NUPHB116771116771S0550-3213(24)00337-710.1016/j.nuclphysb.2024.116771The Author(s)High Energy Physics – TheoryHigh Energy Physics – TheoryGeneralized Extended Uncertainty Principles, Liouville theorem and density of states: Snyder-de Sitter and Yang modelsAnnaPachołanna.pachol@usn.noDepartment of Microsystems, University of South-Eastern Norway, Campus Vestfold, NorwayDepartment of MicrosystemsUniversity of South-Eastern NorwayCampus VestfoldNorwayDepartment of Microsystems, University of South-Eastern Norway, Campus Vestfold, NorwayEditor: Stephan StiebergerAbstractModifications in quantum mechanical phase space lead to changes in the Heisenberg uncertainty principle, which can result in the Generalized Uncertainty Principle (GUP) or the Extended Uncertainty Principle (EUP), introducing quantum gravitational effects at small and large distances, respectively. A combination of GUP and EUP, the Generalized Extended Uncertainty Principle (GEUP or EGUP), further generalizes these modifications by incorporating noncommutativity in both coordinates and momenta. This paper examines the impact of GEUP on the analogue of the Liouville theorem in statistical physics and density of states within the classical limit of non-relativistic quantum mechanics framework. We find a weighted phase space volume element, invariant under the infinitesimal time evolution, in the cases of Snyder-de Sitter and Yang models, presenting how GEUP alters the density of states, potentially affecting physical (thermodynamical) properties. Special cases, obtained in certain limits from the above models are discussed. New higher order types of GEUP and EUP are also proposed.Data availabilityNo data was used for the research described in the article.1IntroductionThe search for the fundamental theory of Quantum Gravity has been supported by the development of phenomenological models that explore possible modifications to the known quantum mechanical or gravitational phenomena. In this context the various purely phenomenological proposals have appeared aiming to capture potential signatures of quantum effects of gravity, hinting towards possible experimental set-ups which would help guide the way in the full formulation of the theory. What is known is that one needs to challenge the concepts of classical space-time. The idea that the structure of space-time should be modified is a common feature of various approaches to Quantum Gravity, such as String Theory, Loop quantum gravity, Causal Dynamical Triangulations, Asymptotically Safe Quantum Gravity, Horava-Lifshitz Gravity and Noncommutative Geometry, just to name a few [1–6]. Many of these share also the concept of minimum length [7]. Such minimal length can be implemented in quantum mechanics through introducing the modifications in Uncertainty Principles (UP)11For simplicity, to illustrate the main points, we use 1-dimensional case throughout the introduction.:(1)ΔxΔp≥12|〈[x,p]〉|where Δ denotes standard deviation and 〈〉 the quantum expectation value on a given state. From the above relation, it is clear that any modifications in the canonical Heisenberg commutation relations between coordinates and momenta will imply changes in the UP. Various generalizations of UP have been considered in the literature. The most common type of such modifications is the Generalized Uncertainty Principle (GUP) with the momenta-dependent right hand side of (1). It was firstly proposed in [8] and then related to the specific algebraic structure of quantum phase space [9–12]. The right hand side of (1) may include quadratic terms in momenta (resulting in QGUP), see e.g. [11–13] or have linear and quadratic terms (LQGUP), see e.g. [14–18]. In momenta-dependent GUP relations one introduces the parameter, usually denoted by β, which is related to the Planck (length) scale lP, hence linking GUPs with Planck scale physics and quantum gravitational effects. Such phenomenological models