Breakdown of Special Relativity at the Length Scale below Subhadronic Region
E. J. Jeong
Department of Physics, The University of Michigan, Ann Arbor,
Michigan 48109
Abstract
According to the Einstein’s mass energy relations,, is a fixed constant with no known parametric dependences. However, the recent study of the scale dependent mass relations suggests that the rest mass of the elementary particles may not be, in general, constants. Although special relativity was developed as a theory with no interactions present, the notion that the rest mass is an indisputable constant and there is only the corresponding interaction potential may not be justified. In the conventional field theoretical consideration, the interaction potential energy is not directly translated into the rest mass of the particles. However, in the recent result of the study of the quark confinement problem which can easily be explained by the consideration of the scale dependent rest mass, it is suggested that the interaction potential energy may represent the major part of the rest mass of the elementary particles.
1. Introduction
There have been many suggestions and suspicions about the validity of special relativity in the small length scale (3). However, the experimental results have been in good agreement with the theoretical predictions that such considerations have largely been ignored. In this connection, we have discussed and explained the quark confinement problem using the scale dependent mass relations (2) without any specific contradictions (4). Since there is no such known phenomenon as the scale dependent variation of the rest mass of a particle in special relativity, we suspect that special relativity may be an approximation of a theory that incorporates the scale dependencies in it.
2. Einstein’s Mass Energy Relation
Einstein has published the theory of special relativity (1) in 1905. One of the results of his theory was the mass energy relation given by. [1]
The validity of this relation has been verified with great accuracy (5). The above equation can also be written
, [2]
where is tacitly assumed to be a constant. Since the mass energy relation was initially developed for the macroscopic objects, there was no question regarding the constancy of the rest mass [6]. Even in the strong gravitational field, the rest mass would be a constant and according to general relativity, it is the warping of the space time that causes the motion of the stellar objects. However, there are two unnatural factors about the equation [2].
1. There is no room for the varying potential energy.
2. There is no concept of the interaction between the particles which is thought, in general, to be related to the energy of the particles under investigation.
In fact, 1 and 2 are closely related to each other and make us feel that may well not be a constant, so that if special relativity has to be broken down, it will start with the redefinition of . In most of the cases of the non interacting particles, there is no issue of the contradiction, however, since we have observed this clue in our previous notes (2) (4), we claim that is not a constant contrary to the tacit assumption of special relativity as it was originally formulated.
3. Definition of the Scale
We define the scale as an arbitrary quantity with the dimension of mass. We say the scale is large if the interacting charges are close together and small if they are far apart. Therefore, the scale is, in general, dependent on the relative distance between the two interacting charges. And also the scale depends on the relative momentum of the charged particles due to the uncertainty relation
[3]
In essence, our definition of the scale is the same as the one commonly named “momentum scale” (7). It is emphasized that when we mention the scale, we have in mind the scale dependent running coupling constant which has been derived using the renormalization group equations(8).
4. Scale Dependent Rest Mass Formula
We have shown in the previous notes(2) that the rest mass of a particle can be written for an electron,
, [4]
and for a quark
, [5]
where the constancy of has been demonstrated in our discussion on the quark confinement problem(4). The definitions [4] and [5] were derived using the first loop self energy diagram(2) in the limit , , . Therefore, we effectively singled out the pure self energy contribution which does not include the kinetic energy. We call this mass the “potential mass” to distinguish it from the conventional rest mass. In the macroscopic scale, the potential mass is approximately the same as the rest mass.
Since the coupling constants in quantum field theory depend, in general, (7) on the arbitrary scale parameter as shown from the study of the renormalization group equation(8), the relations [4], [5] indicate that the potential masses are not in general constants.
We have shown in the previous notes (4), how this scale dependent rest mass concept can be used to explain the problem of the quark confinement. Since this potential mass excludes the kinetic portion of the energy involved, we claim that it should be in fact the conventional rest mass in special relativity. Therefore, we are confronted with a contradiction between special relativity and the running coupling constant in quantum field theory. In fact, we have two choices either to discard the concept of the running coupling constant as false or to revise special relativity on the ground that the constant rest mass term is unnatural. We take the view that special relativity needs modification despite all the beauty and successes of its predictions. The reasons for this choice are;
1. The concept of the running coupling constant explained the asymptotic freedom (9).
2. There exists experimental evidence that the coupling constants do depend on the scale (10).
3. The quark confinement problem can be explained easily by using the concept of the scale dependent rest mass (4).
One may say that the concept of the running coupling constant is all right but the relation [4[, [5] are wrong. However, it must be noted that the conventionally known potential energy can not be translated into a mass in the classical treatment of the problem, the concept of which was essential for the explanation of the quark confinement.
