1


Electromagnetic structure
of hadrons

Alexander G. Kyriakos


Saint-Petersburg State Institute of Technology,
St.Petersburg, Russia



Present address:
Athens, Greece
e-mail: lelekous@otenet.gr



Abstract

In the previous papers [1,2,3], based on Dirac's equation, we have
considered the electromagnetic structure of the leptons. In the
present paper, using the Yang-Mills equation, we will analyse the
electromagnetic structure of the hadrons.


PASC 12.10.-g Unified field theories and models.
PASC 12.90.+b Miscellaneous theoretical ideas and models.



Contents

1.Introduction
1.1 The Yang-Mills equation and QCD
1.2 The investigation object choice
1.3 Problem statement
2."One quark" theory
2.1 Inner and external fields
2.2 The "one quark" equation
2.3 Electromagnetic form of "one quark" equation
2.4 The "one quark" linear equation Lagrangian
2.5 Lagrangian of non-linear "one quark" equation
3.Quantum and electromagnetic forms of "three quark"
equations
3.1 Electromagnetic forms of quantum equations
3.2 Rings or knots?
3.3 "Three quarks" equation without interaction.
3.4 Interaction appearance
4. Electromagnetic "two quark" equations
5. About symmetry breaking and mass appearance
Appendix 1. Hadron models
A1 "Three quarks" model
A2 "Two quarks" model
Conclusion
References


2


1. Introduction


As it is known the modern Standard Model Theory of elementary
particles is described by the Yang-Mills equation [4].


1.1 The Yang-Mills equation and QCD


The Standard Model Theory contains the electroweak theory,
which is based on the SU(2)xU1 symmetry group, and the strong
interaction theory (SU(3) symmetry group). Both theories are built
on the base of the Yang-Mills equation in the same way.
Let's introduce briefly the QCD theory [4,5,6]. As it is
known, the QCD theory and the QED theory forms are alike.
Particularly, the Lagrangian and the equation of the QCD have the
same form as the Lagrangian and the equation of the QED [4,5].
Since it is important for our theory understanding, we
consider at first the theory structure of QED.
In the usual notations [5] the Lagrangian of QED is
represented by the sum:
1
L = + - -
QED ([i eA m
  )  ] 
F 
F , (1.1)
4
where =
 / x is the partial derivative with respect to 4-co-
ordinate x = -
 , F A A
   is the operator of electromagnetic

field strength, - e, m are the electrical charge and the mass of

the electron, respectively;  are the Dirac's matrices, is the

hermitian wave function, 
A is the potential of the

electromagneticl field (the summation is always taken with respect
to same indeces). The first and the third summands of the
Lagrangian describe the free motion of electron, the last summand
describes the same of photons, and the term

A describes its

interaction. Using the covariant derivative
D = - ieA
   , (1.2)
we can write the Lagrangian in the form:
1
L = - -
QED [D m
  ] 
F 
F , (1.3)
4
The physical means of the present theory will be understood
better, if we use the our notations [1]. In this notations we have
for QED Lagrangian:
L = L + Lint + L
QED D M , (1.4)

where
L = + + ! ! +
D ( ^ ^
^
^ ^ 2

c p m c
e
0 ) ,
(1.5)


is the free electron Lagrangian.
The Lagrangian:
L = - + +
int (0 c !^
^ p!
ex ex )
, (1.6)

is the interaction Lagrangian, and


3


1 ! !
2 2
L = -
M (E H )
8 , (1.7)
is the Maxwell theory Lagrangian.
Thus, the full Lagrangian of QED theory is:
! ! ! ! !
L = +
- + - + + -
QED [^ ( ex) ^
^ c (^p pex) 2 1
^ m c E H
0 e ] ( 2 2), (1.8)
8
^! ^ ! !
^
where ^ , ^ = i" , p = -i
0 , - are Dirac's matrices,
"
- the
t
+
operators of energy and momentum, is the Hermitian wave
e
! !
function, = e, p = A
ex ex are the external electron energy and
c ! !
momentum , c is the light velocity, m E,
e is the electron mass, H
( !
, A)
are the electric and magnetic field, respectively, and is 4-
^! ^
potential of external field. The matrices ^ ,
0 , are the

