“ON THE MATHEMATICAL FORMULATION OF A TOPOLOGICAL MODEL FOR THE STRUCTURE OF ELMENTARY PARTICLES OF PHYSICS ACCORDING TO THE G

“ON THE MATHEMATICAL FORMULATION OF A TOPOLOGICAL MODEL FOR THE STRUCTURE OF ELMENTARY PARTICLES OF PHYSICS ACCORDING TO THE GHASSEMI’S SPACE-TIME-MASS THEORY OF THE UNIVERSE”

ABSTRACT

A topological model for the structure of elementary particles is proposed. Topological particles from which matter is to be created fill the space-time and are to be the building blocks of space-time. This model only focuses on a kinematics viewpoint.

INTRODUCTION

This paper is an enlargement of the first one I wrote which had to see with a topological model for the structure of elementary particles of physics according to the Ghassemi’s space-time-mass theory of the universe. From that paper we can establish the following problem:

¿Which ones should be the last and fundamental building blocks from which matter is made taking into account a topological model and according to the Ghassemi’s space-time-mass theory of the universe?

The answer to this problem was already established in the same one as it follows:

There are only two kinds of elementary particles being the building blocks of space-time: the first one, whose topological structure has positive gaussian curvature and the second one whose topological structure has negative gaussian curvature.

With this hypothesis it was possible to explain matter creation, electrostatic interaction of charged particles, etc.

Topological particles have been called the particles in the hypothesis above proposed.

1. TOPOLOGICAL CHANGE

Consider the two-dimensional surface of a sphere with radius whose gaussian curvature is constant and equal to

(1.1)

( Figure 1.1.a ). Consider also the surface of torus there where its gaussian curvature is negative, say,

, (1.2)

( Figure 1.1.b , shaded part ) ( see details of 1.1 and 1.2 in Do Carmo, 1976, chapter 3 ).

(a) ( b )

Figure 1.1

In this sense it is not difficult to realize that topology of figures 1.1.a and 1.1.b ( shaded part ) are different.

Make and build up the following figure

Figure 1.2

Say, we have joined the sphere with the shaded part of the surface of torus.

Now, we are in conditions of formulating the definition of topological change.

DEFINITION 1.1 We say that a topological change between sets of points belonging to any surfaces has performed, when as a result of the union of them, results a set of points belonging to a surface whose topology is well defined.

From the union made in figure 1.2 the immediate and more simple change of topology that it can be performed is that which gives as a result the surface of torus ( Figure 1.3 ).

Figure 1.3

2. TRAVELING LIKE WAVES

Let us consider a one-dimensional transverse like wave*) which travel along the x-axis. Let us suppose the like wave has wavelength and frequency .

Suppose at the shape of the like wave is given by

(2.1)

as it can be shown by a solid curve in figure 2.1.

Figure 2.1

is the displacement of the like wave at and, the amplitude ( maximal displacement ) of the like wave. This relation gives a shape that it repeats itself each like wavelength ( what we want ) since the displacement is the same. For instance at , , and so on ( since ).

Suppose now that the like waves moves to the right with velocity . Then, after a time , each part of the like wave has been displaced to the right an amount without deformation (dashed curve in figure 2.1 ).

Consider any point at on the like wave ( for instance a crest in a position ). After a time , that crest will have displaced a distance times greater than its previous position. In order to describe the same point on the like wave, the argument of the function must be the same in such a way that we can replace by in equation 2.1 to obtain

(2.2)

Taking into account the following relations

___________________________________________________________________

I have called like waves those ones with period

, , (2.3)

where is the period of oscillation at each point the angular frequency and the like wave number. Equation 2.2 can also be written in the form

(2.4)

Furthermore

(2.5)

The amount is called the phase of the like wave.

The relation 2.4 is solution of the wave equation

(2.6)

where is the velocity of the like wave.

3.DIMENSIONAL STRUCTURE OF THE TOPOLOGICAL PARTICLES

Let us see the way we can generate the dimensional structure of the topological particles in using a spatial two-dimensional surface and a temporal 1 – dimensional one.

For instance, we can take the surface of the sphere and make its radius varying in time in the interval . This will cause that the two-dimensional surface generates a volume, the three dimensional volume that we know. But taking into account the relation 1.1, it gives the same thing if we make it to vary in time the gaussian curvature ( Figure 3.1 ).

Figure 3.1

Then we can say that we have generated a spatial two-dimensional volume and temporal 1-dimensional one.

4.ANNIHILATION AND COMPACTIFICATION BETWEEN TOPOLOGICAL PARTICLES.

Let us suppose that a topological change has performed. Let us further suppose that the torus ( as a result of the topological change ) deforms to a sphere continuously while . In these circumstances we say that an annihilation process has performed.

DEFINITION 4.1 An annihilation process between topological particles is that for which when a topological change has performed.

DEFINITION 4.2 A compactification process between topological particles is an annihilation process between them.

DEFINITION 4.3 A compactification is the result of a compactification process that has performed in such a way one of the gaussian curvatures of the topological particles taking part in the process has prevailed ( Figure 4.1 ).

Figure 4.1

In technical paper was obtained a fundamental relation in the form

(4.1)

With the help of this mathematical relation was possible to explain the electrostatic interaction but now it is possible to explain a new fact what is well known. It is the existence of positron. Following the explanations about electrostatic interaction given in technical paper, the schema including the existence of positron is

R :

A :

Where

The existence of positron has sense due that for each electron of the group of electrons, the pair must be different in such a way they can not experiment a compactification process, say, they repel each other, although, they must have the same number of compactifications in order they must have the same mass. In this context there is other possibility for the parameters , say, there are pairs of topological particles with the same values ( so they experiment attraction ), even though the number of compactifications are the same for those pairs of topological particles as well as for the electrons. This means that one of them is a positron. Even more, following the same reasoning above, it is not difficult to show that each particle has its proper antiparticle.

From equation (4.1) and by differentiation we can derive a pair of partial differential equations in the unknown functions ,, y as follows.

Differentiating with respect to x we obtain

then

(4.2)

Again from (4.1)

Then equation (4.2) splits into two equations as follow

(4.3)

(4.4)

Ifandare sufficient small,

~1 and similarly ~1

So equations (4.3) and (4.4) become

In a similar way by differentiating equation (4.1) with respect to t, we can derive the following pair of equations

Say we have obtained a set of four non - homogeneous linear partial differential equations in the unknown functions ,,and. However we can only take two of them, for instance