quarksand antiquarks. Considering the annihilations of other fourflavor quarks and antiquarks, we can have many other types ofleptons that are corresponding to the second and third genera-tions of quarks and antiquarks. These leptons must be hardlygenerated and observed because a higher energy is required[4].

4 Quark Annihilation and Pair Production: NeutrinoOscillation

The complete (or type-II) annihilation between a quark andits corresponding antiquark forms a colorless and electricallyneutral neutrino. On the contrary quark-antiquark annihila-tion, a neutrino, when it collides with a nucleon, may bedisintegrated into a quark-antiquark pair. The disintegratedquark-antiquark pairs can be excited if it absorbs energy (e.g.,γ + u0 −→ u1) and changed in flavor if it exchanges a weakparticle (e.g., u0 + W− −→ d0) during the disintegration. Theexcited and/or flavor changed quark-antiquark pair then ei-ther annihilates into another type of neutrino or interacts withthe nucleon to form hadrons and electrically charged leptons.This provides a possible explanation for neutrino oscillations[19-20]. This scenario of neutrino oscillations does not needneutrinos to have mass and thus does not conflict with thestandard model of particle physics.

Figure 4 and 5 show all possible oscillations among thefour types of neutrinos. An electron neutrino can oscillate

Fig. 4: Neutrino oscillations. (a) Oscillation between electron andtau neutrinos. (b) Oscillation between electron and muon neutrinos.(c) Oscillation between electron and lambda neutrinos.

into a tau neutrino if the disintegrated quark-antiquark pair(u0ū0) is excited into (u1ū1) (Figure 4a), a muon neutrinoif the disintegrated quark-antiquark pair (u0ū0) is changedin flavor into (d0d̄0) (Figure 4b), and a lambda neutrino ifthe disintegrated quark-antiquark pair (u0ū0) is excited into(u1ū1) and then changed in flavor into (d1d̄1) (Figure 4c).Similarly, A muon neutrino can oscillate into a tau neutrinoif the disintegrated quark-antiquark pair (d0d̄0) is excited andchanged in flavor into (u1ū1) (Figure 5a) and a lambda neu-trino if the disintegrated quark-antiquark pair (d0d̄0) is ex-cited and changed into (d1d̄1) (Figure 5b). A tau neutrino canoscillate into a lambda neutrino if the disintegrated quark-antiquark pair (u1ū1) is changed in flavor into (d1d̄1) (Figure5c). All these oscillations described above are reversible pro-cesses. The right arrows in Figures 4 and 5 denote the neu-trino oscillations when the disintegrated quark-antiquark pairabsorbs energy to be excited or capture weak particles to bechanged in flavor. Neutrinos can also oscillate when the dis-integrated quark-antiquark pair emits energy and/or releasesweak particles. In this cases, the right arrows in Figure 4and 5 are replaced by left arrows and neutrinos oscillate fromheavier ones to lighter ones.

The recent OPERA experiment at the INFN’s Gran Sassolaboratory in Italy first observed directly a tau particle in amuon neutrino beam generated by pion and kaon decays andsent through the Earth from CERN that is 732 km away [21-23]. This significant result can be explained with a muonneutrino disintegration, excitation, and interaction with a nu-

T. X. Zhang. Quark Annihilation and Lepton Formation versus Pair Production and Neutrino Oscillation 23


Fig. 5: Neutrino oscillations. (a) Oscillation between muon andlambda neutrinos. (b) Oscillation between muon and tau neutrinos.(c) Oscillation between tau and lambda neutrinos.

cleon. Colliding with a neutron, a muon neutrino νµ is disin-tegrated into a ground state down-antidown quark pair (d0d̄0),which can be excited into (d1d̄1) and (u1ū1) when the flavor isalso changed. The excited down-antidown quark pair (d1d̄1)can either completely annihilate into a lambda neutrino νλ(Figure 5a) or interact with the neutron to generate a nega-tive tau particle τ− when the excited antidown quark degen-erates into a ground state antidown and a ground state up-antiup quark pair, d̄1 −→ d̄0 + (u0ū0) (Figure 6a). As shownin Figure 6a, the excited down quark in the neutron can in-completely annihilate with the ground state antiup quark intoa negative tau particle τ− and the ground state antidown quarkcan incompletely annihilate the ground state up quark into apositron e+. Interacting with a proton (Figure 6b), the ex-cited down-antidown quark pair (d1d̄1) can generate a posi-tive tau particle τ+ when the excited up quark in the protondegenerates into a ground state up quark and a ground stateup-antiup quark pair, u1 −→ u0 + (u0ū0). The excited antid-own quark can incompletely annihilate with the ground stateup quark into a positive tau particle τ+ and the ground stateantiup quark can completely annihilate with the ground stateup quark into an electron neutrino νe. If the excited up quarkis not degenerated but directly annihilate with the excited an-tidown quark, a lambda particle λ+ is produced (as shown inFigure 3).

On the other hand, for an electron neutrino beam, collid-ing with a nucleon, an electron neutrino νe is disintegratedinto a ground state up-antiup quark pair (u0ū0) and excited

Fig. 6: Production of tau particles by a muon neutrino beam. (a) Anegative tau particle is produced when an excited quark-antiquarkpair, which is disintegrated from a muon neutrino and excited, in-teracts with a neutron. (b) A positive tau particle is produced whenan excited quark-antiquark pair, which is disintegrated from a muonneutrino and excited, interacts with a proton.

into (u1ū1), which may be also from the disintegration of amuon neutrino with the favor change. This excited up-antiupquark-antiquark pair can either completely annihilate into atau neutrino as shown in Figure 1 or interact with the nucleonto generate a muon particle (Figure 7). If the flavor is alsochanged, the annihilation and interaction with nucleons willproduce the tau particles and neutrinos as shown in Figure 6or lambda particles and neutrinos as shown in Figure 3.

Therefore, with the lepton formation and quark-antiquarkpair production model developed in this paper, we can under-stand the recent measurement of a tau particle in a muon neu-trino beam as well as the early measurements of muon parti-cles in electron neutrino beams. More future experiments ofthe Large Electron-Positron Collider at CERN and measure-ments of neutrino oscillations are expected to validate thislepton formation and quark-antiquark pair production modeland detect the fourth generation of leptons.

5 Conclusions

This paper develops a quark-antiquark annihilation and pairproduction model to explain the formation of leptons and theoscillation of neutrinos and further predict the fourth gener-ation of leptons named as lambda particle and neutrino. It



Fig. 7: Production of muon particles by an electron neutrino beam.(a) A negative muon particle is produced when an excited up-antiupquark pair, which is disintegrated from an electron neutrino and ex-cited, interacts with a neutron. (b) A positive tau particle is producedwhen an excited quark-antiquark pair, which is disintegrated from amuon neutrino and excited, interacts with a proton.

is well known that all known or discovered le

including related themes from mathematics progress in …fs.unm.edu › pip-2011-02.pdfvolume 2...

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