have attracted a lot of attention [9–35], see also [36] for a recent review and more references on the topic.On the other hand, the symmetry between the position and momenta in the canonical (quantum mechanical) commutation relations, as well as Born reciprocity [37], suggests the possibility of introducing corrections to the Heisenberg uncertainty principle by including the modifications proportional to coordinates instead, i.e. so that the RHS of (1) is quadratic or linear in coordinates, instead of momenta. This complimentary type of uncertainty relation has been called an Extended Uncertainty Principle (EUP) [38,39]. Here, the parameter of the model, usually denoted by α, is related with the non-vanishing cosmological constant Λ and this way it can be embedded in the non-relativistic quantum mechanics. It has been shown [40] that this type of modification to the UP is related with the (Anti-)de Sitter geometric background, and the parameter α is then naturally linked with the (Anti-)de Sitter radius. While GUP, with β∼lP, exposes the gravitational modifications in quantum mechanics at the small distances; EUP, with α∼Λ introduces the idea of modifications at large distances. Relying on the analogy with GUPs, various types of such coordinate-dependent models have been introduced leading to different interesting physical effects, see e.g. [38,39,41,42].Combination of GUP and EUP (i.e. considering both coordinate and momenta dependent terms on the RHS of (1)) appeared firstly in the construction of noncommutative quantum mechanics with quantum groups as a symmetry [43], where both nonzero minimal uncertainties have appeared for position and momentum observables, respectively. Later on the relation of the type:(2)ΔxΔp≥ħ2(1+α2(Δx)2+β2(Δp)2) under the name of Generalized Extended Uncertainty Principle (GEUP, or alternatively EGUP), has been studied in various contexts, see e.g. [38–40,44], for more recent work see e.g. [45] where Born reciprocity is also promoted or see e.g. [46] for applications in statistical physics.Even though the origins of GUPs, EUPs and GEUPs can be tracked to the noncommutative geometry and quantum groups as the underlying mathematical frameworks [10,43], these models are mainly treated as purely phenomenological in the majority of the literature related to this subject. They have been used (and have been quite fruitful) in providing predictions for various phenomenological effects. However, such departure from the underlying mathematical framework resulting in the specific form of GUP, EUP or GEUPs together with the non-uniqueness of the defining commutation relations between coordinates and momenta has lead to many conceptual shortcomings and ambiguities (see e.g. the recent review [36] or [47] for discussion of some of the arising issues). Moreover, it is worth to point out that the minimal length may not necessarily appear in such models if one bases only on the modifications of the quantum mechanical phase space, see e.g. [48–51].In this paper, following the point of view that the noncommutative geometry is the underlying framework (mathematical language) of the possible fundamental theory, we assume that the quantization process of general relativity, includes the quantization of space-time [52,53], i.e. requiring the space-time coordinates to become noncommutative [xˆμ,xˆν]≠0. The noncommutativity of space-time introduces corrections to the canonical quantum-mechanical phase space relations. Hence modifications to the UP appear as a natural consequence. In this view, quadratic GUPs have been mainly linked with the Snyder model [10,34,35], with noncommuting coordinates and commuting momenta. EUPs, on the other hand can be linked with the (anti)-de Sitter geometric background and algebraic structure with commuting coordinates but noncommuting momenta [40]. The main aim of this paper is to show how the noncommutativity of both coordinates [xˆi,xˆj]≠0 and momenta [pˆi,pˆj]≠0 leading to GEUPs (and in some cases, the appearance of the nonzero minimal uncertainties in positions and momenta separately) affects the density of states and the analogue of the Liouville theorem in statistical mechanics. For this reason we limit ourselves to the case of non-relativistic quantum mechanics. We find the new form of weighted phase space volume element in the presence of GEUPs in the specific cases of Snyder-de Sitter (SdS) and Yang models. We show that noncommuting coordinates and noncommuting momenta with the corresponding GEUP require introducing the modification in the density states and this may impact various physical effects and thermodynamical properties (as it has been shown in the case of various GUP models).In the next section, we summarise the framework generalized to the case where both coordinates and momenta do not commute and present the general formulae for the Jacobian arising from the variable change under the infinitesimal time evolution. In Sec. 3, we specify the model to the Snyder-de Sitter (SdS) algebra, as only then we can identify the required weight factor (which depends on both coordinates and momenta) for the phase space volume element. The factor we obtain in this case does not depend on the dimension D. We also discuss the special cases obtained in the certain limits of the parameters, giving GUP or EUP relations (by reducing the starting SdS algebra to Snyder or dS algebras, respectively) and we give the expressions for the phase space volume element in these cases. In Sec. 4, we discuss the Yang model and consider specific realizations of its generators on the canonical phase space leading to another type for (higher order) GEUP relation. The weighted phase space volume element is obtained for this case in 1 dimension by adapting the result from SdS model. Special cases, obtained in certain limits are also discussed, one provides the known “square-root” (or Maggiore) GUP and the other leads to the new higher order (“square-root”) type of EUP. In Sec. 5, the Lie algebraic case with commuting momenta is briefly considered with the fuzzy sphere as an example. It is shown that in this case the phase space volume element stays invariant under the time evolution and there is no change in the density of states.2PreliminariesIn this section we set up the most general framework for investigating the effects of noncommuting coordinates and noncommuting momenta (and in principle the appearance of the nonzero length and momentum uncertainties) on the density of states in the phase space so that we can adapt the Liouville theorem in statistical physics to this new scenario. For this reason we do not fix the specific choice for the noncommutativity of space-time or momenta and the resulting deformation of the quantum phase space commutation relations at this point yet.We start with the following (most general) set of commutation relations describing the noncommutative quantum mechanical phase space algebra:(3)[xˆi,xˆj]=iħaij(xˆ,pˆ),[pˆi,pˆj]=iħbij(xˆ,pˆ)(4)[xˆi,pˆj]=iħcij(xˆ,pˆ),where aij(xˆ,pˆ),bij(xˆ,pˆ),cij(xˆ,pˆ) are functions which may include all kinds of terms (linear, quadratic, higher order etc.) in space-time coordinates xˆ and momenta pˆ, such that the Jacobi identities are satisfied. We will focus on the non-relativistic general case in any dimension D with i,j=1,2,…,D. When none of the above commutators are zero (see e.g. [45,54–57]), this will lead to the interesting types of the GEUPs: ΔxˆiΔpˆj≥ħ2|〈cij(xˆ,pˆ)〉| with the specific form of RHS depending on the concrete choice of the algebra (3), (4).In the phenomenological approaches, one considers the right hand side of (4) as the definition of a new effective value of ħ which (in the most general case) may depend on both coordinates and momenta. This means that the size of the unit cell that each quantum state occupies in the phase space can be