following Dirac's matrices:
1 0 0 0 0 0 0 1

0 1 0 0 0 0 1 0
^
= , ^
=
0 1 ,
0 0 1 0 0 1 0 0 1 0 0 0

0 0 0 1 1 0 0 0 ^ 0 1 0 0
! = ,(1.9)
0 0 0 - i 0 0 1 0 4
0 0 -1 0

0 0 i 0 0 0 0 -
1 0 0 0 -1
^
=
, ^
=
2 3
0 - i 0 0 1 0 0 0

i 0 0 0 0 -1 0 0

In Quantum Chromodynamics (QCD) we have quarks instead of
electrons and gluons instead of photons, between which there are
the strong interactions instead of electromagnetic interactions.
The strong interactions among quarks and gluons are described by
the non-abelian gauge theory, based on the gauge group SU )
3
( C ,
instead of the abelian gauge theory of the EM theory, based on the
gauge group U )
1
( . Each quark sort (flavour) corresponds to a
colour (strong interaction charge) triplet in the fundamental
representation of SU )
3
( and the gauge fields needed to maintain
the gauge symmetry, the gluons, are in the adjoint representation
of dimension 8. Gauge invariance ensures that gluons are massless
whose spin is equal to 1, as the photon. Since the gluons masses
are equal to zero, the quark interaction radius is equal to
infinity, but the sizes of the quark system (particles) are very
-
small (in order 10 13 cm). The explanation of this fact is not
known (the confinement problem).
The QCD Lagrangian may by written in ordinary notations
[4,5,6] as
1
L = - -
QCD i[D m
  ] i 
F 
F , (1.10)
4


4




a
where a
F = F
  , (1.11)
2

stands for the gluon field tensor, i are the quark fields and the
covariant derivative is defined by
D = -
  
igA , (1.12)

The strong coupling is represented by g and indices are summed

over a = 8
,...,
1 and over i = ,
1 3
,
2 . Finally, a 2 and f SU
abc are the )
3
(

group generators and structure constants, respectively, which are
related to the commutator:
[ , = if
2
a b ] c
abc , (1.13)
For the gluons the field strength has the view:
F = A - - -
  
A ig[ 
A
A
A 
A ], (1.14)
or in the vector form:
! ! ! ! !
F = A - A + gA  A
    , (1.15)

The photon potentials A are the numbers and in the QED case
the above commutator is equal to zero. In the case of the non-
abelian gauge fields (gluons) A are the matrices, and the above
comutator is not equal to zero. Thanks to this commutator, the
non-linear interaction of the gluons appears.

The matrices a are named Gell-Mann matrices and have the
view:
0 1 0 0 i 0 1 0 0
-
= , = i , = ,
3 0 -1 0
2 0 0
1 1 0 0

0 0 0
0 0 0
0 0 0

0 0 1 0 0 - i 0 0 0

= 0 0 0 , = 0 0 0 , = 0 0 1 , (1.16)
4 5 6
1 0 0 i 0 0 0 1 0
0 0 0 1 0 0

= 0 0 - i , = 0 1 0 .
7 8
0 i 0 0 0 - 2
If we consider the quark as real fermion, the contradiction
with Pauli principle appears. To eliminate this contradiction some
inner degree of freedom is introduced, named colour (or colour
charge).
According to modern theory in strong interactions the colour
charges play the same role as the electrical charges in the
electromagnetic interactions. The interaction between electrons
takes place through the photon exchange. In this case gluons play
the role of photons, which are also electrically neutral vector
particles. The basic difference between gluons and photons is that
the photon is one, but gluons are 8 and have colour charges.


5


Thanks to these colour charges the gluons interact strongly with
one another and can radiate one another.


1.2 The investigation object choice


Apparently the simplest of the hadrons must be the scope of
our investigation. There is a reason to believe that the simplest
particles are particles with the lower mass of the mass spectrum
of each particle family. In the case of the baryons the simplest
particles are nucleons - proton (antiproton) and neutron
(antineutron); among the mesons these are the pi mesons. We will
name these particles the basic particles.
As it is known, the basic particles consist only of two
quarks u and d:
1
+ - 0
= ud , = du, = (uu - dd ), p = uud, n = ddu
2

which have the following charges:
2 1
q = e, q = - e
u d
3 3

The quark masses don't depend on their colour, but depend on the
flavour. The u and d quarks have aproximetly the following masses:

m 4 MeV , m 7 MeV
u d


1.3 Problem statement


The problem of the present paper is to show the
electromagnetic structure of the hadrons. As it is followed from
the Standard Model Theory (SMT), the quark family is analogue to
the lepton family.
Based on this analogy we suppose that:
1) The electromagnetic structure of the hadrons is similar to the
lepton structure;
2) the above hadron Lagrangian (and equations) is composed from
three Lagrangians (equations) of the lepton type, i.e. by three
Dirac's equation Lagrangians (three Dirac's equations).
In other words, we must show that the lepton Lagrangian (also
lepton equation) is the one quark Lagrangian (equation).



