Dirac Oscillator Under the New Generalized Uncertainty Principle From the Concept Doubly Special Rel
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S. Sargolzaeipor,1,*, H. Hassanabadi1, W. S. Chung2Faculty of Physics, Shahrood University of Technology, Shahrood, Iran Department 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. Keywords：Dirac equation;generalized uncertainty principle;doubly special relativity
PDF (798KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article S. Sargolzaeipor, H. Hassanabadi, W. S. Chung. Dirac Oscillator Under the New Generalized Uncertainty Principle From the Concept Doubly Special Relativity. [J], 2019, 71(11): 1301-1303 doi:10.1088/0253-6102/71/11/1301
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
$$ \left[ {X\,,\,P} \right] = i\,\hbar \,\left( {1 + \beta \,{P^2}} \right). $$ We set $\left( {\hbar = 1} \right)$, $X$ and $P$ are the position operator and the momentum operator; respectively, and $\beta$ is the generalized uncertainty principle parameter given by
$$ \beta = \frac{{{\beta _0}}}{{{{\left( {{M_{\rm pl}}c} \right)}^2}}}, $$ where ${M_{\rm pl}}$ is the Planck mass and ${\beta _0}$ is of the order of the unity.
The modified commutation relation can be written as[21-23]
$$ \left[{X, P}\right] = i\,{\rm K}\left[ P \right]. $$ In Eq. (3), ${\rm K}\left[ P \right]$ 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
$$ P = p\,,\quad X = i\,{\rm K}\left( p \right){\partial _p}\,, $$ where $\left( {X\,,\,P} \right)$ implies the position and momentum operators at the high energy while $\left( {x\,,\,p} \right)$ 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]
$$ X = x = i\,\frac{\partial }{{\partial \,p}}\,\,,\,\,P = p\,\alpha \left( p \right), $$ where $P$ is the generalized uncertainty principle momentum. The inverse relation of Eq. (5) becomes
$$ \label{a6} p = P\,\gamma \left( P \right), $$ and by inserting in Eq. (5), one can easily find
$$ \alpha \left( p \right)\,\gamma \left( P \right) = 1. $$ From Eq. (5), we obtain
$$ \left[ {X\,,\,P} \right] = i\left[ {\alpha \left( p \right) + p\,\alpha '\left( p \right)} \right]. $$ Differentiating Eqs. (5), (7) with respect to $p$ we obtain
$$ \alpha '\left( p \right)\,\gamma \left( P \right) + \alpha \left( p \right)\,\gamma '\left( P \right)\frac{{d\,P}}{{d\,p}} = 0, \nonumber\\ \frac{{d\,P}}{{d\,p}} = \alpha \left( p \right) + p\,\alpha '\left( p \right). $$ Combination of above equations gives
$$ \alpha '\left( p \right) = - \frac{{\gamma '\left( P \right)}}{{{\gamma ^2}\left( P \right)\left[ {\gamma \left( P \right) + {{P\,\gamma '\left( P \right)}}/{{\gamma \left( P \right)}}} \right]}}. $$ By substitution of Eq. (7) and Eq. (10) in Eq. (8), we obtain
$$ \left[ {X, P} \right] = \frac{i}{{\gamma \left( P \right) + P\,\gamma '\left( P \right)\,}}. $$ Based on the doubly special relativity, we consider the following relation
$$ P = \frac{p}{{1 + {p}/{\kappa }}}, $$ where $\kappa $ is the Planck momentum. Equation (5) gives
$$ \alpha \left( p \right) = \frac{p}{{1 + {p}/{\kappa }}}. $$ The inverse transformation is
