Open Journal of Microphysics, 2011, 1, 28-31
doi:10.4236/ojm.2011.12005 Published Online August 2011 (http://www.SciRP.org/journal/ojm)
Copyright © 2011 SciRes. OJM
A Unified Equation of Interactions
Physics Department, Bingöl University, Bingöl, Turkey
Received July 13, 2011; revised August 14, 2011; accepted August 25, 2011
The aim of this study is to combine four fundamental forces in a single equation. Dirac equation is written by
putting the Yukawa potential as a representation of the strong and gravitational forces. The ordinary terms
seen in the Dirac Equation are treated as the representations of the electromagnetic forces. The Lagrangian of
the weak local interaction of the charged particles is converted to the energy representation according to the
virial theorem and is put in the equation. Thus four fundamental forces are combined in a unique equation.
Keywords: GUT Scale, Fundamental Forces, Dirac Equation, Virial Theorem
After the foundations of the Maxwell’s equations, physi-
cists are trying to combine the four fundamental forces in
a unique equation. Albert Einstein studied on this subject
for a long time but he didn’t obtain a unified equation .
In , the classical solutions of a unified theory is writ-
ten. A unified description of long-ranged interactions is
done in . Some other studies related to a unified Equa-
tion are [4-6].
The models at Grand Unified Theories (GUT) implies
that at high energy level of the universe the four funda-
mental forces (weak, electromagnetic, strong, gravita-
tional forces) have the same validity. Therefore, the four
fundamental forces must be written in a single represent-
tation to describe the true universe.
The weak interactions are the results of the charged
particles and W, and Z bosons are the force carriers of
the weak interactions. The weak interactions are short
ranged interactions. Photons are the force carriers in the
electromagnetic interactions, the electromagnetic inter-
actions are the long ranged interactions. The strong in-
teractions are confined in the nucleus, it holds nucleons
together, three colored gluons are the force carriers in the
strong interactions. The gravitational forces are the re-
sults of the mass effects of the particles and they are the
long ranged interactions.
The non-relativistic Schrödinger equation is written in
relativistic form by Dirac in 1928 so named as Dirac
equation. It contains electromagnetic interactions of par-
Yukawa potential is used to describe the strong inter-
actions at nuclei distances and the gravitational inter-
actions at large distances.
In the Standard Model, the weak interactions are de-
scribed by the Lagrangian of the interacting charged par-
In this study, the Yukawa potential is placed in the
Dirac equation and it is solved for the characteristic wave
function of the strong and the gravitational interactions.
The strong and the gravitational interactions are thought
to be combined in a space of higly mass concentrated.
Later, the Lagrangian of the weak interactions is written
in terms of energy by using the virial theorem. This term
is put in the equation and four fundamental forces are
written in a single equation.
2. Materials and Methods
Previous works and known equations of interactions in
physics are used in this study to combine the four fun-
damental forces. Mathematical calculations are done.
The literature is searched. Theoretical assumptions are
done. As a result, the combination of four fundamental
forces in a single equation is written.
The two strong candidates for a unified theory of interac-
tions are the Klein-Gordon equation and the Dirac equa-
tion. The Dirac equation includes the electromagnetic
interactions. The Dirac equation is derived in [7,8] as
are the 2 × 2 Pauli matrices. The first and sec-
ond terms on the right of the Equation (1) includes kin-
ematical interactions. Since the electromagnetic forces
are the result of the kinematical interactions, it is the
representation of the electromagnetic forces in the equa-
tion. The last term is for the rest mass of the particle.
In , the strong and the gravitational interaction po-
tential is given in the form
here q is the radius related to the particle under con-
sideration. The total energy is given in the article  by
where R is the Schwarzschild radius of the particle. Q is
given by the equation
The Yukawa potential given by Equation (3) can be
directly placed in the Dirac equation;
When the kinematical and charge effects are much
higher than the strong and the gravitational effects, then
the Yukawa potential must have small effects on the mo-
tion of the particles. In this situation, the Yukawa poten-
tial term can be omitted and the original Dirac equation
explains the motion of the particles.
When q is too large, the effective energy comes
from the gravitational forces. The other terms in Equa-
tion (6) are negligible. In this case, the Dirac equation
reduces to the form;
Now, the distances can be taken as . Then, the
Equation (7) reduces to the equation below
Integrating Equation (8) we obtain the wave function
This is the wave character of the gravitational forces.
When q is too small, the effective force on the mo-
tion is the strong interaction. Again in this case, .
The Dirac equation for this condition is the same as
Equation (7). Therefore, the wave function given by
Equation (9) describes the wave structures of both the
gravitational and the strong interactions.
In a highly charged and highly mass concentrated
space, the gravitational interaction valid for the large
distances and the strong interaction valid for the small
distances have the same equivalence. The interaction of
the particles for a so much charged, mass concentrated,
and higly energitic space can be described by a single
where the factor 2 describes the combination of the
strong and the gravitational forces for such a space.
In , the interaction of a field with an electromag-
netic potential for an electron is given by
here I is a 4 × 4 unit matrix and
represents the Dirac
matrices. The second term on the right of Equation (12)
is seen in the Equation (10). The first term on the right of
the Equation (12) can be replaced instead of the term
. Then Equation (10) can be written as
In [11,12], the Lagrangian of the weak local interact-
tion of the charged leptons is given by
where G is the universal Fermi constant and the four
for electron, muon, tau respectively
for their neutrinos.
The virial theorem is written in the relativistic form in
the articles [13,14]. In these articles it was proven that
Copyright © 2011 SciRes. OJM
30 H. ARSLAN
the virial theorem has the same validity in the relativistic
applications as the applications of it in the classical
physics. Therefore, the virial theorem have to be used to
convert the Lagrangian of the weak interactions to the
form of energy.
In [15,16], the virial theorem states that the potential
energy is two times of the kinetic energy in minus sign.
Since the Lagrangian in classical mechanics is written by
The total energy is E = T + V and by the virial theorem
V = 2T. Therefore, the total energy is (3T) and the La-
grangian is (-T). By using these in the Equation (14), the
total energy of the weak interactions, in terms of the La-
grangian, can be written as
As a result, the unified equation of interactions is
The last term in this equation describes the weak in-
teractions. The Equation (18) includes the weak, elec-
tromagnetic, gravitational and the strong interactions
where the strong and the gravitational interactions are
thought to be combined for the considered space.
4. Results and Discussions
The wave function of the gravitational and the strong
interactions is derived in Equation (7) by putting the
Yukawa potential in the Dirac equation, the weak inter-
actions’ energy is written by Equation (17) according to
the virial theorem, and the unified equation of interactions
is derived as Equation (18) to describe the true universe.
At GUT scale, the high energy structure of the universe
needs the validity of the four fundamental forces with the
same importance. Since the kinematical and charge ef-
fects on the motion could not be eliminated for highly
energetic and charged particles at too large or too small
distances, the electromagnetic and weak interactions
have the same validity at these distances like the gravita-
tional and the strong interactions. In the same way, in the
highly energetic space, for large distances the space be-
haves like a nucleus. Therefore, the strong interactions
must be used to define the motion of the particles. For
small distances, the highly concentrated mass needs the
gravitational forces to explain the attractive forces. Such
a true space may be considered as the space the Bing
Bang began in. In the world we live in, there is no need
to use this equation, as one knows, in special conditions
some of these interactions are effective and others are
negligible. The application of this equation must be be-
yond the Standard Model.
As a result, the Dirac equation is a strong candidate for
unification of the four fundamental forces, as one is done
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