2. "One quark" theory


2.1 Inner and external fields


The comparison between the QCD and the QED Lagrangians, that
we made above, is not entirely right. The QED Lagrangian is not
the free lepton Lagrangian, but the Lagrangian with interaction
between leptons and photons. On the other hand, the above
Lagrangian of hadrons is indeed the Lagrangian of the free hadron.
This distinction is bounded with the distinction among inner
and external fields of the particles. The external field, used in


6


the QED for the description of the interaction among the electron
and other charge particles, is in the case of hadron the inner
field, describing the quark-quark interaction.
Thus, it follows from above that the "one quark" Lagrangian
and equation must be the QED Lagrangian (equation) for external
field, but inside the hadron this "external field" describes the
quark-quark interaction.



2.2 The "one quark" equation


For the solution of the present problem we use here some
results, which were represented in detail in the paper [1].
Particularly, to understand the "one quark" equation form, we
consider the electromagnetic and quantum forms of electron-positron
pair production theory.
Let's consider the particle-antiparticle production
conditions. One -quantum cannot turn spontaneously into the
electron-positron pair, although it interacts with the electron-
positron vacuum. For the pair production, at first, the following
2
mass correlation is necessary: 2m c
p e (where p is the photon

energy, me - the electron mass and c - the light velocity). At
second, the presence of the other particle, having the
electromagnetic field, is needed. It can be some other -quantum,
- +
the electron e , e , an atom nucleus Ze etc. For example, we have

the typical reaction:
+ -
+ Ze Ze + e + e , (2.1)
which means that, while moving through the particle field (as in
medium with high refraction index) the photon (or maybe virtual
photon) takes some transformation, which corresponds to the pair
production. Considering the fact that Pauli's matrices describe
the vector rotations and also taking in account the optical-
mechanical analogy analysis, we can assume that the above
transformation is a field distortion. From this follows the
distortion hypothesis:
By the fulfilment of the pair production conditions
the distortion of the electromagnetic field of the photon can take
place; as a result photon is able to move along the closed
trajectory, making some stable construction named elementary
particle.
About the electron, as the simplest particle, we can suppose
that the photon trajectory is circular.
Consider the linear photon, moving along - y -axis (of course,
we can also use any other direction [2]). In a general case it has
the two possible polarisations and contains the field vectors
E , E , H , H (E H
y = y =
x z x z )
0 . Such photon can form the ring only on the
(x, ,
o y) or the ( y, ,
o z) plains.
The bispinor form of free lepton Dirac's equations can be
written as one equation [7]:


7


^! ^
^ ! ^ 2
+
c p + m c
e = ,
0 (2.2)

(the notations see above).
Put the following semi-photon bispinor:
E
z
Ex
= iH (2.3)
z
iH x
Using (2.3), we can write the equation of the electromagnetic
wave moved along any axis in form:
(# 2 2
- c # 2
p! ) = ,0
(2.4)

The equation (2.4) can also be written in the following form:

2 2
2 ! !
( #
- c p 0
o #
) ( # #)
= , (2.5)
where 1^
^o = is the unit matrix. In fact, taking into account that
2
2 2 ! ! !2
( #
= p = p
o #
) # , ( # #) # ,
we see that equations (2.4) and (2.5) are equivalent.
Factorising (2.5) and multiplying it from left on Hermithian-

conjugate function + we get:
+ (# # - c#! #p!) (# # + c#! #p! 0
o o ) = , (2.6)

The equation (2.6) may be disintegrated on two equations:
+
( ^! ^
^
^ !

o - c p)= 0, (2.7)
( # +c ! p! = 0
o #
#
#) , (2.8)

It is not difficult to show (using (2.2)) that the equations (2.7)
and (2.8) are Maxwell's equations without current and, at the same
time, are Dirac's electron-positron equations without mass.
In accordance with our assumption, the reason for current
appearance must be the electromagnetic wave motion along a
curvilinear trajectory. We will show the appearance of the current,
using the general methods of the distortion field investigation
[8], but the same result can be obtained simplier in the vector
form (see [1] Appendix 1, chapter A1.2.). The question is about the
tangent space introduction at every point of the curvilinear space,
in which the orthogonal axis system moves. This corresponds to the
fact, that the wave motion along a circular trajectory is
accompanied by the motion of the rectangular basis, built on
! ! ! !
vectors ( E, S , H ), where S is the Poynting vector.