thought of as being also coordinates and momenta dependent. This will have an effect on the density of states and as a consequence affect physical, for example thermodynamical, properties. For this interpretation to be valid, the volume of phase space must evolve in such a way that the number of states does not change with time, in other words we are looking for the analogue of the Liouville theorem in statistical physics. Similar investigation has been done in the case of the quadratic GUP [12,13] or GUPs with higher order terms [58]. Note that here we assume that both coordinates and momenta do not commute. Nevertheless, later on we shall see how this more general case can be reduced to the special cases like the GUP or EUP (which include some commutative generators).Starting with the quantum mechanical commutation relations (3), (4), these will correspond to the Poisson brackets in classical mechanics:(5)1iħ[Aˆ,Bˆ]→classicallimit{A,B}.Therefore, in the classical limit22It is worth to note that the classical limit in GUP models may be more involved than just applying the above transformation, see e.g. [59] where a possible way out is suggested to derive the GUP relations from the (explicitly state dependent) deformed commutators between coordinates and momenta. Here, on the contrary to the phenomenological GUP approaches, we are starting from the noncommutative model as a possible description of the quantization of space-time, hence the starting algebra is fixed from the beginning and modified uncertainty relations and all effects are a consequence of the starting choice of quantum-deformed phase space. we obtain (in our shortcut notation):(6){xi,xj}=aij(x,p),{pi,pj}=bij(x,p),(7){xi,pj}=cij(x,p).The time evolution of the coordinates and momenta is governed by the equations (where the summation convention is assumed)(8)x˙i={xi,H}={xi,pj}∂H∂pj+{xi,xj}∂H∂xj=cij∂H∂pj+aij∂H∂xj,(9)p˙i={pi,H}=−{xj,pi}∂H∂xj+{pi,pj}∂H∂pjδt=−cji∂H∂xj+bij∂H∂pj.The Liouville theorem requires that the phase space volume is invariant under time evolution. Hence we consider an infinitesimal time interval δt and the evolution of the coordinates and momenta during δt is:xi′=xi+δxi,pi′=pi+δpiwhere(10)δxi=x˙iδt=(cij∂H∂pj+aij∂H∂xj)δt,δpi=p˙iδt=(−cji∂H∂xj+bij∂H∂pj)δt. The infinitesimal phase space volume after this infinitesimal time evolution will be:dDx′dDp′=JdDxdDpwith the Jacobian(11)J=|∂(x1′,…,xD′,p1′,…,pD′)∂(x1,…,xD,p1,…,pD)|=1+(∂δxi∂xi+∂δpi∂pi)+…where we used:(12)∂xi′∂xj=δij+∂δxi∂xj,∂xi′∂pj=∂δxi∂pj,(13)∂pi′∂xj=∂δpi∂xj,∂pi′∂pj=δij+∂δpi∂pj.Up to the first order in δt (based on (10)) we get:(14)(∂δxi∂xi+∂δpi∂pi)=∂∂xi(cij∂H∂pj+aij∂H∂xj)δt+∂∂pi(−cji∂H∂xj+bij∂H∂pj)δt=[(∂∂xiaij−∂∂picji)∂H∂xj+(∂∂xicij+∂∂pibij)∂H∂pj]δtwhere the terms with mixed derivatives have cancelled each other and the antisymetricity of the Poisson brackets was used to cancel the remaining terms. This way we find the expression for the time evolved infinitesimal phase space volume (in the first order of δt) as:(15)dDx′dDp′=dDxdDp(1+[(∂∂xiaij−∂∂picji)∂H∂xj+(∂∂xicij+∂∂pibij)∂H∂pj]δt+O(δt2))valid for any Poisson algebra (6), (7) obtained as the classical limit of any noncommutative model (3), (4). Since in general the terms in the brackets will not cancel out, already here we see the need to introduce the weight to the phase space volume element so that the analogue of the Liouville theorem is satisfied. The specific factor has to be chosen in such a way that the weighted phase space volume is invariant under the time evolution, i.e. so that:(16)dDx′dDp′F(x′,p′)∼dDxdDpF(x,p).It is straightforward to notice from (15) that the noncanonical Poisson brackets for coordinates and for momenta (6), as well as the mixed relation between coordinates and momenta (7) are crucial in the choice of the weighted phase space volume. Hence the underlying noncommutative geometry (and the choice of (3), (4)) is the intrinsic feature of phenomenological models