Euclidean Equivalent of Minkowski’s Space-Time Theory and the Corresponding Model of Elementary Particles (“Cuantex” Model)

This document communicates some of the main results obtained from a theoretical work which performs a type of Wick’s rotation, where Lorentz’s group is connected in the resulting Euclidean metric, and as a consequence models the particles with rest mass as photons in a compacted additional dimension (for a photon of the ordinary 3-dimensional space, they do not go through the 4-dimension due to null angle in this dimension). Among its reported results are new explanations, much more elegant than the current ones, of the material waves of De Broglie, the uncertainty principle, the dilation of the proper time, the Higgs field, the existence of the antiparticles and specifically of the electron-positron annihilation, among others. It also leaves open the possibility of unifying at least three of the fundamental forces and the different types of particles under a single model of photon and compact dimension. Additionally, two experimental results are proposed that can only currently be explained by this theory.

1. Euclidean Equivalent of Minkowski’s Space-Time Theory and the Corresponding Model of Elementary Particles (“Cuantex” Model)
JPAR











Euclidean Equivalent of Minkowski’s Space-Time Theory
and the Corresponding Model of Elementary Particles
(“Cuantex” Model)

Juan Carlos Alcerro Mena
Department of Mathematics, Universidad Cristiana Evangélica Nuevo Milenio
Faculty of Engineering, Universidad Jesús de Nazareth.
Department of Mathematics, Universidad de San Pedro Sula.
Email: [email protected], [email protected]

This document communicates some of the main results obtained from a theoretical work which
performs a type of Wick’s rotation, where Lorentz’s group is connected in the resulting Euclidean
metric, and as a consequence models the particles with rest mass as photons in a compacted
additional dimension (for a photon of the ordinary 3-dimensional space, they do not go through
the 4-dimension due to null angle in this dimension). Amon g its reported results are new
explanations, much more elegant than the current ones, of the material waves of De Broglie, the
uncertainty principle, the dilation of the proper time, the Higgs field, the existence of the
antiparticles and specifically of the electron-positron annihilation, among others. It also leaves
open the possibility of unifying at least three of the fundamental forces and the different types of
particles under a single model of photon and compact dimension. Additionally, two experimental
results are proposed that can only currently be explained by this theory.

Key words: Wick’s rotation, Lorentz’s group, Special relativity, Photons, Elementary Particles, Four-vectors, Space-time,
Higgs’ boson, Compact dimension.


INTRODUCTION

Minkowski’s space-time theory is a mathematical
mechanism that provides a metric to special theory of
relativity. However, this abstract reality is based in a metric
of complex values, causing that in various theoretical
topics in physics, you need to perform Wick’s rotation.
Although Minkowski’s metric works mathematically, this
does not provide a complete physical comprehension
concerning its meaning. For example, the spatial
components of the four-vector velocity in an ordinary three
dimensions space, presents an unavoidable incoherence
with the ordinary ones used in physics
1
, it also happens
with the components in an ordinary three dimensions
space of the four-vector force. On the other hand, the

1
Einstein wrote “it is the only four-vector that can be formed with the
velocity components of the matter particle…” in his book “The meaning
of relativity”.
theory of relativity and the wave-particle duality of quantum
mechanics are theoretical constructs that although
compatible, are independent of each other, leaving open
the problem of finding a more general theoretical
framework that deduces both. The above are just some of
the needs that the present work tries to solve, but it also
introduces the discovery that all elementary particles with
rest mass can be modeled like a photon traveling in a
fourth compact spatial dimension, in such a way that the
corpuscular existence in ordinary space of these particles
(with mass) happens instantaneously and intermittently
2

and with a defined frequency.

