Investigation of Heisenberg algebra for a modified anti-de Sitter and modified anti-Snyder models

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Nasrin Farahani^,¹, Hassan Hassanabadi¹, Won Sang Chung², Leyla Naderi³¹Faculty 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, Republic of Korea
³Faculty of Physics, Semnan University, PO Box 35195-363, Semnan, Iran

Received:2019-11-23Revised:2020-01-6Accepted:2020-01-7Online:2020-02-27

Abstract
In this paper, we consider a possible modification of the de Sitter and anti-de Sitter space for the extended uncertainty principle. For the modified anti-de Sitter model we discuss the representation and wave functions of the momentum operator for a one-dimensional box problem. Also, we consider modified Snyder and anti-Snyder models for the generalized uncertainty principle. Then, we assume the Hamiltonian with different potential and solve the Heisenberg algebra for the modified (anti)-de Sitter and (anti)-Snyder models in both position and in the momentum space.
Keywords： modified (anti)-de Sitter model;extended uncertainty principle;generalized uncertainty principle;Heisenberg algebra

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Nasrin Farahani, Hassan Hassanabadi, Won Sang Chung, Leyla Naderi. Investigation of Heisenberg algebra for a modified anti-de Sitter and modified anti-Snyder models. Communications in Theoretical Physics, 2020, 72(3): 035404- doi:10.1088/1572-9494/ab6914

1. Introduction

Any attempt to measure precisely the velocity of a subatomic particle, such as an electron, which is unpredictable, means that the simultaneous measurement of its position has no validity. This is the result of the principle of quantum mechanics (QM), formulated by Heisenberg [1], that accurate measurement of the one of two related observable quantities, as position and momentum or energy and time, produces uncertainties in the measurement of the other one, such that the product of the uncertainties of both quantities is equal to or greater than ℏ/2π, where ℏ is the Planck’s constant. Heisenberg's uncertainty principle is often expressed as the limitation of operational possibilities imposed by QMs. The Heisenberg’s uncertainty principle (HUP) has been extended to contain the existence of a minimal uncertainty in momentum. Therefore, we have another modified form of Heisenberg uncertainty principle in an (anti)-de Sitter background in which the Heisenberg uncertainty principle should be modified by introducing some corrections proportional to the cosmological constant with the (anti)-de Sitter radius. The de Sitter space was introduced by Willem de Sitter [2]. In the de Sitter background, space has a positive radius and in the anti-de Sitter background it has a negative radius. The modified form of the Heisenberg relations in both spaces is referred to as extended uncertainty principle (EUP). Then, we know that EUP has a minimal uncertainty in the momentum. The HUP should break down at energies near to Planck scale [1], and produce a sentence of Planck scale, that is a modified Heisenberg uncertainty relation and leads to deformed canonical commutator due to generalized uncertainty principle (GUP). On the other hand, GUP predicts the existence of a minimal length. Some approaches in physics predict this minimal length, Gedanken experiment [3], String theory [4–6], and black hole [7]. In the String theory, a GUP approach was introduced by Amati and et al [8]. This approach leads to finding a distance smaller than characteristic string length [4–6, 9]. Kempf introduced a minimal length scale to the mathematical basis of QM [10–13]. Thus, QM and quantum gravity are in agreement for the existence of a minimal uncertainty in position. These results are consistent with the non-commutative space–time discussed in [12, 14, 15] in the Minkowski spacetime.

In this paper we investigate the Heisenberg algebra using momentum representation by modified (anti)-Snyder based on GUP. In the second section, we introduce different modified anti-de Sitter and de Sitter models. In section 3, we obtain the different representations in the position space and the momentum space for the introduced commutation relations. Then, we obtain eigenstates of momentum in the position space in section 4. In section 5, we assume a new algebraic approach for these commutation relations and obtain the wave function in the momentum representation, and we calculate the Heisenberg algebra for a free particle in the modified anti-de Sitter and de Sitter models in section 6. We calculate the Heisenberg algebra by solving through the successive approximation method with Hamiltonian by considering harmonic oscillator potential for all commutation relations. In section 7, we introduce the GUP in Snyder and anti-Snyder space and obtain the modified kinetic energies, velocities and coordinates from the Poisson bracket in Snyder background and we obtain the modified frequency from solving Heisenberg algebra by considering Hamiltonian with Harmonic oscillator potential in the momentum representation for GUP in the Snyder and anti-Snyder space.

2. General extension of the (anti)-de Sitter model

We have another modified relation of the HUP [1] in an (anti)-de Sitter background. The HUP should be modified by introducing some corrections proportional to the cosmological constant $3{\lambda }^{2}=\left|{\rm{\Lambda }}\right|={10}^{-52}\ {{\rm{m}}}^{-2}$ , ${\lambda }^{2}=\tfrac{1}{{L}_{H}^{2}}$ , with L_H the (anti)-de Sitter radius and $\tfrac{1}{{L}_{H}^{2}}={\alpha }^{2}$ [10, 16]. Also we know de Sitter’s space–time is an exact solution of the equations of ordinary general relativity discovered in 1917 [2]. Also it can be defined as a sub manifold of a generalized Minkowski space of one higher dimension. In quantum field theory based on a curved space, the anti-de Sitter space–time (Ads) is a maximally symmetric with a negative cosmology constant (negative radius with L_H>0) and the cosmology constant is positive in the de Sitter space–time. In the case of zero curvature of this space–time, we have Minkowski space. As such, they are exact solutions of Einstein’s field equations for an empty universe with a positive, zero or negative cosmological constants, respectively. This space–time was named after Willem de Sitter [17]. Modified Heisenberg equation under the EUP in the de Sitter space is represented as [18](1)$\begin{eqnarray}\left[X,P\right]={\rm{i}}(1+{\alpha }^{2}{X}^{2}),\,\,\,\,\,{\alpha }^{2}\geqslant 0.\end{eqnarray}$Also, in the anti-de Sitter space for EUP we have [19, 20](2)$\begin{eqnarray}\left[X,P\right]={\rm{i}}(1-{\alpha }^{2}{X}^{2}),\,\,\,\,\,{\alpha }^{2}\geqslant 0.\end{eqnarray}$In this representation for EUP in the de Sitter and anti-de Sitter we have(3)$\begin{eqnarray}\hat{X}=X(x),\,\,\,\,\hat{P}=p.\end{eqnarray}$This representation leads to(4)$\begin{eqnarray}P=-{\rm{i}}\displaystyle \frac{\partial }{\partial x}.\end{eqnarray}$The relation between the EUP position and the canonical position in anti-de Sitter model is obtained as(5)$\begin{eqnarray}\displaystyle \frac{{\rm{d}}X}{{\rm{d}}x}=(1-{\alpha }^{2}{X}^{2}).\end{eqnarray}$Then, we have the modified position and its ordinary form as(6)$\begin{eqnarray}X=\displaystyle \frac{1}{\alpha }\tanh (\alpha x),\,\,\,x=\displaystyle \frac{1}{\alpha }{\tanh }^{-1}(\alpha X),\end{eqnarray}$in which X and x represent the modified position and its ordinary form, respectively. We can find the limits from above equations as(7)$\begin{eqnarray}-\displaystyle \frac{1}{\alpha }\leqslant X\leqslant \displaystyle \frac{1}{\alpha }.\end{eqnarray}$The EUP in the de Sitter background can be generalized into the following form [21](8)$\begin{eqnarray}\left[X,P\right]={\rm{i}}{\left(1-{\alpha }^{2}{X}^{2}\right)}^{a},\,\,\,\,\,a\gt 0,\end{eqnarray}$where a=1 gives the ordinary EUP model. The relation between the EUP position and canonical position is(9)$\begin{eqnarray}\displaystyle \frac{{\rm{d}}X}{{\rm{d}}x}={\left(1-{\alpha }^{2}{X}^{2}\right)}^{a},\end{eqnarray}$Then, we can write the extended following relations for EUP(10)$\begin{eqnarray}a=\displaystyle \frac{1}{2},\,\,\,\,\,X=\displaystyle \frac{1}{\alpha }\sin (\alpha x),\,\,\,\,x=\displaystyle \frac{1}{\alpha }{\sin }^{-1}(\alpha X),\end{eqnarray}$

