Abstract
Minimal length uncertainty is used to study the fundamental interactions range. Both the quantum mechanical and the quantum gravitational effects are employed to calculate the range of exchanging interactions. It is shown that at the Planck scale, the range of interactions is 2- factor larger than the usual range of interactions.
Introduction
Study of unification of all fundamental interaction in the early universe is quite interesting problem in the both modern cosmology and quantum field theory point of view [1,2,3]. The modern scenario of the big bang cosmology at the early universe covered in the Weinberg’s First Three Minutes [4], and certain time regimes have been proposed with type of events as, before one Planck time, era of a Planck time, separation of the strong interaction, inflationary period, quark-antiquark period and finally the quark confinement.
In this scenario, before the Planck time all of the four fundamental interactions are unified as one interaction etc [5]. Fundamental interactions are containing gravitation, electromagnetism, weak and strong interaction. However, because the effective QCD coupling is not small, performing a precise calculation f long-range strong interaction effects is problematic and we cannot use the perturbation theory [6].
At current analysis, the range of exchanging interaction [3] is calculated by taking into account the Heisenberg uncertainty principle that only considers the quantum effects. It is believed that the unification of fundamental interactions may accurse at the Planck regime. At the Planck regime, both the quantum and quantum gravitational effects are important. To have a complete analysis of exchanging interaction we must consider both quantum and quantum gravitational effects to our analysis. In this letter we have developed an approach related to the unification focusing on the effective range of interactions. Quantum field theory explains the exchanging of energy during the interactions via the interaction carriers, called bosons. The mechanism is based on the emitting particles which have no reality except to push or pull matter with the exchanging momentum. All fundamental interactions are involves the exchanging of one or more particles. For example the underlying color is involved an exchanging of particles labeled by gluons. Such exchanging interaction may be either attractive or repulsive, but is limited in the range by the nature of exchanging interaction that is constrained by the uncertainty principle.

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The separation of fundamental interactions in the early universe may be related to the separation of exchanging range of virtual particles based on the spontaneous symmetry breaking mechanism [1-5]. Alternatively, this may be viewed as a mechanism to separation of fundamental interactions. For example the range of color interaction is the shortest range and the range of gravity is the longer range. But in the early universe or in a very high energy probe there is unification between fundamental interactions. To obtain a complete picture of the range of exchanging interactions in a high-energy probe, it is important to consider both the quantum mechanical and quantum gravitational effects, by imposing the minimal length uncertainty relation. The problem is related to consider the quantum gravity effects on the exchanging interactions range.
To study the quantum gravity effects on the exchanging particles, we can use the minimal length uncertainty [6-9],
(1)
Where is the Planck length. Dividing both side of relation (1) to the speed of light, we obtain a deformed form of usual time energy uncertainty as [9],
(2)
Where is the Planck time. Putting the natural units as, eq. (2) reads,
(3)
Solving (3) to minimum energy we obtain,
(4)
Expanding (4) around t’=0, obtains,
(5)
The energy of interaction which involves the exchanging particles is constrained by the uncertainty principle. According to special relativity a particle with mass of m has a rest energy as. So in the exchanging process, the particle does not go outside the constraints of uncertainty principle,
(6)
Combining (5) and (6) we give,
(7)
Theoretically, the exchanging particle “virtual particle” cannot exceed the speed of light and cannot travel faster than the speed of light c times than lifetime. Since the maximum range of a interaction () would be
(8)
The r.h.s of relation (8) have two term, the first term is the usual range of exchanging interaction and is a new term, this new term is obtained from the correction based on the minimal length uncertainty analysis. An important problem in the standard model is study of the unification of all fundamental interactions at the Planck time. If the usual range of fundamental interactions was compared with the range of fundamental interactions at the Planck scale, w obtain a surprising result. According to (8) each interaction contains two ranges of exchanging, and. If , then eq. (8) reads the usual range as,
R usual (9)
At the Planck time when, eq. (8) reads,
(10)
Comparing (9) with (10) we have,
R planck = 2 R usual (11)
From (11) it may be concluded that at the Planck scale, the range of interactions is 2- factor larger than the usual range of interactions.
Conclusion
A complete picture of the range of fundamental interactions may be obtained considering both the quantum mechanical and quantum gravitational effects. Using minimal length uncertainty the range of exchanging virtual particles is calculated. As shown by (11) at the Planck regime, the range of interactions is 2- factor larger than usual one. It is found that in the exchanging process of fundamental interactions, the mass of the carrier interactions (bosons) is an effective parameter.
References
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