2
Hence the name of the cuantex model, which is the abbreviation of the
phrase "quanta of existence".
Journal of Physics and Astronomy Research
Vol. 4(1), pp. 079-086, September, 2018. © www.premierpublishers.org, ISSN: 9098-7709

Research Article

2. Euclidean Equivalent of Minkowski’s Space-Time Theory and the Corresponding Model of Elementary Particles (“Cuantex” Model)
Alcerro-Mena 080


With the purpose of clearly developing the communication
of this theoretical work, this document is composed of
three sections. The current section 1 gives a brief
introduction of the two concepts involved (Wick’s rotation
and Lorentz’s group), it also introduces the formalism of
the main proposed hypothesis and develops the metric
and the resulting elementary particle model
3
. Section 2
communicates some of the consequences of adopting
particles model implicit in the proposed hypothesis and,
finally, section 3 shows some of the final conclusions of
this work and experimental predictions. It is also important
to mention that this analysis is made under the special
relativity theory (Lorentz group), pending the development
of the corresponding general relativity theory.

CONCEPTS

Wick Rotation

(A) It is a method to find a solution on a mathematical
problem on Minkowski´s space-time, starting from the
solution of an analog problem on a Euclidean space,
through the transformation that substitutes an imaginary
variable a real one. Specifically talking, in this work, the
replacement of “t” for “it”, is studied. This is the variable of
time.

(B) In quantum theory of fields, it is the rotation from a real
time axis to an imaginary time axis on a complex time level
(Liu, 1997).

Lorentz group. It refers to the group of all the
transformations in Minkowski’s space-time.

HYPOTHESIS FORMALISM

Being:
ds
2
= -c
2
dt
2
+ dx1
2
+ dx2
2
+ dx3
2
(1)

The distance measurements of Minkowski’s space-time
(Stephani, 2004), that in tensor form is:

ds
2
=dx

 dx



Where c is the speed of light in the vacuum and:

-10 0

 0 0 1 0
0 0 0 1



If Wick’s rotation is applied (tw= i t), with the special
condition that Lorentz’s group is compatible with the
Euclidean metric
4
, then we have as result the following
metric structure:

3
Cuantex model.
4
The reasoning of this condition will be seen in section 1.3.
ds
2
= c
2
dt
2
+ dx1
2
+ dx2
2
+ dx3
2
+ dx4
2
(2)
Which in tensor form:
ds
2
=dx
U
UV dx
V

Where:
10 0 0

UV  0 0 1 0 0
0 0 0 1 0

0 0 0 0 1


Lorentz’s group connection in this Euclidean metric
and its four-vectors in x1-x2-x3-x4.

This development was made apart from the ct dimension
in the metric of the expression (2).
The following figure 1 shows the connection of
Minkowski’s coordenate axes rotation
5
with the metric (2):


Figure 1. Connection of the rotating space-time axes of
Minkowski with the metric of the expression (2).
Notice that, it is only feasible to represent the dimensions
x1, ict and x4.

As you can see, the invariant in the rotating space-time
axes of Minkowski correspond themselves with the
absence of effective rotation of the axe x4, since this axe
rotates only around itself. Keeping in consideration only ict,
x1 and x4, it was found that the coordinate transforming
matrix, for this system, is:

cos sin 0
sincos (3)
0 0 1


Where: is the complex angle for a reference frame
rotates with respect of the other in Minkowski’s space-
time, cos = 1 / (1-v
2
/c
2
)
1/2
, sin= i(v/c) / (1-v
2
/c
2
)
1/2
and v
is the speed of the frame of reference in relative movement
(along x1) in respect of the one in relative rest.
5
Do not confuse with Wick’s rotation.

3. Euclidean Equivalent of Minkowski’s Space-Time Theory and the Corresponding Model of Elementary Particles (“Cuantex” Model)
J. Phys. Astron. Res. 081


Coordinates transformation in the system (ict, x1, x4)
and four-vector of displacement in x1-x2-x3-x4.