(11)$\begin{eqnarray}a=1,\,\,\,\,\,X=\displaystyle \frac{1}{\alpha }\tanh (\alpha x),\,\,\,\,x=\displaystyle \frac{1}{\alpha }{\tanh }^{-1}(\alpha X),\end{eqnarray}$

(12)$\begin{eqnarray}a=\displaystyle \frac{3}{2},\,\,\,\,\,X=\displaystyle \frac{x}{\sqrt{1+{\alpha }^{2}{x}^{2}}},\,\,\,\,x=\displaystyle \frac{X}{\sqrt{1-{\alpha }^{2}{X}^{2}}}.\end{eqnarray}$Now, we obtain another different representation for EUP(13)$\begin{eqnarray}\left[X,P\right]={\rm{i}}{\left(1-{\alpha }^{2}{X}^{2}\right)}^{\tfrac{1}{2}},\,\,\,\,\,{\rm{\Delta }}X{\rm{\Delta }}P\geqslant \displaystyle \frac{1}{2}\left\langle {\left(1-{\alpha }^{2}{X}^{2}\right)}^{\tfrac{1}{2}}\right\rangle ,\end{eqnarray}$where(14)$\begin{eqnarray}P\to -{\rm{i}}\sqrt{1-{\alpha }^{2}{X}^{2}}\displaystyle \frac{\partial }{\partial X}.\end{eqnarray}$From the self-adjointness of the operator P we have the following inner product(15)$\begin{eqnarray}(\phi | \psi )={\int }_{\tfrac{-1}{\alpha }}^{\tfrac{1}{\alpha }}\displaystyle \frac{{\rm{d}}X}{\sqrt{(1-{\alpha }^{2}{X}^{2})}}{\phi }^{* }(X)\psi (X).\end{eqnarray}$And the expectation value of the operator $\hat{A}$(16)$\begin{eqnarray}(\phi | A\phi )={\int }_{\tfrac{-1}{\alpha }}^{\tfrac{1}{\alpha }}\displaystyle \frac{{\rm{d}}X}{\sqrt{(1-{\alpha }^{2}{X}^{2})}}{\phi }^{* }(X)A\phi (X),\end{eqnarray}$where φ(X) obey(17)$\begin{eqnarray}\phi \left(\pm \displaystyle \frac{1}{\alpha }\right)=0.\end{eqnarray}$and the normalization condition(18)$\begin{eqnarray}(\phi | \phi )={\int }_{\tfrac{-1}{\alpha }}^{\tfrac{1}{\alpha }}\displaystyle \frac{{\rm{d}}X}{\sqrt{(1-{\alpha }^{2}{X}^{2})}}{\left|\phi (X)\right|}^{2}=1,\end{eqnarray}$first we will incorporate that(19)$\begin{eqnarray}\left\langle \sqrt{1-{\alpha }^{2}{X}^{2}}\right\rangle =L\ne 0.\end{eqnarray}$which implies that $\left\langle \sqrt{1-{\alpha }^{2}{X}^{2}}\right\rangle $ cannot be zero for all normalized wave functions φ(X). Let us assume that L=0. Then we have(20)$\begin{eqnarray}\left\langle \sqrt{(1-{\alpha }^{2}{X}^{2})}\right\rangle ={\int }_{\tfrac{-1}{\alpha }}^{\tfrac{1}{\alpha }}{\rm{d}}X{\left|\phi (X)\right|}^{2}=0.\end{eqnarray}$Because $| \phi (X){| }^{2}\geqslant 0$, we have φ(X)=0 for all $X\in \left[-\tfrac{1}{\alpha },\tfrac{1}{\alpha }\right]$. Then, we have(21)$\begin{eqnarray}(\phi ,\phi )=0.\end{eqnarray}$That contradicts equation (18). Therefore $L\ne 0$ and 0< L<1. Because ${X}^{2}\leqslant \tfrac{1}{{\beta }^{2}}$ for all $X\in \left[-\tfrac{1}{\alpha },\tfrac{1}{\alpha }\right]$ we get(22)$\begin{eqnarray}{\rm{\Delta }}X\leqslant \displaystyle \frac{1}{\alpha }.\end{eqnarray}$Thus, we have(23)$\begin{eqnarray}{\rm{\Delta }}P\geqslant \displaystyle \frac{L\alpha }{2}.\end{eqnarray}$Therefore, it has been shown that the modified (anti)-de Sitter model (13) has the non-zero minimal momentum.

3. Three representations of the modified (anti)-de Sitter model

For the modified anti-de Sitter and de Sitter model we have three types of representations.

