A New Approach to the Field Theory

Ali Rıza ŞAHİN

Gazi Mahallesi İsmet Paşa Caddesi 1347 Sokak No:50 D:3

34090 Küçükköy İstanbul

TURKEY

ars1945@veezy.com

We propose interaction space model for field theory and we focus on this model in order to give details about it. Later, we will show that Maxwell Equations can be derived exactly, by using the interaction space concept.

1.) Interaction Space Concept

This study is based on completely new ideas in order to achieve a new point of view for the field theory, elementary particles and interactions. Although the ideas are new, we are familiar them because of classical mechanics. In construction of this assumption, there are important motivations of Einstein’s General Relativity Theory and Kaluza-Klein Theory, especially the latter. Certainty and objectivity of classical mechanics and similarities between classical mechanics and field theory lead us to suggest there may or must be a special space for field theory. This space is defined once; it is easier to get results of it. Consequently; we define interaction space and try to find equations of motion by receiving inspirations from General Relativity and Kaluza-Klein, although we do not use these theories in appearance.

Before we define the interaction space; we will remind some points about classical mechanics. In classical mechanics; the motion of any point is described by a position vector (). means distance of the point from an O origin of a fixed three-dimensional Euclidean space. We can show any point of this space by unique vector (Figure 1.a) in terms of coordinates and unit vectors as

where are coordinates and are unit vectors of the space (). Unit vectors satisfy

For a moving point in a fixed space, coordinates will be some functions of the time () as

If a rotating frame of reference with respect to a fixed space, unit vectors become some functions of the time (Figure 1.b) also

In order to determine motion of an object which refers we must calculate equations of the motion. There are several methods to do this. However; we want to establish a relation between geometrical structure and motion of the object, similar to General Relativity and Kaluza-Klein Theory. There is a well-known way in order to do this; it is geodesic motion of the system. By determining geodesics of a system we can find equations of motion. Consequently, metric of the system must be found firstly. The metric is

. (1)

By skipping some details; for a classical mechanics system

can be written, where is the kinetic energy and mass of the object. Geodesics of the system can be derived from

where and are endpoints of the action.

Now; we will try to adopt these classical mechanics’ variational principles of the geodesics to field theory. But; we must determine a coordinate system or space for field theory first of all. In classical mechanics; systems can be observable and all things depend on the time, so defining an exact coordinate system is very easy. In field theory; systems can not be observed so determination of coordinate system is difficult. At this point, we will try to determine it by using similarity between classical mechanics and field theory. In the classical mechanics; equations of motion of coordinates are tried to determine as functions of the time while equations of motion of fields are tried to find as functions of the space-time in the field theory. This situation gives an important idea about coordinate systems of the field theory; fields of the field theory must be coordinates of the coordinate systems. Plus for a moving point in this space these coordinates are some functions of the space-time. Naturally; we assume that this system is Euclidean similar to the classical systems and in general it can be -dimensional. Figure 2.a shows a 4-dimensional Euclidean space (Suppose each axis is perpendicular to each other). In this figure are coordinates and are unit vectors of this space . We call this space as (4-dimensional) interaction space.

We can show any point in the -dimensional interaction space with a position vector

where . Unit vectors of the interaction space satisfy

(2)

If this point moves in the interaction space

can be written (Figure 2.a). In addition the previous situation, if the interaction space rotates with respect to a fixed space, unit vectors will be observed some functions of the space-time as

Similar to eq. (1), the metric of the interaction space is

where is the complex conjugate of . For the geodesics of the interaction space we need to calculate

In appendix I section, detailed calculation of geodesics and equations of motion are given. By using results of these, we can use a Lagrangian to find geodesics of the interaction space as

, (3)

where

Hereafter, superscripts denote complex conjugate of a quantity, exception of the space-time.

2.) Applications of the Interaction Space

For applications we choose 4-dimensional interaction space since results of this is easier to understand and explain.

For the first application we assume that we have a fixed 4-dimensional interaction space and its unit vectors are real. For this space position vector of any point is

, (4)

and the complex conjugate of eq. (4) is

From eq. (3), Lagrangian is

(5)

eq. (5) leads to the following equations of motion

(6)

and the conserved quantity from these is

In eq. (6)

We see that eq. (6) represents some massless particles. We do not make any interpretation for charge and spin for this particles unless we get to results of next example of the interaction space.

Our next example is the same 4-dimensional interaction space again. But this time, it is not a fixed space and it rotates with respect to a fixed space. Because of this, the position vector of any point will be

This time unit vectors are some functions of the space-time and this leads to an important result: Since are unit vectors of the interaction space, is perpendicular to and therefore must lie in the plane of the other unit vectors. Because of this we can write for

, (7)

where are some coefficients and they are important for the Langrangian. We give detailed information about derivation of the Lagrangian for this 4-dimensional interaction space in the Appendix II section. This Lagrangian is

, (8)

where

Lagrangian (8) gives following equations of motion

, (9)

, (10)

. (11)

We show in Appendix III section that eq. (9) can be written as

, (12)

and is a conserved quantity which can be written in a form (Appendix III section)

, (13)

where

Another conserved quantity can be derived from equations (10) and (11) as

(14)

Now; turn back eq. (12). It is nothing but Maxwell’s Equations in a different form. The well-known Maxwell’s equations can be derived by writing definition of in terms and by writing components of the electric and magnetic fields vectors E and B as follows

, , ,

, , .

Here and are components of the electric and magnetic fields vectors E and B respectively.

3.) Conclusions

There are many results of interaction space assumption. But; since they are extremely speculative now, we try to give the most obvious results.

If eq. (12) is Maxwell’s equations, we see that photons appear as a result of the relative motion of the interaction space. Unless, interaction space rotates, photons are not observable.