investigating the possible effects arising from such modified phase space volume element (and consequently affecting the thermodynamical properties of physical systems). To be able to investigate the time evolution of the weight factor F(x,p), we need to consider the concrete noncommutative model, which we shall do in the next section.3Snyder-de Sitter model and the density of statesSnyder-de Sitter (SdS) model, which includes the noncommutative space-time coordinates and the noncommutative momenta, was proposed [54] as a generalization of the Snyder model [60] to a space-time background of constant curvature and it was investigated in many contexts, see e.g. [61], [62]. In SdS model, the noncommutativity among space-time coordinates corresponds to the curved momentum space, and vice-versa noncommutative momenta lead to the curved space-time. Since we are interested in investigating how the noncommutativity of both coordinates and momenta affects the density of states we choose SdS set of commutation relations which exhibit these features:(17)[xˆμ,xˆν]=iħβ2Mˆμν,[pˆμ,pˆν]=iħα2Mˆμν,(18)[Mˆμν,xˆρ]=iħ(ημρxˆν−ηνρxˆμ),(19)[Mˆμν,pˆρ]=iħ(ημρpˆν−ηνρpˆμ),(20)[Mˆμν,Mˆρτ]=iħ(ημρMˆντ−ημτMˆνρ+ηντMˆμρ−ηνρMˆμτ),with the modified quantum mechanical phase space relation:(21)[xˆμ,pˆν]=iħ(ημν+α2xˆμxˆν+β2pˆμpˆν+αβ(xˆμpˆν+pˆμxˆν−Mˆμν)).Here ημν is the flat metric with Lorentzian signature and μ,ν=0,1,…,D. Depending on the sign33For the Jacobi identities to be satisfied both coupling constants α2 and β2 must have the same sign. Following the convention of [61] when α2 and β2 are negative then αβ is negative. of α2 the Lorentz generators Mˆμν and momenta pˆμ generate de Sitter (dS) or Anti-de Sitter (AdS) subalgebras. This model involves, besides the speed of light,44We set c=1 throughout the paper. two other observer-independent constants,55SdS is also known as doubly special relativity (DSR) in de Sitter space [56] or triply special relativity (TSR) model [54]. the Planck length as well as the de Sitter radius which is related to the cosmological constant. More precisely, the parameter |α2| has the dimension of the inverse of the square of length and it can be identified with the (Anti-)de Sitter radius and the cosmological constant as its inverse, while the parameter |β2| has the dimension of the inverse square of mass and it can be is identified with 1/MP2=lP2, where lP and MP are the Planck length and mass, respectively. In general, α2 and β2 can take positive or negative value, which generates models with very different properties [61].In the following we focus on the non-relativistic quantum mechanical counterpart (which in D=3 would give the Snyder model restricted to a three-dimensional sphere) however we keep unspecified dimension D, i,j=1,2,…,D. The GEUP corresponding to (21) has the form similar to (2) [40],66Note that the same GEUP can be realized by a different deformed Heisenberg algebras, see e.g. [45]. but additional (mixed) terms may appear on the RHS depending on the choice of the representation of the angular momentum used, see e.g. [61]. Without any additional assumptions we can, at most, write:(22)(Δxˆi)(Δpˆi)≥ħ2|1+α2(Δxˆ)2+β2(Δpˆ)2+γ| where we used (Δxˆ)2=〈xˆ2〉−〈xˆ〉2, (Δpˆ)2=〈pˆ2〉−〈pˆ〉2 and γ=α2〈xˆ〉2+β2〈pˆ〉2+αβ(〈xˆpˆ〉+〈pˆxˆ〉) as well as the standard notation xˆ2=xˆixˆi, pˆ2=pˆipˆi, xˆpˆ=xˆipˆi was assumed, while on the LHS there is no summation. In 1-dimensional case, one can show [61] that, for both α2>0 and β2>0 the bounds on the localization in position and momentum space arise, while when α2<0 and β2<0 a combination of spatial and momentum coordinates becomes bounded instead.The classical (non-relativistic) Snyder-de Sitter Poisson algebra is given by77Where we used the fact that the above algebra (and relation (21)) can be implemented on the canonical phase space with Lorentz generators defined as Mμν=xμpν−xνpμ.:(23){xi,xj}=β2(xipj−xjpi)≡aij,{pi,pj}=α2(xipj−xjpi)≡bij,{xi,pj}=δij+α2xixj+β2pipj+2αβpixj≡cij. and