3.1. EUP position representation

The EUP position representation for algebra (13) is as [22](24)$\begin{eqnarray}\hat{X}=x,\,\,\,\,\,\,\,\hat{P}=-{\rm{i}}\sqrt{(1-{\alpha }^{2}{X}^{2})}{\partial }_{X}.\end{eqnarray}$The EUP position representation acts on the square integrable functions(25)$\begin{eqnarray}\psi (X)\in L\left(\displaystyle \frac{-1}{\alpha },\displaystyle \frac{1}{\alpha };\displaystyle \frac{{\rm{d}}X}{\sqrt{1-{\alpha }^{2}{X}^{2}}}\right),\end{eqnarray}$and the norm of ψ(X) is given by(26)$\begin{eqnarray}{\left|\left|\psi \right|\right|}^{2}={\int }_{\tfrac{-1}{\alpha }}^{\tfrac{1}{\alpha }}\displaystyle \frac{{\rm{d}}X}{\sqrt{1-{\alpha }^{2}{X}^{2}}}{\left|\psi (X)\right|}^{2}.\end{eqnarray}$The Schrödinger equation reads(27)$\begin{eqnarray}\left[\displaystyle \frac{-1}{2m}{\left(\sqrt{1-{\alpha }^{2}{X}^{2}}\displaystyle \frac{\partial }{\partial X}\right)}^{2}+V(X)\right]\psi (X)=E\psi (X).\end{eqnarray}$

3.2. EUP momentum representation

The momentum representation for the EUP (13)(28)$\begin{eqnarray}\hat{P}=p,\,\,\,\,\,\,\hat{X}=\displaystyle \frac{1}{\alpha }\sin (\alpha x)=\displaystyle \frac{1}{\alpha }\sin ({\rm{i}}\alpha {\partial }_{p}).\end{eqnarray}$The momentum representation acts on the square integrable functions(29)$\begin{eqnarray}\phi (p)\in {L}^{2}(-\infty ,\infty ;{\rm{d}}p),\end{eqnarray}$and the norm of φ is given by(30)$\begin{eqnarray}{\left|\left|\phi \right|\right|}^{2}={\int }_{-\infty }^{\infty }{\rm{d}}p{\left|\phi (p)\right|}^{2}.\end{eqnarray}$The Schrödinger equation reads(31)$\begin{eqnarray}\left[\displaystyle \frac{{P}^{2}}{2m}+V\left(\displaystyle \frac{1}{\alpha }\sin (\alpha x\right)\right]\phi (p)=E\phi (p).\end{eqnarray}$

3.3. Canonical position representation

The canonical position representation for the (13) algebra is obtained from the momentum representation with replacing $\hat{P}=\hat{p}=-{\rm{i}}\tfrac{\partial }{\partial x}$ , as(32)$\begin{eqnarray}\hat{P}=\hat{p}=\displaystyle \frac{1}{{\rm{i}}}{\partial }_{x},\,\,\,\,\,\,\hat{X}=\displaystyle \frac{1}{\alpha }\sin (\alpha x).\end{eqnarray}$The canonical position representation acts on the square integrable functions(33)$\begin{eqnarray}\psi (x)\in {L}^{2}(-\infty ,\infty ;{\rm{d}}X),\end{eqnarray}$and the norm of ψ is given by(34)$\begin{eqnarray}{\left|\left|\psi \right|\right|}^{2}={\int }_{-\infty }^{\infty }{\rm{d}}X{\left|\psi (x)\right|}^{2},\end{eqnarray}$the Schrödinger equation reads(35)$\begin{eqnarray}\left[\displaystyle \frac{-1}{2m}\displaystyle \frac{{\partial }^{2}}{\partial {x}^{2}}+V\left(\displaystyle \frac{1}{\alpha }\sin (\alpha x\right)\right]\psi (x)=E\psi (x).\end{eqnarray}$

4. Eigenstates of momentum operator in the EUP position space

The momentum operators generating EUP position-space eigenstates are given by(36)$\begin{eqnarray}P{\psi }_{p}(x)=p{\psi }_{p}(x),\end{eqnarray}$or(37)$\begin{eqnarray}-{\rm{i}}\sqrt{1-{\alpha }^{2}{X}^{2}}\displaystyle \frac{\partial }{\partial x}\psi (X)=p\psi (X).\end{eqnarray}$This differential equation can be solved to obtain formal momentum eigenvectors(38)$\begin{eqnarray}\psi (x)=C\exp \left(\displaystyle \frac{{\rm{i}}p{\sin }^{-1}(\alpha X)}{\alpha }\right),\end{eqnarray}$for the case of $| X| \lt \tfrac{1}{\alpha }$ the equation (36) reduces to ${{C}{\rm{e}}}^{-{pX}}$, that is the plane wave. The wave function ψ_p(X) obeys the normalization condition(39)$\begin{eqnarray}\left\langle {\psi }_{p}(x)| {\psi }_{p}^{{\prime} }(x)\right\rangle =\delta (p-{p}^{{\prime} }),\end{eqnarray}$or(40)$\begin{eqnarray}{\int }_{\tfrac{-1}{\alpha }}^{\tfrac{1}{\alpha }}\displaystyle \frac{{\rm{d}}X}{\sqrt{(1-{\alpha }^{2}{X}^{2})}}{\psi }_{p}^{* }(X){\psi }_{{p}^{{\prime} }}(X)=\delta (p-{p}^{{\prime} }),\end{eqnarray}$which means $C=\tfrac{1}{\sqrt{2\pi }}$. Then, the normalized plane wave in the modified anti-de Sitter model is(41)$\begin{eqnarray}\psi (X)=\displaystyle \frac{1}{\sqrt{2\pi }}\exp \left(\displaystyle \frac{{\mathrm{ipsin}}^{-1}(\alpha X)}{\alpha }\right).\end{eqnarray}$Also, we can be calculate the Wave Function in a momentum representation and deformed exponential function by using the Jackson's q-derivative [23–25] or Tsallis's q-derivative [26–28].

5. Heisenberg algebra for a free particle by modified (anti)-de Sitter models

In this section, we write the Heisenberg algebra for both anti-de Sitter and de Sitter models for a free particle.

We assume the following algebra in anti-de Sitter space(42)$\begin{eqnarray}\left[X,P\right]={\rm{i}}(1-{\alpha }^{2}{X}^{2}),\end{eqnarray}$and we have the following representation for above introduced commutation(43)$\begin{eqnarray}\hat{X}=x,\,\,\,\,\hat{P}=-{\rm{i}}(1-{\alpha }^{2}{X}^{2}){\partial }_{x}.\end{eqnarray}$On the other hand, we know the relation between commutator brackets in QM and Poisson brackets in CM is as follow(44)$\begin{eqnarray}\displaystyle \frac{1}{{\rm{i}}}[X,P]=\left\{X,P\right\}.\end{eqnarray}$That $\left\{X,P\right\}$ is Poisson brackets.