When applying the transformation coordinates of the
expressions (3) we found the following result
6
:

x1´ = (x1-vt)/(1-v
2
/c
2
)
1/2

ict´= i(ct-x1v/c) /(1-v
2
/c
2
)
1/2
(4)
x4´= x4

We can appreciate that the transformations for x1´ and ict´
are the same of the space-time of Minkowski.

On the other hand, the four-vector of a particle’s
displacement observed in the space x1-x4, it was
reasoned
7
that the component for x4 needed to have the
same magnitude of the invariant interval in the space-time
of Minkowski and we obtained:

x = x1 i + ct/ 

The infinitesimal form for the space x1-x2-x3-x4 is:

dx = dx1 i + dx2 j + dx3k +cdt/

Where i, j, k and  are unit vectors, = 1/(1-u
2
/c
2
)
1/2
and u
is the speed of the observed particle.

This and the following four-vectors are spatial and should
not be confused with the common four-vectors of space-
time.

Four-vector of velocity in x1-x2-x3-x4

Of the expression (5-B) the following four-vector U was
deducted for the velocity of a particle observed in the
space x1-x2-x3-x4:

U = u1 i + u2 j + u3 k + c/  (6-A)

When squaring the magnitude of this four-vector, it
results:
|U |
2
= c
2


This states that the magnitude of the four-vector of velocity
U is the speed of light, in other words, the particle with rest
mass is a photon transiting in the fourth spatial dimension.
This conclusion will be taken up later.


6
An alternate way to get to the same result, it is through the reference
frame translation x1´-x4´ respect of x1-x4, when the ict dimension is
eliminated in the coordinate system of the figure 1.

Figure 2: Representation in the plane x1-x4 of the four-
vector of velocity U, which can also be called C because it
is the same speed of light in a vacuum.  is the angle
between said vector and the x1 axis.

Based on the component for x4 of the expression (6-A) and
the component on x4 of Figure 2, the following equation is
valid:
c/csen
From which it follows that:
1/sen

And therefore (5-B) can be rewritten as:

dx = dx1 i + dx2 j + dx3k +cdt sin

Where  is the angle formed between the vector x or dx
and the axe x1 (the direction of 4D-motion). This symbol
must not be confused with Lorentz velocity factor β=v/c, in
fact, in this document  means inverse cosine of v/c.

On the other hand, (6-A) can be rewritten as:

U = u1 i + u2 j + u3 k +c sen  (6-B)

Four-momentum in x1-x2-x3-x4

When applying the transformation matrix (3) to the linear
momentum measured by the two frames of reference for
x1-ict and x1´-ict´ in the system of the figure 1 and
reasoning that the component of the lineal moment in x4
should have the same magnitude of the invariant space-
time of Minkowski, the following four-vector was obtained
for the linear moment in x1-x2-x3-x4:

P = m0u1i + m0u2j + m0u3k +m0c

Where mo is mass in rest of the observed particle.





7
Reasoning based on the existing correspondence between the
invariance of the rotation of axes of the space time of Minkowski and
the absence of rotation of the axis x4.

4. Euclidean Equivalent of Minkowski’s Space-Time Theory and the Corresponding Model of Elementary Particles (“Cuantex” Model)
Alcerro-Mena 082


Four-vector of current density

When applying the transformation matrix (3) to the current
density measured by the frames of reference for x1-ict and
x1´-ict´ in system of figure 1 and reasoning that the
component in x4 of the current density should have the
same magnitude of the invariant in the space-time of
Minkowski, the following four-vector of current density was
obtained for x1-x2-x3-x4:

cj1

i + j2

j + j3k

c

Where and  are the volumetric density of electric
charge in motion and resting respectively and j1, j2 and j3
are the current densities in x1, x2, and x3 respectively.