For the generalisation of Dirac's equation in Riemann's
geometry it is necessary [8] to replace the usual derivative
/
 x (where x are the co-ordinates in the 4-space) with the
covariant derivative: D = +  =
   ( ,
0 ,
1 ,
2 3 are the summing

indexes), where  is the analogue of Christoffel's symbols in the
case of the spinors theory. When a spinor moves along the beeline,

all  = 0 , and we have a usual derivative. But if a spinor moves


8



along the curvilinear trajectory, then not all  are equal to zero
and a supplementary term appears. Typically, the last one is not
the derivative, but it is equal to the product of the spinor itself

with some coefficient  . Thus we can assume that the supplementary
term is a longitudinal field, i.e. a current. So from (2.7-2.8) we
obtain:
 
D
 = (  + )
 = 0 ,
According to the general theory [8] the increment in spinor 
has the form of the energy-momentum 4-vector. It is logical (see

also [1], Appendix 1, chapter A1.2.) to identify  with 4-vector
of energy-momentum of the electron's own field:

=
 { p
c!
,
s s }, (2.9)

Then equations (2.7) and (2.8) in the curvilinear space will have
the view:
+
[ ( ^! ^
^
^ c p! c !
p!
o - ) -( ^
^o s - s ) ] = 0 ,

(2.10)
[ ( ^! ^
^
^ c p! c ! p!
o + ) +( ^
^
os + s ) ] = 0, (2.11)
According to the energy conservation law we can write:
! ! 2
^ ^
^
 c p = $ m c
o s s e , (2.12)

Substituting (2.12) in (2.10) and (2.11) we will arrive at the
usual kind of Dirac's equation with the mass:
+
[ ( ^! ^
^
^ !
o - c p) ^ 2
- m c
e ]= 0 , (2.13)
[ ( ^! ^
^
^ !
o +
c p) ^ 2
+ m c
e ] = 0, (2.14)

Figure 1 illustrates the above description of the process of
electron-positron pair generation:





Fig.1



It is not difficult to see that the pair production process
corresponds to the photon division on two circular semi-periods
(semi-photons). According to our supposition the "one quark" is the
ring or knot (loop) of the same type.


9





2.3. Electromagnetic form of "one quark" equation


Taking into account that = (y) , from (2.13) using (2.14) we
obtain:
!
! !
rot E + 1 H = i H,
c t c
! , (2.15)
! 1 E !
rot H - = -i E,
c t c
2
m c
where e
= . The equations (2.15) are the Maxwell equations with
"
current [9]. It is interesting that along with the electrical
current the magnetic current also exists here. This current is
equal to zero by Maxwell's theory, but its existence by Dirac
doesn't contradict to the quantum theory. (As we have showed [3]
the magnetic current appearance relates to the initial photon
circular polarisation and integrally this current is equal to
zero).


2.4. The "one quark" linear equation Lagrangian


As it is known [9], the Lagrangian of the free field Maxwell's
theory is:
1 ! !
2 2
L = -
M (E H )
8 (2.16)
The following expression can be taken as Lagrangian of Dirac's
theory [7]:
L = + + ! ! +
D ( ^ ^
^
^ ^ 2

c p m c
e
0 )
, (2.17)

For the wave moving along the y -axis the equation (2.17) can be

written:
1 m c
L + +
# i e
= - - +
# ,
D (2.18)
c t y y "
Transferring each term of (2.18) in electrodynamics' form (see [1])
we obtain for the semi-photon or "one quark" equation the following
Lagrangian:
! ! ! !
L = +
+ - -
D ( ^ ^
^ ^ c p i s E H
0 ) 2 2
( ), (2.19')
8
or in ordinary form:
! !
s 2 2
L = +

 - i -
D (E H ), (2.19'')
8
2
2m c
where e
=
s (note that we must differ the complex conjugate
" ! ! ! !
field vectors E * , H * and E , H ).
As we have supposed, the Lagrangian (2.19'') is the
Lagrangian of "one quark", and it is actually similar to (1.10).


10


We can say that the Lagrangian (1.10) is constructed from
Lagrangians of three interacting quarks. Therefore, to obtain
(1.10) we must sum three Lagrangians (2.19'') and "turn on" the
interactions among quarks (i.e. to go over from simple derivative
to the covariant derivative).
1
Now we will show that the photon interaction term 
F 
F can
4
be represented as non-linear expression the same type as (1.15),
which describes the photon-photon (i.e. gluon-gluon) interactions.



2.5. Lagrangian of non-linear "one quark" equation


Using (2.10) we can write (see [1]) the Lagrangian of "one
quark" equation in the form:

L = + - c !^
^ p!^ + + - c !^ p!
N ( ) (s s )
, (2.20)

Let's show that the expression (2.20) represents the common form of
Lagrangian of non-linear "one quark" (or "one ring", or "one knot",
or electron Dirac's) equation.
Taking into account that the free electron Dirac's equation solution
is the plane wave:
( )

= -
ei t ky
0 , (2.21)
we can represent (2.20) in the approximate quantum form:
+ s
LN = ^ !