Hamiltonian for a free particle is $H=\tfrac{{P}^{2}}{2m}$. Then, we obtain time evolution equations for position and momentum operators as [29](45)$\begin{eqnarray}\dot{X}=\displaystyle \frac{1}{{\rm{i}}{\rm{\hslash }}}\left[X,H\right]=(1-{\alpha }^{2}{X}^{2})\displaystyle \frac{P}{m}.\end{eqnarray}$(46)$\begin{eqnarray}\dot{P}=\displaystyle \frac{1}{{\rm{i}}{\rm{\hslash }}}\left[P,H\right]=0\,\,\Rightarrow \,\,P=\mathrm{constant}.\end{eqnarray}$Then, by using equation (46) and substituting in equation (45) we obtain the following differential equation(47)$\begin{eqnarray}\int \displaystyle \frac{{\rm{d}}X}{(1-{\alpha }^{2}{X}^{2})}=\int \displaystyle \frac{P}{m}{\rm{d}}t,\end{eqnarray}$By solving this differential equation, we obtain(48)$\begin{eqnarray}X=\displaystyle \frac{1}{\alpha }\tanh \left(\displaystyle \frac{P}{m}\alpha t\right),\end{eqnarray}$which leads to(49)$\begin{eqnarray}X=\displaystyle \frac{P}{m}t\left(1-\displaystyle \frac{{\alpha }^{2}}{3}{\left(\displaystyle \frac{P}{m}t\right)}^{2}\right).\end{eqnarray}$Now, we solve (anti)-de Sitter model for a free particle by using Heisenberg algebra(50)$\begin{eqnarray}\left[X,P\right]={\rm{i}}(1+{\alpha }^{2}{X}^{2}),\end{eqnarray}$we obtain time evolution equations as(51)$\begin{eqnarray}\dot{X}=\displaystyle \frac{1}{{\rm{i}}}\left[X,H\right]=(1+{\alpha }^{2}{X}^{2})\displaystyle \frac{P}{m},\end{eqnarray}$(52)$\begin{eqnarray}\dot{P}=\displaystyle \frac{1}{{\rm{i}}}\left[P,H\right]=0,\,\,\Rightarrow \,\,P=\mathrm{constant},\end{eqnarray}$and substituting equation (52) into equation (51) leads to(53)$\begin{eqnarray}X=\displaystyle \frac{1}{\alpha }\tan \left(\displaystyle \frac{P}{m}\alpha t\right).\end{eqnarray}$For small values of α in tan we have(54)$\begin{eqnarray}X=\displaystyle \frac{P}{m}t\left(1+\displaystyle \frac{{\alpha }^{2}}{3}{\left(\displaystyle \frac{P}{m}t\right)}^{2}\right).\end{eqnarray}$In figure 1, we have plotted X versus t by using of equations (49) and (54).

Figure 1.

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Figure 1.Plot of the equation of motion under the EUP in the de Sitter space (black line) and the equation of motion under the EUP in the anti-de Sitter (dashed line) where we set P=m=1 and α=0.5.

Then, by considering following commutation(55)$\begin{eqnarray}\left[X,P\right]={\rm{i}}\sqrt{(1-{\alpha }^{2}{X}^{2})},\end{eqnarray}$From the Heisenberg algebra for a free particle, we write the time evolution equation as(56)$\begin{eqnarray}\dot{X}=\displaystyle \frac{1}{{\rm{i}}}\left[X,H\right]=\sqrt{(1-{\alpha }^{2}{X}^{2})}\displaystyle \frac{P}{m},\end{eqnarray}$(57)$\begin{eqnarray}\dot{P}=\displaystyle \frac{1}{{\rm{i}}}\left[P,H\right]=0,\,\,\Rightarrow \,\,P=\mathrm{constant},\end{eqnarray}$and by substituting equation (57) into equation (56) we obtain(58)$\begin{eqnarray}X=\displaystyle \frac{1}{\alpha }\sin \left(\displaystyle \frac{P}{m}\alpha t\right),\end{eqnarray}$Then, by considering small values α for sin(59)$\begin{eqnarray}X=\displaystyle \frac{P}{m}t\left(1-\displaystyle \frac{{\alpha }^{2}}{3!}{\left(\displaystyle \frac{P}{m}t\right)}^{2}\right).\end{eqnarray}$Also, in de Sitter model for a free particle(60)$\begin{eqnarray}\left[X,P\right]={\rm{i}}\sqrt{(1+{\alpha }^{2}{X}^{2})}.\end{eqnarray}$From the Heisenberg algebra, we write the time evolution equations as(61)$\begin{eqnarray}\dot{X}=\displaystyle \frac{1}{{\rm{i}}}\left[X,H\right]=\sqrt{(1+{\alpha }^{2}{X}^{2})}\displaystyle \frac{P}{m},\end{eqnarray}$(62)$\begin{eqnarray}\dot{P}=\displaystyle \frac{1}{{\rm{i}}}\left[P,H\right]=0,\,\,\Rightarrow \,\,P=\mathrm{constant}.\end{eqnarray}$Then, by substituting equation (62) into equation (61) we obtain(63)$\begin{eqnarray}X=\displaystyle \frac{1}{\alpha }\sinh \left(\displaystyle \frac{P}{m}\alpha t\right),\end{eqnarray}$and for small values of α for sinh, we have(64)$\begin{eqnarray}X=\displaystyle \frac{P}{m}t\left(1+\displaystyle \frac{{\alpha }^{2}}{3!}{\left(\displaystyle \frac{P}{m}t\right)}^{2}\right).\end{eqnarray}$Then, we know in equations (49), (54), (59) and (64) if α tends to zero we obtain the standard equation.

6. Heisenberg algebra for Harmonic oscillator by modified (anti)-de Sitter models

In this section, we would like to obtain Heisenberg algebra by considering a harmonic oscillator potential [30, 31].