Space time topology and cuantex model of the
elementary particle

All of the above proposed the following physical-
mathematical properties:

(A) It was found that when u=0 and therefore  is /2 in
the expressions (6-A) and (6-B), then U = c(and
generally when squared the magnitude of the four-
vector 6-A result |U|
2
= c
2
), which takes us to the
conclusion that all elementary
8
particle with mass at
rest is a photon in the space defined by the metric (2).
And where u = c cos. The implicit reasoning
9
in the
conclusion above deducted that the topology in space
–time results to be M
4
x S
1
, this is a Cartesian product
between an Euclidean space-time M
4
(the first four
dimensions, ct-x1-x2-x3, of the metric 2) and a circle
S
1
(the fifth dimension, x4, of the metric 2) of radius R.
This is to have a cylindrical space-time, where for the
metric (2) dx4 = RdBeing dthe angular
infinitesimal displacement around the cylinder and
where -2< < 2In other words, the dimension x4
is compact (see figure 4).

(B) Since all particle with mass at rest m0 must be treated
as a photon that moves through the cylindrical surface
defined by M
4
x S
1
, then it was found that the length
of the trajectory of any elementary particle, in M
3
x S
1

(x1-x2-x3-R), is defined by :

∫(&#3627408465;??????
1
2
+ &#3627408465;??????
2
2
+ &#3627408465;??????
3
2
+ ??????
2
&#3627408465;??????
2
)
(
1
2
)
=&#3627408464; ∫???????????? (9)

Where – Infinite < < + Infinite. (S
1
it is rolled up, but
the orthogonal nature is the same as if it were an
extended dimension).

And therefore in M
4
x S
1
by:
2
1/2
c∫dt=∫dL (10)


8
One of the reasons why the reference is to elementary particles, is that
photons are elementary particles.
9
If all elementary particle with mass at rest in the ordinary space, is
equivalent to a photon going through the dimension x4, then this
dimension should be compact, so that this particle keeps its existence
Notice that the photons of the ordinary tridimensional
space M
3
(the commonly known) it has to be =0 and
therefore d= 0, thus they do not go through the
dimension x4= R.

(C) With the intension to define the value of R, the path
(world line) of the equivalent photons were visualized
over the cylindrical space M
3
xS
1
(by simplicity the
temporal dimension was omitted), as if they were on
an extended equivalent plane (see figure 3). When
you assume this, then the periodicity over S
1
of the
photon’s path, was replaced by a continuous
emission of equivalent photons of a same elementary
particle.
On this basis, it was easy to calculate the wavelength
 (distance between two flat waves) of these photons,
giving as result:
 = h (1-u
2
/c
2
)
1/2
/m0c = h / (γm0c) (11)

Where h is the Planck constant.

Here it is supposed
10
that = h/p, where p is the
lineal moment of the photon. The perimeter of S
1
must match the wavelength of the equivalent photon
of the particle observed when in rest, by which if it is
supposed u = 0 then:
R = u=0 
Therefore: R= (1/2) (h/m0c) (12)

This means that the radius R of the cylinder
circumference must depend of the mass at rest of the
elementary particle observed. This is so that each type
of elementary particle, according to its mass at rest,
must exist a proper cylindrical Surface, but keeping the
space-time M
4
in common.


Figure 3: Equivalent extended space to M
3
xS
1
(x1-x2-x3-R), that
shows two fronts of flat waves of the equivalent photon of an
elementary particle with mass at rest, which has a lineal moment
P, and a wavelength . This also shows a geometrical
interpretation of the wavelength of De Broglie DB of the
elementary particle with mass at rest.
(although instantaneously and intermittently) in the ordinary space x1-x2-
x3.
10
Although it is not shown here, this relation and the constant of Planck
may be rigorously deducted in this proposed model.

5. Euclidean Equivalent of Minkowski’s Space-Time Theory and the Corresponding Model of Elementary Particles (“Cuantex” Model)
J. Phys. Astron. Res. 083


(E) Graphic interpretation of the cylindrical space
proposed.