We propose here a set of the general rules that the concept of the scale dependent rest mass brings into the new physics.
5. Postulates
(1) The potential mass of a charged particle depends on the coupling constants the particle is subjected to.
(2) All the coupling constants of the renormalizable gauge theories depend on the scale.
(3) The scale is an unknown function of the relative 4 coordinates or the 4 momentum between the given interacting charges.
(4) The total energy of the system in consideration is independent of the scale.
One can rephrase (1), (2), (3) simply, “The potential mass of a charged particle, in general, depends on the relative 4 coordinates or the momentum between the interacting charged particles”.
6. Modification to Einstein’s Mass-Energy Relation
Suppose that a particle m is subjected to a coupling constant g, then, we can write the total energy of the system, according to our postulate
[6]
where is the potential mass and is the spatial momentum. The dependence of is a necessary consequence of our postulates. For a macroscopic body, the coupling constants involved are usually more than one. Even for an electron, we may not know how many coupling constants are involved depending on the composite nature of its substructures. The only possible guiding principle is the constancy of (4).
Therefore, we can write the energy equation for an electron
[7]
and for a quark
[8]
If we suggest that the scale dependent running coupling constant is valid beyond the perturbation calculation limit (7), we may write for an electron
[9]
and for a quark, if the electric contribution can be ignored,
[10]
Where and are the coupling constants at respectively, and , and are the group structure dependent constants (7).
Now, we have two nontrivial energy equations, which have additional parameter dependence. Equation [9] shows that in the electron-electron collision experiment, the potential mass of each of the electrons increase when they come close together, which will be subsequently reduce the kinetic mass( defined in [13]) of the electron at the same time. Equation [10] says instead that in the quark-quark interaction case, the potential mass of each quark decrease when they come close together, which is in the asymptotic region, however, if they are about to be separated wide, the potential mass increase drastically (4), which is the cause of the quark confinement. According to the equation [9], the increase of potential mass is observable only when the two charges are close together, which may well have escaped the experimental discrimination. Therefore, rhw success of QED does not violate the equation [9]. On the other hand, the asymptotic freedom and the quark confinement are the natural consequences of the equation [10]. To see the scale variation dependence of, we take a partial derivative over on the equation [6].
[11]
which shows that as the coupling constant increases, the corresponding momentum decreases along the variation of the scale. Leaving behind the many potential applications of this result on the cosmological problems, we define for convenience,
= potential mass [12]
= kinetic mass [13]
In terms of these definitions, the equation [6] can be written,
[14]
, [15]
where M is the usual relativistic mass, which doesn’t depend on the scale. As we have suggested in the beginning, the unnaturalness of the constancy of the rest mass and that there is no room for the potential energy variation are remedied by this prescription. We may be able to state a law at this point “A particle moves in space toward the region where the potential mass is the minimum.”
7. Conclusions
Since the equation [6] is valid for a particle charged only with one type of interaction and the corresponding coupling constant, a generalization is necessary. For a particular case of the proton and the neutron, one can express the total energy by the sums of the three quarks’ potential mass and the kinetic mass contributed from both the color and electric charges respectively. However, we doubt the necessity of describing a macroscopic body in terms of quark coupling constant. At the level of the gravitation, the only charge that matters would be the mass of the macroscopic body. In this case we can write,
[16]
where is the rest mass and is the potential mass contribution due to the external gravitational interaction and the momentum due to the external interaction of the gravitation. include the internal gravitational potential mass and also the kinetic mass coming from the internal motion of the constituents. An easy place to test the theory would be in the subhadronic behavior of the quarks using the equation [10] and [11].
There remain several questions regarding on how the Lorentz transformation has to be reconciled with this variation and how this result will affect quantum field theory especially in the renormalization program. And also finally how general relativity will be affected by this result? We leave these questions for the future research.
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