Therefore, we consider the following modified (anti)-de Sitter models(65)$\begin{eqnarray}\left[X,P\right]={\rm{i}}(1\pm {\alpha }^{2}{X}^{2}).\end{eqnarray}$For this commutation we have bellow representation(66)$\begin{eqnarray}\hat{P}=-{\rm{i}}(1\pm {\alpha }^{2}{X}^{2}){\partial }_{X},\,\,\,\,\,\,\hat{X}=x.\end{eqnarray}$Considering the harmonic oscillator potential, Hamiltonian takes the following form(67)$\begin{eqnarray}H=\displaystyle \frac{{P}^{2}}{2m}+\displaystyle \frac{1}{2}m{\omega }_{0}^{2}{X}^{2}.\end{eqnarray}$We write the Heisenberg equations of motion for X and P by using this representation and Hamiltonian(68)$\begin{eqnarray}\dot{X}=\displaystyle \frac{1}{{\rm{i}}}\left[X,H\right]=(1\pm {\alpha }^{2}{X}^{2})\displaystyle \frac{P}{m}.\end{eqnarray}$(69)$\begin{eqnarray}\dot{P}=\displaystyle \frac{1}{{\rm{i}}}\left[P,H\right]=-m{\omega }_{0}^{2}X(1\pm {\alpha }^{2}{X}^{2}).\end{eqnarray}$Now, we again use the Heisenberg equations of motion and Hamiltonian in order to obtain $\ddot{X}$(70)$\begin{eqnarray}\ddot{X}=\displaystyle \frac{1}{{\rm{i}}}\left[\dot{X},H\right]=-{\omega }_{0}^{2}X{\left(1\pm {\alpha }^{2}{X}^{2}\right)}^{2},\end{eqnarray}$in case α=0, equation (70) leads to the ordinary relation(71)$\begin{eqnarray}\ddot{X}+{\omega }_{0}^{2}X=0.\end{eqnarray}$For nonzero values of α, we have(72)$\begin{eqnarray}\ddot{X}+{\omega }_{0}^{2}X\mp 2{\alpha }^{2}{\omega }_{0}^{2}{X}^{3}=0.\end{eqnarray}$For solving this equation using the successive approximation method, we consider X as a function of the frequency ω as [32](73)$\begin{eqnarray}X={X}_{0}\cos (\omega t).\end{eqnarray}$Substituting this term in equation (72)(74)$\begin{eqnarray}-{X}_{0}\cos (\omega t){\omega }^{2}+{X}_{0}\cos (\omega t){\omega }_{0}^{2}\mp 2{\alpha }^{2}{X}_{0}^{3}{\cos }^{3}(\omega t){\omega }_{0}^{2}=0.\end{eqnarray}$On the other hand, we have used expansion for the third term and considering it equals to zero(75)$\begin{eqnarray}\begin{array}{l}(-{\omega }^{2}+{\omega }_{0}^{2}\mp \displaystyle \frac{3}{2}{\alpha }^{2}{X}_{0}^{2}{\omega }_{0}^{2}){X}_{0}\cos (\omega t)\\ \quad \mp \,\displaystyle \frac{1}{2}{\alpha }^{2}{X}_{0}^{3}{\omega }_{0}^{2}\cos (3\omega t)=0,\end{array}\end{eqnarray}$where(76)$\begin{eqnarray}\omega ={\omega }_{0}{\left(1\mp \displaystyle \frac{3}{2}{\alpha }^{2}{X}_{0}^{2}\right)}^{\tfrac{1}{2}}.\end{eqnarray}$We see that deformed frequency in α=0 leads to the ordinary relation and is equal to ordinary frequency ${\omega }^{2}={\omega }_{0}^{2}$.

Now, in order to calculate the deformed frequency, we consider higher order X in equation (74) and we obtain the high order deformed frequency for EUP. Therefore, we add ${X}_{1}\cos (3\omega t)$ term to equation (73) and by substituting it in equation (72) we have(77)$\begin{eqnarray}\begin{array}{l}-9{X}_{1}\cos (3\omega t){\omega }^{2}-{X}_{0}\cos (\omega t){\omega }^{2}+{X}_{1}\cos (3\omega t){\omega }_{0}^{2}\\ \quad +\,{X}_{0}\cos (\omega t){\omega }_{0}^{2}\mp 2{\alpha }^{2}{X}_{0}^{3}{\cos }^{3}(\omega t){\omega }_{0}^{2}\\ \ \mp \ 2\beta {X}_{1}^{3}{\cos }^{3}(3\omega t){\omega }_{0}^{2}\mp 6\beta {X}_{0}^{2}{X}_{1}{\cos }^{2}(\omega t)\cos (3\omega t){\omega }_{0}^{2}\\ \quad \mp \,6\beta {X}_{0}{X}_{1}^{2}{\cos }^{2}(3\omega t)\cos (\omega t){\omega }_{0}^{2}=0.\end{array}\end{eqnarray}$We neglect the sixth, seventh and eighth terms We regulate the terms versus coefficient ${X}_{0}\cos (\omega t)$ and $\cos (3\omega t)$ . Then, setting the first quantity in parentheses equal to zero gives the previous value for ω in equation (77)(78)$\begin{eqnarray}\begin{array}{l}\left(-{\omega }^{2}+{\omega }_{0}^{2}\mp \displaystyle \frac{3}{2}{\alpha }^{2}{X}_{0}^{2}{\omega }_{0}^{2}\right){X}_{0}\cos (\omega t)\\ \quad +\,\left(-9{X}_{1}{\omega }^{2}+{X}_{1}{\omega }_{0}^{2}\mp \displaystyle \frac{1}{2}{\alpha }^{2}{X}_{0}^{3}{\omega }_{0}^{2}\right)\cos (3\omega t)=0,\end{array}\end{eqnarray}$where(79)$\begin{eqnarray}-9{X}_{1}{\omega }^{2}+{X}_{1}{\omega }_{0}^{2}\mp \displaystyle \frac{1}{2}{\alpha }^{2}{X}_{0}^{3}{\omega }_{0}^{2}=0,\,\,\,\,\quad {X}_{1}=\mp \displaystyle \frac{{\alpha }^{2}{X}_{0}^{3}}{16}.\end{eqnarray}$Considering equation (79) we obtain(80)$\begin{eqnarray}X(t)={X}_{0}\cos (\omega t)\mp \displaystyle \frac{{\alpha }^{2}{X}_{0}^{3}}{16}\cos (3\omega t).\end{eqnarray}$Equation (80) can be used to obtain the dynamical properties of the system.

In addition, we can use this method for solving the another commutation relation as(81)$\begin{eqnarray}\left[X,P\right]={\rm{i}}\sqrt{(1\pm {\alpha }^{2}{X}^{2})}.\end{eqnarray}$Thus, we obtain(82)$\begin{eqnarray}\dot{X}=\displaystyle \frac{1}{{\rm{i}}}\left[X,H\right]=\sqrt{(1\pm {\alpha }^{2}{X}^{2})}\displaystyle \frac{P}{m},\end{eqnarray}$(83)$\begin{eqnarray}\dot{P}=\displaystyle \frac{1}{{\rm{i}}}\left[P,H\right]=-m{\omega }_{0}^{2}X\sqrt{(1\pm {\alpha }^{2}{X}^{2})}.\end{eqnarray}$Then, we obtain the modified equation(84)$\begin{eqnarray}\ddot{X}+{\omega }_{0}^{2}X\mp {\alpha }^{2}{\omega }_{0}^{2}{X}^{3}=0.\end{eqnarray}$From the successive approximation method we have(85)$\begin{eqnarray}\omega ={\omega }_{0}{\left(1\mp \displaystyle \frac{3}{4}{\alpha }^{2}{X}_{0}^{2}\right)}^{\tfrac{1}{2}}.\end{eqnarray}$And ${X}_{1}=\mp \tfrac{{\alpha }^{2}{X}_{0}^{3}}{32}$. Then, we obtain the dynamical properties of the system as(86)$\begin{eqnarray}X(t)={X}_{0}\cos (\omega t)\mp \displaystyle \frac{{\alpha }^{2}{X}_{0}^{3}}{32}\cos (3\omega t).\end{eqnarray}$Also, we use this method for solving the following commutation(87)$\begin{eqnarray}\left[X,P\right]={\rm{i}}{\left(1\pm {\alpha }^{2}{X}^{2}\right)}^{\tfrac{3}{2}},\end{eqnarray}$

(88)$\begin{eqnarray}\dot{X}=\displaystyle \frac{1}{{\rm{i}}}\left[X,H\right]={\left(1\pm {\alpha }^{2}{X}^{2}\right)}^{\tfrac{3}{2}}\displaystyle \frac{P}{m},\end{eqnarray}$