The following figure 4 illustrates a segment of the path
of the equivalent photon of any elementary particle:


A B
Figure 4: Representation of M
3
xS
1
where only x1-R
(that is x
1
-x
4
) is shown. (A) Angular view.
(B) Transverse view, here the vertical and horizontal
axes refer to the geometry of the circle

Observations:

(1) For this illustration it is supposed that the spatial
displacement, of the elementary particle, is only in the
axe x1. However, its equivalent photon should also
have a trip in S
1
, it means the compact axe x4.
(2) Notice in figure 4-A that commonly a fourth spatial axe
would be noticed as extended (pointed lines), but this
model refers to a rolled up or compact as it is seen in
the figure.
(3) The trip of the equivalent photon (drawn as an arrow)
is made on a cylindrical surface.


RESULTS AND PREDICTIONS OF THE CUANTEX
MODEL OF AN ELEMENTARY PARTICLE

Wavelength of De Broglie explained by this cuantex
model

At applying a simple geometrical analysis on an extended
equivalent plane to M
3
xS
1
(x1-x2-x3-R in figure 3 and
assuming an angle between the equivalent photons
direction and M
3
(this is that the elementary particle
equivalent to these photons, is in movement), then the
formation of a composed triangle rectangle by three sides
can be verified (see the zoom): the wavelength of the
equivalent photons, a flat wave front of the equivalent
photons and, forming an angle with the first one, the
wavelength of De Broglie DB. This last one results to be
the length between the flat waves fronts of the equivalent
photons, but measured in M
3
. It can be verified that from
the analysis of this triangle it is obtained that:

DB = / cos 
And therefore
DB = h (1-u
2
/c
2
)
1/2
/(m0 u) = h /(γm0 u)
Where u is the speed of the particle.
Notice that this expression is the same for the wave length
of De Broglie, however, it only applies to elementary
particles, unlike what the current theory assumes since
Louis V. de Broglie (De Broglie, 1924). This is because
replacing mo, in that expression, by the total mass of a
composite particle would imply the existence of its own
compact dimension, calculated by the expression (12). All
this implies the observable fact that, if the wavelength of a
composite particle was accurately measured, then it would
be observed that it does not exactly match the de Broglie
wavelength.

Particle-antiparticle annihilation explained by cuantex
model

Consider the following ratios

dx4/dx = dp4/dp = cc= -sin

This is now combined to expressions (7), (8) and
(5-B) respectively:

-m0c / P = (-cdt sin) / dx = -cc= -sin

Where P, dx and care the magnitudes of the
corresponding four-vectors.

This suggests that the change of signal in the charge,
implies to change the sense to the component of the fourth
axis in the four-momentum (7) it also involves a change in
the sense for the displacement of the photon along S
1
.
Considering this, and if we proceed to verify what happens
when an electron (e
-
) collides against its antiparticle, the
positron (e
+
), being these the negative and positive charge
and if we suppose the movement of the equivalent
photons only on the axis x1 and x4=Rand on opposite
senses (in the axis x4 due to opposite charges) therefore
the following system of equation of momentum and
energy is deduced (before and after the collision):

x1 axis Pe
-
i1 - Pe
+
i1 = Pe
-
f1 - Pe
+
f1 = 0
x4 axis Pe
-
i4 - Pe
+
i4 = Pe
-
f 4 - Pe
+
f 4 = 0
Energy ((Pe
-
i1)
2
+(Pe
-
i4)
2
)
1/2
c + ((Pe
+
i1)
2
+(Pe
+
i4)
2
)
1/2
c =
((Pe
-
f1)
2
+(Pe
-
f 4)
2
)
1/2
c + ((Pe
+
f1)
2
+(Pe
+
f 4)
2
)
1/2
c

Where Pe
-
i1, Pe
+
i1,Pe
-
f1 y Pe
+
f1 are the components on the
axis x1 of the four-momentum initial and final for the
equivalent photons from the positron and the electron
respectively; Pe
-
i4, Pe
+
i4,Pe
-
f4 and Pe
+
f4 are the four-
momentum first and final in the axis x4 of the equivalent
photons from the electron and positron respectively.