supplemented by the relations with Lorentz generators. By using the general set up from the previous section and now the explicit form of SdS model (23) we obtain that the infinitesimal phase space volume element after the infinitesimal time evolution as:(24)dDx′dDp′=dDxdDp(1+2[(α2xj+αβpj)∂H∂pj−(β2pj+αβxj)∂H∂xj]δt+O(δt2))where we explicitly used (23) in (15) with the following:(25)∂∂xiaij=β2(D−1)pj,∂∂xicij=α2(D+1)xj+2αβpj,(26)∂∂pibij=α2(1−D)xj=−α2(D−1)xj,∂∂picji=β2(D+1)pj+2αβxj.In the remaining part of this section we will show that for the SdS Poisson algebra (23) the analogue of the Liouville theorem is satisfied when we consider the following weighted phase space volume(27)dDxdDp(1+α2x2+β2p2+2αβx⋅p)which is invariant under the infinitesimal time evolution.To show this, we consider the time evolution of each of the terms in the proposed factor F(x,p)=1+α2x2+β2p2+2αβx⋅p during the infinitesimal time interval δt. At first we keep all the formulae as general as possible and express them in terms of aij, bij and cij and only at the end we will specify to SdS case (23). We get the following expressions:(28)x′2=(xi+δxi)2=x2+2xiδxi+O(δt2)=x2+2xi(cij∂H∂pj+aij∂H∂xj)δt+O(δt2),(29)p′2=(pi+δpi)2=p2+2piδpi+O(δt2)=p2+2pi(−cji∂H∂xj+bij∂H∂pj)δt+O(δt2),and for the last term(30)x′⋅p′=x′p′=(xi+δxi)(pi+δpi)=xp+piδxi+xiδpi+O(δt2)=xp+pi(cij∂H∂pj+aij∂H∂xj)δt+xi(−cji∂H∂xj+bij∂H∂pj)δt+O(δt2).Therefore the time evolution of the whole factor F(x,p) can be first written generally as (where for simplicity we use 2αβ=γ):(31)1+α2x′2+β2p′2+γx′p′=1+α2x2+β2p2+γxp++[2α2xicij+2β2pibij+γ(picij+xibij)]∂H∂pjδt++[2α2xiaij−2β2picji+γ(piaij−xicji)]∂H∂xjδt+O(δt2).Now specialising this to the case of SdS (23), after plugging in the expressions for aij, bij and cij, we obtain (in the first order of δt):(32)1+α2x′2+β2p′2+γx′p′=1+α2x2+β2p2+γxp+[2α2(xj+α2x2xj+β2(xp)pj+γ(xp)xj)+2β2α2(xp)pj−2β2α2xjp2+γ(pj+α2x2pj+β2p2pj+γp2xj)]∂H∂pjδt++[−2β2(pj+α2xj(xp)+β2pjp2+γ(xp)pj)+2α2β2x2pj−2α2β2xj(xp)−γ(xj+α2x2xj+β2p2xj+γx2pj)]∂H∂xjδt. Since the aim is to factorise the whole expression (1+α2x2+β2p2+γxp) on the right hand side of this equality, we need to rearrange all the terms in the square brackets in such a way so that we can recognize the whole factor F(x,p). In this way, we obtain (once we returned to the notation γ=2αβ):(33)1+α2x′2+β2p′2+2αβx′p′==1+α2x2+β2p2+2αβ(xp)+[2α2(1+α2x2+β2p2+2αβ(xp))xj+2αβ(1+α2x2+β2p2+2αβxp)pj]∂H∂pjδt++[−2β2(1+α2x2+β2p2+2αβ(xp))pj−2αβ(1+α2x2+β2p2+2αβx⋅p)xj]∂H∂xjδt+O(δt2)=(1+α2x2+β2p2+2αβ(xp))[1+(2α2xj+2αβpj)∂H∂pjδt−(2β2pj+2αβxj)∂H∂xjδt]+O(δt2).We can see that the weighted volume element will stay invariant since the weight factor we introduced produces the same terms as the Jacobian in (24) under the infinitesimal time evolution (up to the first order in δt):(34)dDx′dDp′1+α2x′2+β2p′2+2αβx′p′==dDxdDp(1+2[(α2xj+αβpj)∂H∂pj−(β2pj+αβxj)∂H∂xj]δt+O(δt2))(1+α2x2+β2p2+2αβx⋅p)(1+2[(α2xj+αβpj)∂H∂pj−(β2pj+αβxj)∂H∂xj]δt+O(δ2))=dDxdDp(1+α2x2+β2p2+2αβxp).We point out this result holds to any order of parameters α and β (as we have not done any approximations in the noncommutative parameters).We note that from the SdS model considered above we can obtain two well known special cases. Namely:•Snyder model [60]88Snyder model was the first proposed noncommutative space-time model preserving Lorentz symmetry. is obtained when we take α→0 in SdS algebra (17)-(21), i.e. we obtain the model with noncommutative coordinates and commutative momenta (curved momentum space):(35)[xˆμ,xˆν]=iħβ2Mˆμν,[pˆμ,pˆν]=0,[xˆμ,pˆν]=iħ(ημν+β2pˆμpˆν),supplemented by the Lorentz covariance conditions (18)-(20). Such modification of quantum mechanical phase space relations leads, in the non-relativistic case, to the quadratic GUP (QGUP)99For the simple case in which 〈pˆi〉=0.:(36)ΔxˆiΔpˆi≥ħ2(1+β2(Δpˆ)2).We note that the algebraic set of commutation relations (35) is only one of the many possible realizations of the Snyder model and more general realizations may be considered, see e.g. [34,35]. The non-relativistic classical Poisson algebra:(37){xi,xj}=β2(xipj−xjpi),{pi,pj}=0,{xi,pj}=δij+β2pipjwill result in the following weighted phase space volume element:(38)dDxdDp1+β2p2 (in any dimension D), obtained as the limit α→0 in (27).