(89)$\begin{eqnarray}\dot{P}=\displaystyle \frac{1}{{\rm{i}}}\left[P,H\right]=-m{\omega }_{0}^{2}X{\left(1\pm {\alpha }^{2}{X}^{2}\right)}^{\tfrac{3}{2}}.\end{eqnarray}$Then, we obtain the modified equation(90)$\begin{eqnarray}\ddot{X}+{\omega }_{0}^{2}X\mp 3{\alpha }^{2}{\omega }_{0}^{2}{X}^{3}=0.\end{eqnarray}$From the successive approximation method, we have(91)$\begin{eqnarray}\omega ={\omega }_{0}{\left(1\mp \displaystyle \frac{9}{4}{\alpha }^{2}{X}_{0}^{2}\right)}^{\tfrac{1}{2}},\end{eqnarray}$And ${X}_{1}=\mp \tfrac{3{\alpha }^{2}{X}_{0}^{3}}{32}$. Then, we obtain the dynamical properties of the system as(92)$\begin{eqnarray}X(t)={X}_{0}\cos (\omega t)\mp \displaystyle \frac{3{\alpha }^{2}{X}_{0}^{3}}{32}\cos (3\omega t).\end{eqnarray}$

7. Particle in a uniform gravitational field for the GUP in (anti)-Snyder space

Heisenberg uncertainty principle modifies to GUP, by several investigations in string theory and quantum gravity (see, e.g. [8, 9, 33]) GUP as(93)$\begin{eqnarray}\left[X,P\right]={\rm{i}}{\rm{\hslash }}(1+\beta {P}^{2}).\end{eqnarray}$Where, X and P are position operator and momentum operator. Also, we define generalization parameter as $\beta ={\beta }_{0}\tfrac{{M}_{{\rm{Pl}}}}{{c}^{2}}$. Where β is of order the Planck mass and ${\beta }_{0}$ is of order the unity. Considering GUP and deformed momentum representation and we suggest the existence of the fundamental minimal length ${\rm{\Delta }}{X}_{\min }={\rm{\hslash }}\sqrt{\beta }$, which is of the order of the Planck’s length ${l}_{p}=\sqrt{{\rm{\hslash }}G/{c}^{3}}\simeq 1.6\times {10}^{-35}{\rm{m}}$. The first algebra of this kind in the relativistic case was proposed by Snyder in 1947 [34]. Also, recently Mignemi [35] proposed the new model that is called the anti Snyder model by replacing β with −β as(94)$\begin{eqnarray}\left[X,P\right]={\rm{i}}{\rm{\hslash }}(1-\beta {P}^{2}).\end{eqnarray}$The (anti)-Snyder models can be generalized into the following forms [21](95)$\begin{eqnarray}\left[X,P\right]={\rm{i}}{\rm{\hslash }}\sqrt{(1\pm \beta {P}^{2})},\end{eqnarray}$(96)$\begin{eqnarray}\left[X,P\right]={\rm{i}}{\rm{\hslash }}{\left(1\pm \beta {P}^{2}\right)}^{\displaystyle \frac{3}{2}}.\end{eqnarray}$The Hamiltonian equations of motion in space, in the uniform gravitation field are as follow(97)$\begin{eqnarray}H=\displaystyle \frac{{P}^{2}}{2m}-{mgX},\end{eqnarray}$For anti-Snyder model, Considering Hamiltonian (97), we have the following time evolution relations as [29](98)$\begin{eqnarray}\dot{X}=\{X,H\}=\displaystyle \frac{P}{m}\sqrt{(1-\beta {P}^{2})},\end{eqnarray}$(99)$\begin{eqnarray}\dot{P}=\{P,H\}={mg}\sqrt{(1-\beta {P}^{2})}.\end{eqnarray}$As a result, we obtain the momentum equation as(100)$\begin{eqnarray}P=\displaystyle \frac{1}{\sqrt{\beta }}\sin (\sqrt{\beta }{mgt}).\end{eqnarray}$From the first equation, we obtain velocity as(101)$\begin{eqnarray}\dot{X}=\displaystyle \frac{1}{m\sqrt{\beta }}\sin (\sqrt{\beta }{mgt})\sqrt{1-{\sin }^{2}(\sqrt{\beta }{mgt})}.\end{eqnarray}$As a result, by expanding these equations for small values of $\sqrt{\beta }$, we are able to obtain velocity and coordinate for anti-Snyder model as(102)$\begin{eqnarray}\dot{X}={gt}\left(1-\displaystyle \frac{2}{3}\beta {m}^{2}{g}^{2}{t}^{2}\right),\end{eqnarray}$(103)$\begin{eqnarray}X=\displaystyle \frac{{{gt}}^{2}}{2}\left(1-\displaystyle \frac{1}{3}\beta {m}^{2}{g}^{2}{t}^{2}\right).\end{eqnarray}$When β tends to zero, we will have the well known results(104)$\begin{eqnarray}\dot{X}={gt},\ X=\displaystyle \frac{{{gt}}^{2}}{2}.\end{eqnarray}$Considering Hamiltonian equation (97), we obtain the following equations for Snyder model(105)$\begin{eqnarray}\dot{X}=\{X,H\}=\displaystyle \frac{P}{m}\sqrt{(1+\beta {P}^{2})},\end{eqnarray}$(106)$\begin{eqnarray}\dot{P}=\{P,H\}={mg}\sqrt{(1+\beta {P}^{2})}.\end{eqnarray}$Then, momentum can be obtained as(107)$\begin{eqnarray}P=\displaystyle \frac{1}{\sqrt{\beta }}\sinh (\sqrt{\beta }{mgt}).\end{eqnarray}$By substituting the previous equation in $\dot{X}$, we obtain velocity as(108)$\begin{eqnarray}\dot{X}=\displaystyle \frac{1}{m\sqrt{\beta }}\sinh (\sqrt{\beta }{mgt})\sqrt{(1+{\sinh }^{2}(\sqrt{\beta }{mgt})}.\end{eqnarray}$With using from expansion and Then for small values of $\sqrt{\beta }$ for sinh and the sinh cubic terms, we obtain velocity and coordinate for the Snyder model as(109)$\begin{eqnarray}\dot{X}={gt}\left(1+\displaystyle \frac{2}{3}\beta {m}^{2}{g}^{2}{t}^{2}\right),\end{eqnarray}$(110)$\begin{eqnarray}X=\displaystyle \frac{{{gt}}^{2}}{2}\left(1+\displaystyle \frac{1}{3}\beta {m}^{2}{g}^{2}{t}^{2}\right).\end{eqnarray}$Therefore, we are able to conclude time evolution equations for momentum and position for anti-Snyder model as(111)$\begin{eqnarray}\dot{X}=\{X,H\}=\displaystyle \frac{P}{m}(1-\beta {P}^{2}),\end{eqnarray}$(112)$\begin{eqnarray}\dot{P}=\{P,H\}={mg}(1-\beta {P}^{2}).\end{eqnarray}$Then, from the second equation(113)$\begin{eqnarray}P=\displaystyle \frac{1}{\sqrt{\beta }}\tanh (\sqrt{\beta }{mgt}).\end{eqnarray}$By substituting P in first equation we have(114)$\begin{eqnarray}\dot{X}=\displaystyle \frac{1}{m\sqrt{\beta }}\tanh (\sqrt{\beta }{mgt})\sqrt{(1-{\tanh }^{2}(\sqrt{\beta }{mgt})}.\end{eqnarray}$Therefore, for small values of $\sqrt{\beta }$, we can obtain these results for velocity and coordinate(115)$\begin{eqnarray}\dot{X}={gt}\left(1-\displaystyle \frac{4}{3}\beta {m}^{2}{g}^{2}{t}^{2}\right),\end{eqnarray}$(116)$\begin{eqnarray}X=\displaystyle \frac{{{gt}}^{2}}{2}\left(1-\displaystyle \frac{2}{3}\beta {m}^{2}{g}^{2}{t}^{2}\right).\end{eqnarray}$We obtained the modified velocities and the modified coordinates in anti-Snyder and Snyder models which depends to β parameter and in limited state changes to ordinary form.