It is observed that the system permits the solution where:

Pe
-
f 4 = Pe
+
f 4 = 0

Which implies the reaction:
e
-
+ e
+
 +
Here symbolizes the ordinary space photon M
3
.

6. Euclidean Equivalent of Minkowski’s Space-Time Theory and the Corresponding Model of Elementary Particles (“Cuantex” Model)
Alcerro-Mena 084


This expresses that the result of the collision is the two
contained photons (gamma rays) in the tridimensional
ordinary space, storing all the initial energy, something
that is experimentally observed (and it is also predicted by
the current physics).

Intrinsic magnetic moment from the electron
explained by cuantex model

Something interesting from the structure of this particle
model, is that the path from the equivalent photon of a
particle with reposed mass, through de S
1
, implies a fixed
and determined angular moment that does not disappear
although this particle is in repose on the ordinary space
x1-x2-x3. On the electron case, the value of the magnetic
moment can be obtained, if the following aspects are
considered:

(a) The magnetic momentum for an “I” current around
the A area, is given by:

= I A

(b) The identity generally used for the total charge
density is:
 = n e
Where “n” is the number of electrons per volume unit
and “e” is the electric charge of an electron.

(c ) When the electron is reposed, the expression (8) has
the following form:

c= c0
This is the current density of the equivalent photon,
when the electron is in repose.

(d) The rotation radius of the equivalent photon is the same
R radius, which validity is given by the expression (12).

Therefore, at mixing these four expressions, the following
intrinsic magnetic momentum from the electron is obtained:

= e h_
  m0

This is the same expression for the magnetic moment of
the electron intrinsic spin, foretold by the relativist quantum
physics. Notice that the signal is positive or negative
depending the value of the charge and, therefore, the path
sense of the equivalent photon on S
1
. If the case of moving
electron is analyzed, then we get to the conclusion that this
magnetic moment stays fixed.

Dilatation of proper time explained by the cuantex
model

By making the expression (5-B) correspond with (9) and
figure 4, it is obtained:
∫&#3627408465;??????
4=&#3627408464;/??????∫&#3627408465;??????=??????∫&#3627408465;??????=??????(2????????????)
Where “n” is the number of crosses the equivalent photon
of the elementary particle has in the ordinary space x1-x2-
x3.

The above expression can be rewritten as follows:
 = t / = R

/c= n(2R)/c (13)

Where is the proper time of the observed elementary
particle.

This means that  = n(2R)/c is the proper time of the
elementary particle with mass at rest and corresponds to
the frequency of crosses with the ordinary space x1-x2-x3,

of the equivalent photon of the particle mentioned,
composes its temporal flow. It was deducted from here
that the time dilatation from a particle moving at vi,
velocity, it is the fact that this crossing frequency with the
ordinary space x1-x2-x3

is reduced, due to the  angle
takes different values of /2, since 1/= sen.
In figure 4, the crossing with x1-x2-x3 will be given when
the path of the photon intercept the x1 axis

Uncertainty principle explained by the cuantex model

Consider the following aspects:

(A) If in the expression (13) it is assumed that n = 1, then
it follows that any elementary particle representative of
any clock is associated with a period of time T0:

T0 = h/(m0c
2
)

It is the sensitivity or the minimum uncertainty in the
measurement of time, because on that period the matter
of the clock is not registering events in the tridimensional
ordinary space, because its equivalent photons is not
having contact with the ordinary space x1-x2-x3. In the
case represented by figure 4 this last is illustrated, but for
the equivalent photon of a moving particle on x1.

(B) According to quantum physics, any energy
measurement of a system can finally be reduced to the
frequency measurement of the waves associated to that
quantum system (Serway, 2002).