If eq. (12) is Maxwell’s equations, eqs. (10) and (11) should belong to electrons. We know, electrons are represented by Dirac spinors and Dirac equations. Subsequently, we encounter an interesting situation; eqs. (10) and (11) should be Dirac’s equations in a different and more general form or there is another thing about them. Since this is out of scope of this study we do not give more explanation.

If eq. (12) is Maxwell’s equations, in (13) must be electrical charge-current density. However, we have found another conserved quantity by eq. (14). Two different conserved quantities are not usual situation for electrodynamics. Explanation of this is out of scope this study again.

We can give more consequences of our assumption. But, as we said, they will be extremely speculative. The important thing is the fact that; the interaction space assumption gives us a new way to handle field theory, fields, particles and interactions. However; it needs to be understood first off all.

Appendix I

In this appendix, we will try to find geodesics equations of N-dimensional interaction space and we will show these equations can be derived from a Langrangian.

For N-dimensional interaction space, position vector of any point is

where . The metric this interaction space is

or briefly

(AI.1)

can be written by letting

From definitions of , we see that

Then total derivative of can be written as

(AI.2)

and the metric is

. (AI.3)

By comparing eqs. (AI.1) and (AI.3) we have

. (AI.4)

Using eq. (AI.4) we can write

By letting eq. (AI.2) becomes

where space-time is parameterized byand if variational principles are applied to this metric

(AI.5)

can be written. There is an important point: for this interaction space; only its coordinates vanishes at the endpoints (i.e. ). Eq. (AI.5) leads to the usual Euler-Langrange equations;

(AI.6)

where and . We know; the action (AI.5) must be invariant under any arbitrary reparameterization. As a result of this fact we can choose a reparameterization so that

, (AI.7)

where is a constant. Eq. (AI.7) leads to the Langrangian

Now we will show another thing. Since

(AI.8)

can be written. But normally Lagrangian is . So

. (AI.9)

However; since

we can write from eq. (AI.8)

. (AI.10)

By comparing eqs. (AI.9) and (AI.10) we have

. (AI.11)

Now by writing

eq. (AI.6) becomes

. (AI.12)

Since

eq. (AI.12) becomes

or from eq. (AI.11)

Thus; we can use the last equation to find geodesic motion of the interaction space

Appendix II

In this appendix, we derive Lagrangian for the 4-dimensional interaction space which rotates with respect to a fixed space.

For this 4-dimensional interaction space

From eq. (7)

and

Let

(AII.1)

Using eq. (2)

let

Derivative of the position vector leads us to define a new vector . Here has similar role to the angular velocity vector ω in Classical Mechanics. Similar to Classical Mechanics, derivatives of the position vector can be written as

In the last expression, and denote derivatives of R with respect to fixed and relative interaction spaces, respectively.

In light of above knowledge we can rewrite

(AII.2)

The second term in the right hand of eq. (AII.2) vanishes as result of anti-symmetrical features of Levi Civita’s. So

By performing summation over Levi Civita’s

And since from eq. (AII.1)

we can write

(AII.3)

We can simplify eq. (AII.3) by writing ’s in terms of unit vectors. For this we write

(AII.4)

. (AII.5)

From the last equation, using eq. (2) we can find

By taking differential of the first equation of eq. (2)

(AII.6)

can be found. By replacing eq. (AII.4) into eq. (AII.6) we have

. (AII.7)

By eq. (AII.7), we have found that is anti-symmetric with respect to the last two indices. We can show it is anti-symmetric with respect to the first two indices also. For this, we will multiply eq. (AII.4) by, but we will denote this product process by, since we will find that there must be an anti-commutative product between and. If we multiply eq. (AII.4) by, we find

. (AII.8)

We can rewrite eq. (AII.8), by interchanging m and n indices on the right-hand side of it. Now, we have

. (AII.9)

Eqs. (AII.8) and (AII.9) are the same, so we write

There are two possibilities for product. It is either commutative or anti-commutative. If it is commutative, is symmetric with respect to first two indices. But this option is impossible; because we have found that was anti-symmetric with respect to last two indices. can not be symmetric with respect to first two indices and anti-symmetric with respect to last two indices at the same time. Then product is anti-commutative, consequently is anti-symmetric with respect to first two indices. That is; is totally anti-symmetric. So

. (AII.10)

Using eq. (AII.10)

can be written. Let

Thus

. (AII.11)

We can derive some important relations for . If we multiply eq. (AII.11) by, from eq. (AII.5) we find

. (AII.12)

By multiplying eq. (AII.12) by we have

And using eq. (AII.10)

So using these equations we can find

and the Lagrangian

Appendix III

In this appendix, we will show that is a conserved quantity. We have found that

. (AIII.1)

From eq. (AIII.1)

As a result of summations over anti-symmetrical terms we have

We rewrite it as

By interchanging λ and θ of the first term in the square parentheses we find

And Using eq. (AIII.1)

. (AIII.2)

From eq. (AIII.2), we see that is conserved as a result of mathematical requirement. However; there is another probability for eq. (AIII.2); R may be a constant, for this case is conserved again. Therefore; if R is constant becomes

Figures

Figure 1.a. Three-dimensional Euclidean Space r is position vector of any arbitrary point of this space.

r= x e₁+y e₂+z e₃

Figure 1.b. Relative motion of a 3-dimensional Euclidean Space. Red axes belong to the fixed space.

Figure 2.a. Representative geometrical figure of the 4-dimensional Euclidean Space.

R is the position vector of any arbitrary point in this space

Figure 2.b. Representative geometrical figure of the relative motion of a 4-dimensional Euclidean Space.

Red axes belong to the fixed 4-dimensional Euclidean Space