7.1. Kinetic energy in the (anti)-Snyder space and parameter of deformation

In this section, we assume each particle of the system moves by the same velocity [29]. Let us rewrite the kinetic energy as a function of velocity. From the relation between velocity and momentum for anti-Snyder model in equation (98) we have(117)$\begin{eqnarray}P=m\dot{X}\left(1+\displaystyle \frac{1}{2}\beta {m}^{2}{\dot{X}}^{2}\right).\end{eqnarray}$Then the kinetic energy as a function of velocity in the first order approximation of β becomes(118)$\begin{eqnarray}T=\displaystyle \frac{m{\dot{X}}^{2}}{2}+\displaystyle \frac{\beta }{2}{m}^{3}{\dot{X}}^{4}.\end{eqnarray}$We see that the modified kinetic energy depends on β parameter. Then, we repeat this solution for the equation (105) which is GUP commutator for the Snyder space.(119)$\begin{eqnarray}P=m\dot{X}\left(1-\displaystyle \frac{1}{2}\beta {m}^{2}{\dot{X}}^{2}\right).\end{eqnarray}$Then, the kinetic energy as a function of velocity in the first order approximation over β becomes(120)$\begin{eqnarray}T=\displaystyle \frac{m{\dot{X}}^{2}}{2}-\displaystyle \frac{\beta }{2}{m}^{3}{\dot{X}}^{4}.\end{eqnarray}$The above equation is the modified kinetic energy for the Snyder space and depends on deformation parameter which in the special case has the ordinary form.

Also, from equation (111) we have(121)$\begin{eqnarray}P=m\dot{X}(1+\beta {m}^{2}{\dot{X}}^{2}).\end{eqnarray}$Then, the kinetic energy as a function of velocity for the Snyder space in the first order approximation for β reads(122)$\begin{eqnarray}T=\displaystyle \frac{m{\dot{X}}^{2}}{2}+\beta {m}^{3}{\dot{X}}^{4}.\end{eqnarray}$

7.2. Heisenberg algebra for Harmonic oscillator by modified (anti)-Snyder models

In this section, we consider the following GUP in the Snyder and the anti-Snyder models as(123)$\begin{eqnarray}\left[X,P\right]={\rm{i}}{\rm{\hslash }}\sqrt{(1\pm \beta {P}^{2})}.\end{eqnarray}$Considering GUP and deformed momentum representation as [10](124)$\begin{eqnarray}\hat{X}=x,\end{eqnarray}$(125)$\begin{eqnarray}\hat{P}={\rm{i}}{\rm{\hslash }}\sqrt{(1\pm \beta {p}^{2})}\displaystyle \frac{\partial }{\partial x}.\end{eqnarray}$We consider the Hamiltonian equation (67) for a harmonic oscillator and according to KMM algebra [12], we obtain the Heisenberg equations of motion for X and P, by using this representation and Hamiltonian [15](126)$\begin{eqnarray}\dot{X}=\displaystyle \frac{1}{{\rm{i}}{\rm{\hslash }}}\left[X,H\right]=\sqrt{(1\pm \beta {P}^{2})}\displaystyle \frac{P}{m},\end{eqnarray}$(127)$\begin{eqnarray}\dot{P}=\displaystyle \frac{1}{{\rm{i}}{\rm{\hslash }}}\left[P,H\right]=-m{\omega }_{0}^{2}X\sqrt{(1\pm \beta {P}^{2})}.\end{eqnarray}$Now, applying the solution method in section 7, we will have(128)$\begin{eqnarray}\ddot{P}=\displaystyle \frac{1}{{\rm{i}}{\rm{\hslash }}}\left[\dot{P},H\right]=-{\omega }_{0}^{2}P(1\pm \beta {P}^{2}),\end{eqnarray}$or(129)$\begin{eqnarray}\ddot{P}+{\omega }_{0}^{2}P\mp \beta {\omega }_{0}^{2}{P}^{3}=0.\end{eqnarray}$To solve this equation, we consider P depends on ω frequency as follows(130)$\begin{eqnarray}P={P}_{0}\cos (\omega t).\end{eqnarray}$And substituting this term in the equation (143)(131)$\begin{eqnarray}-{P}_{0}\cos (\omega t){\omega }^{2}+{P}_{0}\cos (\omega t){\omega }_{0}^{2}\mp \beta {P}_{0}^{3}{\cos }^{3}(\omega t){\omega }_{0}^{2}=0,\end{eqnarray}$On the other hand, using the expansion for ${\cos }^{3}(\omega t)$ term, we obtain the deformed frequency [30, 32] as(132)$\begin{eqnarray}\begin{array}{l}(-{\omega }^{2}+{\omega }_{0}^{2}\mp \displaystyle \frac{3}{4}\beta {P}_{0}^{2}{\omega }_{0}^{2}){P}_{0}\cos (\omega t)\\ \quad \mp \,\displaystyle \frac{1}{4}\beta {P}_{0}^{3}\cos (3\omega t){\omega }_{0}^{2}=0,\end{array}\end{eqnarray}$where(133)$\begin{eqnarray}\omega ={\omega }_{0}{\left(1\mp \displaystyle \frac{3}{4}\beta {P}_{0}^{2}\right)}^{\tfrac{1}{2}}.\end{eqnarray}$We see that deformed frequency for the case that β tends to zero leads to ordinary relation and equals to the ordinary frequency ${\omega }^{2}={\omega }_{0}^{2}$.