Considering that and proceeding to calculate the energy
that is measured, from a reference frame added to a clock,
to a wave that intercepts the location of that clock, then
the following results are obtained:

(1) The time interval t taken to measure the wave
frequency, is given by: t = nT0. Where n is a whole
number.

(2) The frequency (fwave) of the wave in study is:

fwave = (s/n) m0 c
2
/h

7. Euclidean Equivalent of Minkowski’s Space-Time Theory and the Corresponding Model of Elementary Particles (“Cuantex” Model)
J. Phys. Astron. Res. 085


Where m0 is the reposed mass from the representative
elementary particle of the clock and s is the number of
cycles or waves that are registered during the t time.

(3) From that it derives that the energy of the wave is
given by:

Ewave = (s/n) E0
Which can be rewritten as:

Ewave = E0/n + E0/n + E0/n +…+ E0/n

It means, the sum of s times E0/n.

(4) The previous series means that E0/n is the sensibility
or the uncertainty in the measurement of energy from
any wave, or along a time interval t = nT0.

(5) If this uncertainty in the energy measurement is called
Em= E0/n and the identities are conveniently
replaced, then we have that:

Em t = h

It is the minimum uncertainty in the energy measurement
of a wave and therefore any E energetic uncertainty will
be:
E t > h

Which is the same uncertainty principle for the energy
measurement used by the quantum physics.
Notice that if T0=0, then we have not discrete, but
continuous, registration of the elapsed time, in that case,
there would not exist an unavoidable minimum uncertainty
in the energy measurement from any system.

Higgs field explained by the cuantex model

The topology of S
1
implies a field of central force that is in
function of R and where the force of its magnitude FH can
be obtained if the following aspects are considered:
(a) A definition of force is: F = dpx/dt i + dpy/dt j

Where i and j are unitary vectors of the x and y axis
respectively, px and py are the components in x and y
respectively, from the P momentum of the equivalent
photon

(b) The component of the momentum in the equivalent
photon that interferes in this calculation is the one of
the projection from the total moment of the photon, on
the transversal section of the cylinder in figure 4-B, this
is P4 (see figure 4-B).

(c) The angle that forms p4 with the axis x is where
is the angle of rotation on S
1
. See figure 4-B.

(d) From the expression (11), it is p4= h/λ.
From the previous it is deduced that the magnitude of F is:
FH = c h / (2R
2
) (14)

Or also:
FH = 2 mo
2
c
3
/ (h)

This field force corresponds to the called Higgs field from
the particles standard model. This is that if a photon does
not enter in S
1
then it is perceived as an ordinary space
photon, it means that Higgs field does not interact with it.
But if a photon travels to S
1
then it is perceived as a
particle with reposed mass from the ordinary space which
reposed mass is obtained from the expression (12), this is
a particle that interacts with Higgs field.
On the other hand, FH does not manifest itself as force on
the ordinary space but as something underlying the mass
from the elementary particles.

Four-force of the cuantex model

The four-vector of force F4R of the space-time of
Minkowski is given by:

F4R = F1 i + F2 j + F3 k + iF1 (u/c) 

Where F1, F2 and F3 are the components in x1, x2 and x3
respectively,  is the corresponding unit vector to the
axis ict and x1 is the axis of motion.

When applying the transformation matrix (3) to this
relativistic four-vector of force and reasoning that the
component of the four-vector of force of the cuantex model
in x4 should have the same magnitude of the invariant in
the space-time of Minkowski, the following four-vector F
was obtained for x1-x2-x3-x4:

F = F1i + F2j + F3k + (F1
2
/
2
+F2
2
+F3
2
)
1/2




CONCLUSIONS AND EXPERIMENTAL PREDICTIONS

(1) The present work is the only one in all the literature of
theoretical physics that models the elementary particle
with mass at rest, as a photon traveling a compact fifth
dimension. There have only been a few approaches to the
idea, for example, according to J.G. Williamson and M.B.
Van der Mark (1997), the electron can be modeled as a
photon confined in a space with toroidal topology, and
however, that space would always be within the three
dimensions of ordinary space. On the other hand, F. G.
Arregui (1989) proposed that any elementary particle with
mass at rest can be modeled as a packet of energy
rotating in a compact axis, however, this particle was not
treated as a photon but as an entity called "energion",
besides that said compact axis was temporary and non-
spatial. This means that the Cuantex model is a whole
land to be explored.

