Dirac Oscillator Under the New Generalized Uncertainty Principle From the Concept Doubly Special Relativity

S.Sargolzaeipor,H.Hassanabadi, and W.S.Chung

1Faculty of Physics, Shahrood University of Technology, Shahrood, Iran

2Department of Physics and Research Institute of Natural Science, College of Natural Science, Gyeongsang National University, Jinju 660-701, Korea

Abstract We discuss one-dimensional Dirac oscillator, by using the concept doubly special relativity.We calculate the energy spectrum by using the concept doubly special relativity.Then, we derive another representation that the coordinate operator remains unchanged at the high energy while the momentum operator is deformed at the high energy so that it may be bounded from the above.Actually,we study the Dirac oscillator by using of the generalized uncertainty principle version and the concept doubly special relativity.

Key words:Dirac equation, generalized uncertainty principle, doubly special relativity

1 Introduction

Very interesting subjects in physics are quantum gravity and quantum groups.The structure and representation theory of the generalized uncertainty principle[1−19]were initially accomplished by Kempf, Mangano, and mann.[20]The generalized uncertainty principle is written by the modified commutation relation between position and momentum operators

whereMplis the Planck mass andβ0is of the order of the unity.

The modified commutation relation can be written as[21−23]

In Eq.(3), K[P]is the generalized uncertainty principle deformation function which reduces to one when the generalized uncertainty principle effect is removed.The standard representation for Eq.(3) is the momentum representation appears as

where (X, P) implies the position and momentum operators at the high energy while (x, p) the position and momentum operators at the low energy.The momentum operator at the high energy should be bounded from the above if we consider the doubly special relativity.[24−29]Indeed the doubly special relativity says that the momentum has the maximum called a Planck momentum, which is another invariant in the doubly special relativity.

In this paper, we investigate another representation where the coordinate remains unchanged at the high energy while the momentum is deformed at the high energy so that it may be bounded from the above.Our generalized uncertainty principle model becomes from the concept doubly special relativity.[30−34]This paper is organized as follows:In Sec.2, we study the new generalized uncertainty principle from the concept of doubly special relativity.In Sec.3, we discuss the Dirac equation in one-dimensional.Finally, we present the results in our conclusion.

2 New Generalized Uncertainty Principle from the Concept of Doubly Special Relativity

Instead of the representation (4) one can consider the following representation[35]

wherePis the generalized uncertainty principle momentum.The inverse relation of Eq.(5) becomes

and by inserting in Eq.(5), one can easily find

From Eq.(5), we obtain

Differentiating Eqs.(5), (7) with respect topwe obtain

Combination of above equations gives

By substitution of Eq.(7) and Eq.(10) in Eq.(8), we obtain

Based on the doubly special relativity, we consider the following relation

whereκis the Planck momentum.Equation (5) gives

The inverse transformation is

Equation (6) gives

In limitp →∞corresponds toP=κ, that there exists the maximum momentum in our model.Then, inserting Eq.(15) into Eq.(11) we have

which gives the generalized uncertainty principle

3 The Dirac Oscillator

The coordinate representation of the algebra (16) is

The one-dimensional Dirac oscillator for a free fermion is written as (=c=1)

which

with Ψ=(ψ, φ)Tin the presence of the coordinate representation Eq.(18), Eq.(19) becomes

Eq.(21) can be rewritten in terms of following coupled equations

From these equations we obtain

Then, the operatorsaanda†act as ladder operators:

the momentumpand positionxoperators are directly written in terms of the boson operatorsaanda†introduced above with

Finally, considering an expansion ofφin terms of|n〉Eq.(23) takes the following form

When 1/κ →0 the well-know relation

is recovered.[36]The energy(+,−)in Eq.(27)are for particle and antiparticle, respectively.Figure 1 shows plots ofEversusn.We know that the energy decreases due to the generalized uncertainty principle effect.We observe that the behavior of the absolute value of the energy increases with the increasing 1/κ, also, the absolute value of the energy increases with the increasingn.

Fig.1 Spectrum of energy versus quantum number for different values of the parameter κ.

4 Conclusion

In this paper,we studied another representation where the coordinate remains unchanged at the high energy while the momentum is deformed at the high energy so that it may be bounded from the above.We investigated one-dimensional Dirac oscillator then we obtained the energy spectrum by using of generalized uncertainty principle version of the concept doubly special relativity and also especially we tested the energy spectrum in(1/κ →0)case in ordinary results were recovered.

Acknowledgment

It is a great pleasure for the authors to thank the referees for helpful comments.