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.

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 $3{\lambda }^2=\left|\mathrm{\Lambda }\right|={10}^{-52}{\mathrm{m}}^{-2}$ , ${\lambda }^2=\frac{1}{L_H^2}$ , with L_H the (anti)-de Sitter radius and $\frac{1}{L_H^2}={\alpha }^2$ [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]Also, in the anti-de Sitter space for EUP we have [19, 20]In this representation for EUP in the de Sitter and anti-de Sitter we haveThis representation leads toThe relation between the EUP position and the canonical position in anti-de Sitter model is obtained asThen, we have the modified position and its ordinary form asin which X and x represent the modified position and its ordinary form, respectively. We can find the limits from above equations asThe EUP in the de Sitter background can be generalized into the following form [21]where a = 1 gives the ordinary EUP model. The relation between the EUP position and canonical position isThen, we can write the extended following relations for EUP Now, we obtain another different representation for EUPwhereFrom the self-adjointness of the operator P we have the following inner productAnd the expectation value of the operator $\hat{A}$ where φ(X) obeyand the normalization conditionfirst we will incorporate thatwhich implies that $⟨\sqrt{1-{\alpha }^2{X}^2}⟩$ cannot be zero for all normalized wave functions φ(X). Let us assume that L = 0. Then we haveBecause $|\varphi (X){|}^2⩾0$, we have φ(X) = 0 for all $X\in [-\frac{1}{\alpha },\frac{1}{\alpha }]$. Then, we haveThat contradicts equation (18). Therefore $L\ne 0$ and 0 < L < 1. Because $X^2⩽\frac{1}{{\beta }^2}$ for all $X\in [-\frac{1}{\alpha },\frac{1}{\alpha }]$ we getThus, we haveTherefore, it has been shown that the modified (anti)-de Sitter model (13) has the non-zero minimal momentum.

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

The EUP position representation for algebra (13) is as [22]The EUP position representation acts on the square integrable functionsand the norm of ψ(X) is given byThe Schrödinger equation reads

The momentum representation for the EUP (13)The momentum representation acts on the square integrable functionsand the norm of φ is given byThe Schrödinger equation reads

The canonical position representation for the (13) algebra is obtained from the momentum representation with replacing $\hat{P}=\hat{p}=-\mathrm{i}\frac{\partial }{\partial x}$ , asThe canonical position representation acts on the square integrable functionsand the norm of ψ is given bythe Schrödinger equation reads

The momentum operators generating EUP position-space eigenstates are given byorThis differential equation can be solved to obtain formal momentum eigenvectorsfor the case of $|X|<\frac{1}{\alpha }$ the equation (36) reduces to ${C\mathrm{e}}^{-pX}$, that is the plane wave. The wave function ψ_p(X) obeys the normalization conditionorwhich means $C=\frac{1}{\sqrt{2\pi }}$. Then, the normalized plane wave in the modified anti-de Sitter model isAlso, 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].

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 spaceand we have the following representation for above introduced commutationOn the other hand, we know the relation between commutator brackets in QM and Poisson brackets in CM is as followThat $\{X,P\}$ is Poisson brackets.

Hamiltonian for a free particle is $H=\frac{P^2}{2m}$. Then, we obtain time evolution equations for position and momentum operators as [29] Then, by using equation (46) and substituting in equation (45) we obtain the following differential equationBy solving this differential equation, we obtainwhich leads toNow, we solve (anti)-de Sitter model for a free particle by using Heisenberg algebrawe obtain time evolution equations as and substituting equation (52) into equation (51) leads toFor small values of α in tan we haveIn figure 1, we have plotted X versus t by using of equations (49) and (54).

Then, by considering following commutationFrom the Heisenberg algebra for a free particle, we write the time evolution equation as and by substituting equation (57) into equation (56) we obtainThen, by considering small values α for sinAlso, in de Sitter model for a free particleFrom the Heisenberg algebra, we write the time evolution equations as Then, by substituting equation (62) into equation (61) we obtainand for small values of α for sinh, we haveThen, we know in equations (49), (54), (59) and (64) if α tends to zero we obtain the standard equation.

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 modelsFor this commutation we have bellow representationConsidering the harmonic oscillator potential, Hamiltonian takes the following formWe write the Heisenberg equations of motion for X and P by using this representation and Hamiltonian Now, we again use the Heisenberg equations of motion and Hamiltonian in order to obtain $\ddot{X}$ in case α = 0, equation (70) leads to the ordinary relationFor nonzero values of α, we haveFor solving this equation using the successive approximation method, we consider X as a function of the frequency ω as [32]Substituting this term in equation (72)On the other hand, we have used expansion for the third term and considering it equals to zerowhereWe 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 haveWe 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)whereConsidering equation (79) we obtainEquation (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 asThus, we obtain Then, we obtain the modified equationFrom the successive approximation method we haveAnd $X_1=\mp \frac{{\alpha }^2{X}_0^3}{32}$. Then, we obtain the dynamical properties of the system asAlso, we use this method for solving the following commutation Then, we obtain the modified equationFrom the successive approximation method, we haveAnd $X_1=\mp \frac{3{\alpha }^2{X}_0^3}{32}$. Then, we obtain the dynamical properties of the system as

Heisenberg uncertainty principle modifies to GUP, by several investigations in string theory and quantum gravity (see, e.g. [8, 9, 33]) GUP asWhere, X and P are position operator and momentum operator. Also, we define generalization parameter as $\beta ={\beta }_0\frac{M_{\mathrm{P}\mathrm{l}}}{c^2}$. 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 $\mathrm{\Delta }{X}_{min}=\hslash \sqrt{\beta }$, which is of the order of the Planck’s length $l_p=\sqrt{\hslash G/c^3}\simeq 1.6\times {10}^{-35}\mathrm{m}$. 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 asThe (anti)-Snyder models can be generalized into the following forms [21] The Hamiltonian equations of motion in space, in the uniform gravitation field are as followFor anti-Snyder model, Considering Hamiltonian (97), we have the following time evolution relations as [29] As a result, we obtain the momentum equation asFrom the first equation, we obtain velocity asAs 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 When β tends to zero, we will have the well known resultsConsidering Hamiltonian equation (97), we obtain the following equations for Snyder model Then, momentum can be obtained asBy substituting the previous equation in $\dot{X}$, we obtain velocity asWith 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 Therefore, we are able to conclude time evolution equations for momentum and position for anti-Snyder model as Then, from the second equationBy substituting P in first equation we haveTherefore, for small values of $\sqrt{\beta }$, we can obtain these results for velocity and coordinate 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.

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 haveThen the kinetic energy as a function of velocity in the first order approximation of β becomesWe 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.Then, the kinetic energy as a function of velocity in the first order approximation over β becomesThe 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 haveThen, the kinetic energy as a function of velocity for the Snyder space in the first order approximation for β reads

In this section, we consider the following GUP in the Snyder and the anti-Snyder models asConsidering GUP and deformed momentum representation as [10] 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] Now, applying the solution method in section 7, we will haveorTo solve this equation, we consider P depends on ω frequency as followsAnd substituting this term in the equation (143)On the other hand, using the expansion for ${\cos }^3(\omega t)$ term, we obtain the deformed frequency [30, 32] aswhereWe 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 obtainWe 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)where Substituting equation (138) into equation (126) we obtainAbove 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).