8. Euclidean Equivalent of Minkowski’s Space-Time Theory and the Corresponding Model of Elementary Particles (“Cuantex” Model)
Alcerro-Mena 086


(2) Considering that the quantum theory of fields assigns
a particle for each considered force field, therefore from
the expression (14) it is deduced that there must exist an
particle associated to the field of this force, from the
exposed on the previous section, this particle must be
Higgs boson.

(4) This elementary particle model proposed, predicts that
the movement inside the tridimensional ordinary
space of an elementary particle, with reposed mass, is
given through “jumps” (as consequence from the
equivalent photon trip on S
1
), in such way that the
length between these jumps is given by:
d = ____h u_____
m0 c
2
(1-u
2
/c
2
)
1/2

This results from including the factors u and and also
assuming that n = 1 in expression (13). This is that with n
= 1, the corpuscular absence of the elementary particle
with mass at rest, is modeled in the ordinary three-
dimensional space.

Now, for an electron each one of these spatial intervals
will be of 1 (nm) when it keeps a velocity corresponding
99.99970611% from the one of the light and which is
related to the injection of a kinetic energy of
3.36894484x10
-11
joules. Up to date there is already
enough technology to access an experimental verification
of this scale, however the means used that show the
“prints” of the electron must be refined so that the diameter
of these, does not match or precisely overcome the length
of the space that corresponds the period of absence from
that particle, which is exactly what would be in search for
verification.

(4) From the four-vector force of the cuantex model,
expression (16), it derives the four-vector impulse of the
cuantex model. This implies that the model predicts the
experimental possibility to transmute the elementary
particles through the component in x4 from some four-
vector impulse resulting, due to the modification of the
mass at rest of the particle affected as long as this
connects to any radius R (expression 12) existing in the
Cosmos. At the moment this is written, the author of this
article is in patenting process of a system that would show
its conclusion and, therefore the validity of this whole
theory.

(5) According to Overduin and Wesson (1998), the five
dimension theory from Kaluza-Klein was mainly
abandoned because it assigns reposed mass values
m0highly elevated to loaded particles as the electron. This
happens because in that theory, the mass is inversely
proportional to the radius values from the compact
dimension.

Related to this, it is very important to observe that in this
proposed model for an equivalent photon, the compact
dimension radius R, given by the expression (12), also
keeping an inverse relationship with m0, however, it
permits mass values in the order of the ones observed in
reality, due to factor h (Planck’s constant), which are very
reduced.

(6) Although it is not shown in this document, it is possible
to get to the same equivalent photon model from a
different analysis made on the special relativity and the
hypothesis of De Broglie.


ACKNOWLEDGEMENT

To Arbab Ibrahim Arbab, Ph.D, for his endorsement.
The reviewers of this work, who kindly suggested
improvements.


REFERENCES

De Broglie L.V. (1924), “Recherches sur la théorie des
quanta”, Ann. de Phys., 10
e
serie.
F. G. Arregui (1989), Los cuatro principios metaempíricos
del ser material: en las fronteras de la física”,
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Manifolds´´, arXiv:hep-th/9707196v1.
Overduin J.M. and Wesson P.S. (1998), ´´Kaluza-Klein
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(1997).

Accepted 19 July 2018

Citation: Alcerro-Mena JC (2018). Euclidean equivalent of
Minkowski’s space-time theory and the corresponding
model of elementary particles (“cuantex” model). Journal
of Physics and Astronomy Research, 4(1): 079-086.



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