$$ \gamma \left( P \right) = \frac{1}{{1 - {P}/{\kappa }}}. $$ In limit $p \to \infty $ corresponds to $P = \kappa $, that there exists the maximum momentum in our model. Then, inserting Eq. (15) into Eq. (11) we have
$$ \left[ {X\,,\,P} \right] = i\,{\left( {1 - \frac{P}{\kappa }} \right)^2}, $$ which gives the generalized uncertainty principle
The coordinate representation of the algebra (16) is
$$ X = x\,,\quad P = \frac{p}{{1 + {p}/{\kappa }}}\,. $$ The one-dimensional Dirac oscillator for a free fermion is written as $\left( {\hbar = c = 1} \right)$
$$ \left[ \alpha \cdot (P -i m\omega \beta X)+\beta m \right]\Psi = E \,\Psi, $$ which
$$ {\alpha _x} = {\sigma _x} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \beta = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, $$ with $\Psi = {\left( {\psi \,,\,\varphi } \right)^{\rm T}}$ in the presence of the coordinate representation Eq. (18), Eq. (19) becomes
$$ \begin{pmatrix} {m - E} & {p\left( {1 - \frac{p}{\kappa } + \frac{{{p^2}}}{{{\kappa ^2}}} - \frac{{{p^3}}}{{{\kappa ^3}}}} \right)} + i m \omega x \\ {p\left( {1 - \frac{p}{\kappa } + \frac{{{p^2}}}{{{\kappa ^2}}} - \frac{{{p^3}}}{{{\kappa ^3}}}} \right)} - i m \omega x & -({m + E}) \end{pmatrix}\begin{pmatrix} \psi \\ \varphi \end{pmatrix}= 0\,, $$ Eq. (21) can be rewritten in terms of following coupled equations
$$ \left( {m - E} \right)\psi +\Big( p\Big( {1 - \frac{p}{\kappa } + \frac{{{p^2}}}{{{\kappa ^2}}} - \frac{{{p^3}}}{{{\kappa ^3}}}} \Big) + i m \omega x \Big) \varphi = 0\,, \quad \Big( p\Big( {1 - \frac{p}{\kappa } + \frac{{{p^2}}}{{{\kappa ^2}}} - \frac{{{p^3}}}{{{\kappa ^3}}}} \Big) - i m \omega x\Big)\psi - \left( {m + E} \right)\varphi = 0\,. $$ From these equations we obtain
$$ a\left| n \right\rangle = \sqrt n \left| {n - 1} \right\rangle \,,\quad {a^\dagger }\left| n \right\rangle = \sqrt {n + 1} \left| {n + 1} \right\rangle, $$ the momentum $p$ and position $x$ operators are directly written in terms of the boson operators $a$ and ${a^\dagger }$ introduced above with
$$ p = i\sqrt {\frac{{m\,\omega \,\hbar }}{2}} \,(a - {a^\dagger }), \quad x = \sqrt {\frac{{\,\omega \,\hbar }}{{2\,m}}} \,(a + {a^\dagger }). $$ Finally, considering an expansion of $\varphi $ in terms of $\left| n \right\rangle $ Eq. (23) takes the following form
$$ E_n\! =\! \pm m\sqrt{1\!+\!\frac{\omega }{m}\left( {n\!+\!\frac{1}{2}} \right)\!+\!\frac{{9\,{\omega ^2}}}{{4\,{\kappa ^2}}}\left( {2{n^2}\!+\!2\,n\!+\!1}\right)}. $$ When ${1}/{\kappa } \to 0$ the well-know relation
$$ E_n = \pm m\sqrt{1 + \frac{\omega }{m}\left( {n + \frac{1}{2}} \right)}, $$ is recovered.[36] The energy $(+,-)$ in Eq. (27) are for particle and antiparticle, respectively. Figure 1 shows plots of $E$ versus $n$. 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 ${1}/{\kappa}$, also, the absolute value of the energy increases with the increasing $n$.
Fig. 1
New window|Download| PPT slide Fig. 1Spectrum of energy versus quantum number for different values of the parameter $\kappa$.
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 $\left( {{1}/{\kappa } \to 0} \right)$ case in ordinary results were recovered.
,1,*, H. Hassanabadi1, W. S. Chung2 
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