 + ( +
)2 - ( + ^

)2 , (2.22)
8
where
s is the "volume" of the particle.
Transform (2.20) into electrodynamics form. For energy and momentum
in the electromagnetic form we have [1]:
1
! !
= U d = +
s
( 2 2
E H )
d , (2.23)
8



1
! ! ! 1 !
p = !
g d = = 
s S d E H d
(2.24)
2 [ ] ,
c 4






Normalising -function by relationship:
' 1
L N = L , (2.25)
2 N
8 m c
e

and using the equations (2.23) and (2.24), we find:
' 2 2 ! 2
L N = +
^
 + 2 (U - c g ), (2.26)
It is not difficult to transform the second summand, using the
known electrodynamics' transformation (that is quantum form of the
known Fierz's correlation):
2 2 2 ! ! ! ! ! ! ! ! !
2 2 2 2 2 2 2 2 2
(8 ) (U - c g ) = (E + H ) - 4(E  H) = (E - H ) + (
4 E H) , (2.27)
So we have: ! ! ! !
'
L N = +
^
 + 2 ([ 2
E + H )2
2 - 4 (E  H )2 ], (2.28)
or


11


! ! ! !
'
L N = +
^
 + 2 ([ 2
E - H )2
2 + 4 (E H )2], (2.28')
We can say that this Lagrangian is the "one quark" non-linear
equation Lagrangian.
Actually, as we have shown in the paper [1], the Lagrangian
(2.28) is similar to the Lagrangian of the photon-photon
interaction [10]:





1 ! ! ! ! ! !
L E H b E H E H
p- p = ( 2 2
- )+ ([ 2 - )22 +7( )2]+...,
8 (2.29)
where b is constant.

Using (2.28) we can build the "three quark" Lagrangian of the

same type as (1.10) with (1.15).


3. Quantum and electromagnetic forms of "three quark"
equations

3.1 Electromagnetic forms of quantum equations


It has been known for a long time [10,11] that the quantum
equations can formally be represented as the Maxwell equation
system. For example, according to [11] the spinor Dirac's equation
system
[ ^! ^! 2
(^ ^
o -
c p) - mc ] = 0
[ , (3.1)
^! ^! 2
(^ ^
o +
c p) + mc ] = 0
becomes the full Maxwell equation system, if we use instead of
and 2x1-matrix wave functions, the 3x1-matrix electromagnetic
field function:
= (! !
E), = (iH ), (3.2)

and instead of 2x2-spinor Pauli's matrices ^ we use the following
3x3-matrices:
0 0 0 0 0 i 0 i 0
-
^
^
^
S i S S i
3 = 0 0
2 = 0 0 0
1 = 0 0 - , , , (3.3)

0 i 0
i 0 0
0 0 0
The zero and unit 3x3-matrices are also useful:
1 0 0 0 0 0

^
S 0^ = 0 0 0
0 = 0 1 0 , , (3.4)

0 0 1
0 0 0
As it
is known [4], these matrices are the rotation
generators of SU(2) group and coincide with some matrices of the
chromodynamics.
We will describe two equations (3.1) as one equation:
for particle


12


[ (6 6 ^! ^
^
^ -c !
o p) 6 ^ 2
- mc ] = 0 , (3.5)


and for antiparticle

+
[ (6 6 ^! ^
^
^ !
o +c p) 6 ^ 2
+ mc ]= 0 , (3.6)


where upper left index "6" means that these matrices are the 6x6-
matrices of the following type:
^!
! ! ! ! ! ^S 0^ ^S 0^
6 ^ 6 6 6 0^ S 6 6 6 ^
= ^
i ^
^

4 = 0
0 = 0
x + ^ j
y + k
z =
^! , ^ , ^ , (3.7)
S 0^ 0^ S 0^ - S
0 0


The wave function has the 6x1-matrix form:
!E

6 = ! , (3.8)
iH
We name the equations (3.5)-(3.6) three-knot equations and 6x6-
matrices - three-knot matrices.

The above matrices give the right expressions of the bilinear
form of the theory (i.e. the basic values of Maxwell's theory), as
it is not difficult to test:

for the energy:
! !
6 + !
6^ = + =
0
6 E 2 H 2
8 U , (3.9)

for the momentum projections of the electromagnetic field:




6 + 6
^ 6
= (
2 E H - E H )
1 y z z y ,

6 + 6
^ 6
= (
2 E H - E H )
2 z x x z , (3.10)
6 + 6
^ 6
= (
2 E H - E H )
3 x y y x ,

or for the Poynting vector:
! ! !
i j k
! 1
S E E E
P = x y z =

4
H H H , (3.11)
x y z

1
= {! ! !
i (E H E H j E H E H k E H E H
y z - )
z y - ( x z - )
z x + ( x y - y x } 1
)
6 +
==
6 !
6

4
8
and for 1st scalar of electromagnetic field:
!
6 + !
6 6 2 2
^
= E - H = 8 I
4 1 , (3.12)

3.2 Rings or knots?


We consider the proton as three "one quark" construction,
i.e. as three bounded lepton constructions, each of which is a
ring.