Now, we calculate the deformed frequency by considering higher order P in equation (129) and we obtain the high order deformed frequency for the GUP. Therefore, we add ${P}_{1}\cos (3\omega t)$ term to equation (130) and we obtain(134)$\begin{eqnarray}\begin{array}{l}-9{P}_{1}\cos (3\omega t){\omega }^{2}-{P}_{0}\cos (\omega t){\omega }^{2}+{P}_{1}\cos (3\omega t){\omega }_{0}^{2}\\ \quad +\,{P}_{0}\cos (\omega t){\omega }_{0}^{2}+2\beta {P}_{0}^{3}{\cos }^{3}(\omega t){\omega }_{0}^{2}\mp \\ \mp \,\beta {P}_{1}^{3}{\cos }^{3}(3\omega t){\omega }_{0}^{2}\mp 3\beta {P}_{0}^{2}{P}_{1}{\cos }^{2}(\omega t)\cos (3\omega t){\omega }_{0}^{2}\\ \quad \mp \,3\beta {P}_{0}{P}_{1}^{2}{\cos }^{2}(3\omega t)\cos (\omega t){\omega }_{0}^{2}=0.\end{array}\end{eqnarray}$We neglect the sixth, seventh and eighth terms Then, we regulate the terms versus ${P}_{0}\cos (\omega t)$ coefficient and $\cos (3\omega t)$. Then, considering the first quantity in parentheses equals zero give us the previous value for ω in equation (133)(135)$\begin{eqnarray}\begin{array}{l}(-{\omega }^{2}+{\omega }_{0}^{2}\mp \displaystyle \frac{3}{4}\beta {P}_{0}^{2}{\omega }_{0}^{2}){P}_{0}\cos (\omega t)\\ \quad +\,(-9{P}_{1}{\omega }^{2}+{P}_{1}{\omega }_{0}^{2}\mp \displaystyle \frac{1}{4}\beta {P}_{0}^{3}{\omega }_{0}^{2})\cos (3\omega t)=0,\end{array}\end{eqnarray}$where(136)$\begin{eqnarray}-9{P}_{1}{\omega }^{2}+{P}_{1}{\omega }_{0}^{2}\mp \displaystyle \frac{1}{4}\beta {P}_{0}^{3}{\omega }_{0}^{2}=0,\end{eqnarray}$

(137)$\begin{eqnarray}{P}_{1}=\mp \displaystyle \frac{\beta {P}_{0}^{3}}{32},\end{eqnarray}$

(138)$\begin{eqnarray}P(t)={P}_{0}\cos (\omega t)\mp \displaystyle \frac{\beta {P}_{0}^{3}}{32}\cos (3\omega t).\end{eqnarray}$Substituting equation (138) into equation (126) we obtain(139)$\begin{eqnarray}\begin{array}{rcl}X(t) & = & \left(\displaystyle \frac{{P}_{0}}{m\omega }\sin (\omega t)+\displaystyle \frac{3\beta {P}_{0}^{3}}{4m\omega }\sin (\omega t)\right.\\ & & \left.\mp \displaystyle \frac{3\beta {P}_{0}^{3}}{32m\omega }\sin (3\omega t\right)+O({\beta }^{2}).\end{array}\end{eqnarray}$Above equation can be used to obtain the dynamical properties of the system. In figure 2, we have plotted X versus t by using of equations (80) and (139).

Figure 2.

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Figure 2.Plot of the EUP oscillator in the de Sitter space (black line) and GUP oscillator in the Snyder space (dashed line) where we set X₀=P₀=m=ω=1 and α=β=0.5.

Then, we repeat this solution for the another commutator in the Snyder and anti-Snyder space. we will be able to obtain time evolution for position and momentum as(140)$\begin{eqnarray}\dot{X}=\displaystyle \frac{1}{{\rm{i}}}\left[X,H\right]={\left(1\pm \beta {P}^{2}\right)}^{\tfrac{3}{2}}\displaystyle \frac{P}{m},\end{eqnarray}$(141)$\begin{eqnarray}\dot{P}=\displaystyle \frac{1}{{\rm{i}}}\left[P,H\right]=-m{\omega }_{0}^{2}X{\left(1\pm \beta {P}^{2}\right)}^{\tfrac{3}{2}}.\end{eqnarray}$Then, we obtain the modified equation(142)$\begin{eqnarray}\ddot{P}+{\omega }_{0}^{2}P\mp 3\beta {\omega }_{0}^{2}{P}^{3}=0.\end{eqnarray}$From the successive approximation method we have [32](143)$\begin{eqnarray}\omega ={\omega }_{0}{\left(1\mp \displaystyle \frac{9}{4}{\alpha }^{2}{X}_{0}^{2}\right)}^{\tfrac{1}{2}}.\end{eqnarray}$And we obtain ${P}_{1}=\mp \tfrac{3\beta {P}_{0}^{3}}{32}$. Then, the dynamical properties of the system becomes(144)$\begin{eqnarray}P(t)={P}_{0}\cos (\omega t)\mp \displaystyle \frac{3\beta {P}_{0}^{3}}{32}\cos (3\omega t).\end{eqnarray}$(145)$\begin{eqnarray}\begin{array}{rcl}X(t) & = & \left(\displaystyle \frac{{P}_{0}}{m\omega }\sin (\omega t)+\displaystyle \frac{9\beta {P}_{0}^{3}}{4m\omega }\sin (\omega t)\right.\\ & & \left.\mp \displaystyle \frac{9\beta {P}_{0}^{3}}{32m\omega }\sin (3\omega t)\right)+O({\beta }^{2}).\end{array}\end{eqnarray}$In fact, we obtained dynamical properties for Gup in the Snyder and anti-Snyder backgrounds and for EUP in the anti-de Sitter and de Sitter space that give us important results in the modified EUP and GUP field.

8. Conclusion

We have shown that there is another modified EUP in anti-de Sitter and de Sitter space–time, which we have written the representation in the position and the momentum space for. Also, we show that Heisenberg algebra for a free particle gives us a modified position equation that in special case ordinary results have been recovered. We have shown the Heisenberg equation of motion for EUP for Harmonic oscillator potential by solving through the successive approximation method, gives us the well-known results in the case of $\alpha =0$. In the non-zero state, we obtained the modified frequency for EUP and with successive approximation method we have obtained an equation in which the dynamical properties of the system are included. Also we introduced the modified anti-Snyder and snyder space for GUP and we have represented the Heisenberg algebra in the gravitational field given a modified kinetic energy for each of the commutations introduced. We are able to interpret this energy and find that it is a deformed form of ordinary energy and in the special case (β tends to zero) changes to previous form. Also, we have obtained modified coordinates and momentum for GUP in the Snyder and anti-Snyder background with harmonic oscillator potential. As we know, if the deformation parameter equals zero, we obtain the basic forms.