According to above suppositions the proton model can have the
following forms (Fig.2):


13





Fig.2


A first question appears: does the proton consist of rings or
of knots?
The following argument exists about the proton consisting of
the three engaged knots: if the proton consisted of rings, the
proton charge would be equal to e
3 . Therefore, proton has the
third scheme of the figure 2. (On the other hand the asymptotic
freedom corresponds better to the ring structure of the barions).


3.2 "Three quarks" equation without interaction


From the above follows that the proton equation contains
three "one quark" equations, i.e. three electron equations or
three pair of the scalar Maxwell equations (one pair for each co-
ordinate).
Obviously, there is a possibility of two directions of
rotations of each quark. Then there must exist 6+6 scalar
equations for proton description and also 6+6 equations for the
antiproton description.
Let's find these equations without interaction, i.e. putting
the interaction (mass) terms equal to zero. Using the wave
function form (3.8) and the three-knot matrices (3.7) from the
equation (3.5) (or (3.6)) we obtain:
E H H
x 0 - z y

E H H
y z 0
- x

H
E H
z -
1 y x 0
+ +
+ = 0 , (3.13)
c t iH x 0 y iE z iE
z -
y
x
iH
y - iE 0
z iEx

iH iE iE
z
y -
x 0

or


14


1 E
x H H
z y
- - =
0
c t y z
1 Ey Hz
+ - H x
= 0
c t x z
1 E H
z y H
- - x = 0

c t x y
, (3.14)
1 H
x E Ey
z
+ - =
0
c t y
z
1 H y Ez Ex
- - = 0
c t x z
1 H E
z y E
+ - x = 0

c t x y
As we see, the equations (3.13) and (3.14) coincide entirely with
the electromagnetic wave equation system.
Let's consider now the hypothetical three-knot equation of
the three-quark particle. If the equations (3.13) describe the
knots with the x,y,z directions, we can obtain the following
equations:
1 E 1 Ez Hx
x H z
- =
0 a + =
0 a
c t y c t y

1 H 1 H x Ez
z Ex
- = 0 a' + = 0 a'

c t
y c t y

1 E 1 Ex H y
y
- H
x
= 0 b + = 0 b
c t z
c t z
(3.15) (3.15')
1 H E 1 H
x y y Ex
- =
0 b' + = 0 b'
c t
z c t z
1 E H E
y 1 y H
z - = z
0 c + = 0 c
c t x c t x

1 H 1 H E
z y
y
- Ez
= 0 c' + = 0 c'

c t x c t x

As it is not difficult to see, each pair of equations a,b,c
describe a separate knot; the knots of equations (3.15) are rolled
up in the plains XOZ, ZOY, YOX, and the knots of the equations
(3.15') are rolled in the plains XOY, YOZ , ZOX.


15


3.4. Interaction appearance


The interaction term in the electron equation was obtained as
a consequence of the electromagnetic field motion along the
curvilinear trajectory (see also [1]). This coincides with the
conclusions of modern physics.
The modern particle theory is also known as the gauge field
theory. Without a detail discussion we will only underline the
basic property of this theory: the interactions between the
particles are introduced in the field equation via the gauge
transformations and this procedure is equivalent to the field
vector transformations in the curvilinear space [4,12]. These
transformations lead to the covariant derivative appearance
[4,12]:
1
D = - Bkl M
 , (3.16)
u  kl
2
1 kl
which contain the interaction fields B M
 kl , while the usual
2
derivative these fields don't have.
Our theory shows that the appearance of the interaction terms
is not a mathematical transformation, but is bounded with the
vector motion along the curvilinear trajectory.
Actually, in the linear photon equation the interaction terms
don't exist. Therefore, inside of the particle the interaction
among the particle parts appears only in the instant when the
photon begins to roll up. The electromagnetic form of the pair
production theory of electron-positron, stated above, shows that
thanks to the particle trajectory distortion, in the equation the
additional terms appear.
The derivative additional term appearance follows from the
general theory of the vector motion along the curvilinear
trajectory. This theme was studied in the vector analyse, in the
differential geometry and in the hypercomplex number theory
hundred years ago [13,14] and it is well known. It must be noted
that in physics there is no need in using all the mathematical
results, but only the measured physical value (e.g., three co-
ordinates of the space; time; electromagnetic field vectors;
energy, momentum and momentum of rotation, etc.). Below we
consider some conclusions of these theories.
! !
Any vector F (r ) can have the following forms [13]:
! ! ! 1 2 3 1 ! 2 ! 3 ! !1 !2 !3
F(r ) = F(x , x , x ) = F e + F e + F e = F e + F e + F e
1 2 3 1 2 3 , (3.17)
1 2 3
where F , F , F , F , F , F
1 2 3 are the invariant and co-variant vector
i
modulus and e! and e!i are the basis vectors, which in general case
are changed from point to point. When vector moves along the
curvilinear trajectory the partial derivatives get the view:
! i ! !i
F
F
e
! F
e
!
i i i i
= e + F = e + F , (3.18)
j j i j j i j
x
x
x
x
x

where the following notations are used:


16


e!
i k ! i ! k
= e = - e , (3.19)
j ij k kj
x

k
The coefficients ij are named Christoffel symbols or bound
coefficients. Thus, for the y - direction photon
!
E = !3
E e
3 , (3.20)
! !
H = 1
H e
1

we obtain:


1 !E E3 !3 3 !
= e + E
0 3 k
e
k 0
c t x
1 !
H H1 !1 1 !
= e + H
0 1 k
e
c t k
x 0
! , (3.21)
E
= E3 e!3 + E
2 33 ! k
e
y k
x 2
!
H H1 !1 1 ! k
= e + H
2 1 e
y k
x 2

The same we can obtain for the other directions of the photons.

As we see here the additional terms, which the initial linear
equations didn't have, have appeared. Thus, in the general case,
when the electromagnetic field vectors of three-knot particles
move along the curvilinear trajectories, the additional terms of
the same type, which we obtained in the case of Dirac's equation,
appear. Thus, we can obtain equations similar to the Standard
Model Theory equations.

As result we obtain the following equation system:
1 E H H
x
z y
- - = -
i 1 Ex
c t y z c
1 Ey H H
z
+ - x
= -i 1 E
c t x z y
c
1 E H H
z y
- - x = -i 2 E

c t x y z
c
, (3.22)
1 H E E
x y
z 2
+ - =
i H
c t y x
z c
1 H y E E
z x
- - = i 3 H
c t x z y
c
1 H E E
z y
+ - x = i 3 H
z
c t x y c
where
1 , 2 , 3 are the frequencies of each knot. These values

don't entirely define the charges of quarks, as in the case of
electron, since the current values relate also to the rotation and
twisting of field vectors.


17



Note: the Christoffel symbols are not the mathematical
values, but the physical values; namely, they are the currents,
which appeared thanks to the gyration and twisting of the
electromagnetic vectors.
Note also, that these additional terms play a different role
in the physics. For example, in mechanics the additional term is a
mass term; in electrodynamics it is an electrical current; inside
of the particle it is the inner current, i.e. longitudinal non-
linear fields; etc.



4. Electromagnetic "two quark" equations

Obviously, the mesons must contain two knots, i.e. they must
be described by "two engaged quark" equations. In the wave or
oscillation physics two knot figures appear only by summing two
mutually perpendicular oscillations (Fig.3)





Fig.3

!
where the electric field vector E is drawn as strokes. As we see
the two-knot figure has one positive and one negative knot, that
is in accordance with the quark meson model. Note that this meson
model's spin must be equal to zero.
From above we can suppose that:
1) the meson equations must be the equation system which consists
of one electron-like and one positron-like equation; and
2) these equations must be the result of the rolling up of two
initial photons with mutual perpendicular directions,
3) the electric vectors of both photons must lie in one plain,

e.g., of one photon with y -direction, which contains E , H
x z , and

other photon with x -direction, which contains E , H
y z field vectors

(see Fig.4).





Fig.4


Obviously there are a lot of building variants of such pairs.
The simplest way for the two-knot (meson) equation construction


18


is to use the electron-positron equation or the three-knot
(baryon) equations. For example, for the building of a two-knot
particle in the XOY plain it is enough to use the above field
vectors, making equal to zero the field vectors of photon with the
Poynting vector having z -direction.



5. About symmetry breaking and mass appearance


Among the QCD and electroweak theory there is one serious
difference: the interaction particles of QCD (gluons) are
massless, and the interaction particles of electroweak theory
(intermediate bosons) have big masses. The electroweak theory is
built in the same way as the QCD, and until one point the
electroweak interaction particles don't have masses. For the
masses acquirement in the electroweak Lagrangian one function is
introduced - a single complex Higgs doublet, which describes the
Higgs boson  a hypothesised spinless particle H. Then, some
mathematical transformation, which is named the spontaneous
symmetry breaking, is realised. Thanks to the Higgs boson vacuum
interaction, the massless interaction particles acquire the
masses, but the Higgs boson doesn't change; thus the Higgs boson
acts as the reaction catalyst.

The problem exists: is there a particle in our photon
symmetry breaking theory, whose action is similar to the Higgs
boson action?

Let's show that such particle really exists, although it has
other properties from the Higgs boson. Consider newly the pair
production reaction (2.1) in form:
+ -
+ N N + e + e , (4.1)

where N is the nucleus of atom (e.g., proton) or other charged
particle. For the Lagrangian of this reaction we can write
conventionally:
L( + N ) = L( + -
N + e + e ) , (4.2)

or
+ -
L( ) + L(N ) = L(N ) + L(e + e ) , (4.3)

Of course, the nucleus is not the Higgs boson, but it plays the
same role as a catalyst in the particle transformations.
Note also that the relation (4.1) can be considered as a
description of the wave refraction, where the term L(N )

corresponds to the medium with some refraction index and can be
described as the dispersion matrix for the transformation from the
linear wave to the gyration wave.


19


Appendix. Hadron models

We have considered above the hadrons as particles, consisting
of two or three knots. The equation of one knot is the Dirac
equation that has a harmonic solution. Therefore, it can be
supposed that the hadrons are the superposition of two or three
harmonic oscillations. On other words, the hadrons are the space
wave packets. According to the Schreudinger [15] (see also [16],
section 6.1) the wave packets, built from harmonic waves
(oscilations), don't have a dispersion, i.e. they are stable.
Thus, we can, as a fist approximation, build the hadrons model as
the space packet of the harmonics superposition.
Of course, the below models differ a lot from the real models
and can not be used for calculation of the particle features. But
they give some representation about them.
The models were constructed using Mathcad-program.


A1. "Three quarks" model


Thus, we suppose that the three-knot model is built from
three harmonics oscillation. Let's choose the following
oscillation parameters:
= ,
3 = = N := 200
1 ,
2
2 3
3

=
, = = j := 0..
1 , 0 N
2 2 2 3
r = ,
2 r = r = k := 0..N
1 ,
2
2 2
3

t =
: j 2 v =
: k
j N k
X : r t
k j = sin( j - )
, 1 1 1
Y : r t
k j = sin( j - )
, 2 2 2
Z : r t
k j = sin( j - )
, 3 3 3

We obtain the following three-knots figure A1:


2
1
0
1
2 2
2 1
0 1
10 2

1


2

,,X
Y
Z





Fig. A1


20


To show the field plain gyration and twisting we change the
j
parameter t t :=
j to . Then we obtain figure A2:
j 2.2






2
1
0
1
2 2
2 1

0 1
10
2

1


2


,,X
Y
Z





Fig. A2



A2. "Two quarks" model


To build the two-knot (meson) model in the above proton model
equations we choose the following new parameters:
= 1 =
1 and 0
1 ,
and put Z :
k j = 0
, . Then we obtain the figure A3:


2
1
0
1
2 2
,,X
Y
Z
1

0 1

2
0





Fig.A3


We hope that the further investigations will allow us to
build real models, which will give us the opportunity to calculate
the particle features.


21


Conclusion

Our scope to show that Maxwell's equations can describe the
composite space figures, which consist of two or three knots and
have the hadrons properties, is fulfilled.
Of course we can not yet answer all questions about hadrons,
but if the above theory is right we can understand better what are
the quark and gluon, why free quarks and gluons do not exist, why
quarks have fractional charges, why gluons interact with each
other and can produce the particles; and many others.




References

1. Kyriakos, A.G.(http://arXiv.org/abs/, 2002.
2. Kyriakos, A.G.(http://arXiv.org/abs/, 2002
3. Kyriakos, A.G.(http://arXiv.org/abs/, 2002.
4. Lewis H. Ryder. Quantum field theory. Cambridge university
press, 1985.
5. L.B.Okun. Elementary particle physics (in Russian). Moscow,
1988.
6. A.Pich. Aspects of quantum chromodynamics. (arXiv:
 v.1, 13 Jan.2000)
7. L.T. Schiff. Quantum Mechanics. 2nd edition, McGraw-Hill Book
Company, Jnc, New York, 1955.
8. A.Sokoloff, D.Iwanenko. Quantum field theory (in Russian).
Moskwa-Leningrad, 1952.
9. M.-A. Tonnelat. Les Principes de la Theorie Electromagnetique et
de la Relativite. Masson et C. Editeurs, Paris, 1959.
10. W.J.Archibald.Canad. Journ.Phys., 33, 565 (1955)
11. A.I. Achieser, W.B. Berestetski. Quantum electrodynamics.
Moscow,1969.
12. F.A.Kaempffer. Concepts in quantum mechanics. Academic press,
N.Y and London, 1965.
13. G.A. Korn, Th.M. Korn. Mathematical handbook for scientists
and engineers. McGraw-Hill Book Co., 1961.
14. E. Madelung. Die mathematischen hilfsmittel des physikers.
Springer verlag, Berlin, 1957
15. E.Schreudinger. Naturwissenschaften, Bd.14, S.664-666 (1926)
16. M.Jammer. The conceptualdevelopment of quantum mechanics.
McGraw-Hill book Company, 1967.