•(Anti-)de Sitter model (dual Snyder model) is obtained when we take β→0 in SdS algebra (17)-(21), i.e. we obtain the model with commutative coordinates but noncommutative momenta (curved space-time1010We recall, that the parameter α still will play the role of the inverse of the de Sitter radius R i.e. space-time curvature, which is linked to the cosmological constant.):(39)[xˆμ,xˆν]=0,[pˆμ,pˆν]=iħα2Mˆμν,[xˆμ,pˆν]=iħ(ημν+α2xˆμxˆν), with the Lorentz covariance given by (18)-(20). The non-relativistic case results in the quadratic EUP (QEUP)1111For the simple case in which 〈xˆi〉=0.:(40)ΔxˆiΔpˆi≥ħ2(1+α2(Δxˆ)2).The non-relativistic classical Poisson algebra is:(41){xi,xj}=0,{pi,pj}=α2(xipj−xjpi),{xi,pj}=δij+α2xixj. And the weighted phase space volume element (in any dimension D), obtained as the limit β→0 in (27), is:(42)dDxdDp1+α2x2. It is worth to point out that there exists a way to transform SdS algebra (17)-(21) generated by (xˆ,pˆ,Mˆ) into the Snyder algebra (35) generated by (xˆS,pˆS,Mˆ) by the following linear maps:(43)xˆi=xˆiS+βαλpˆiS,pˆi=(1−λ)pˆiS−αβxˆiS where λ is a free parameter and we have temporarily denoted the Snyder algebra (35) generators by the upper index S. This relation of the SdS algebra with the Snyder algebra was first presented in [61]. Through such noncanonical change of basis one can use the already known machinery of realizations in Snyder spaces and consider various applications of SdS algebra, for example to find harmonic oscillator solutions [63] or in applications to quantum field theory [64].One can also consider various generalizations of the SdS model. For example, in [65] the following generalization of the last relation in SdS algebra (21) was proposed:(44)[xˆμ,pˆν]=iħ(ημνφ1+(α2xˆμxˆν+β2pˆμpˆν+αβxˆμpˆν+αβpˆμxˆν)φ2−αβMˆμν)where the functions φ1 and φ2 need to satisfy specific conditions (due to Jacobi identities). When φ1=φ2=1 we get back the original SdS model (21). Various choices of φ1,φ2 are discussed in [65] and for example for one specific choice of φ1 and φ2 we can obtain the following relation:(45)[xˆμ,pˆν]=iħημν1−(α2xˆ2+β2pˆ2+αβxˆpˆ+αβpˆxˆ)−αβMˆμν.Such generalizations would be interesting to investigate further in the context of GEUPs and their influence on the density of states.Before completing this section, since the effects of modifications of UPs on the density of states have been investigated previously few comments are in order. In [12,13], in the case of GUP (36)1212In [12,13] more general case, than (36), is considered with two parameters β, β′. In the comparison we take β′=0 since this is the most similar option to the Snyder model discussed here. Nevertheless, the case with β′≠0 can also be associated with the Snyder model, but requires different (more general) realization see e.g. [34,35]. the invariant weighted phase space volume element is obtained as: dDxdDp(1+β2p2)D, where the power D (dimension) is necessary since the Jacobian obtained is: dDx′dDp′=dDxdDp(1−2β2Dpk∂H∂xkδt+O(δt2)). This is different than the case considered here, where in the limit α→0 we obtain dDxdDp1+β2p2 (38). The difference arises from the fact that commutation relations used in [12] for [xi,pj] have the term proportional to p2 while here we have the terms with pipj instead, cf. (37). In the case of the Snyder model with (37) the terms with D in the Jacobian (24) cancel out, hence the power D is not appearing in the weight factor of the volume form (38).Other important point is that often in the context of modified UP (1) also the inner product on the Hilbert space1313The Heisenberg algebra (3), (4) is represented on the space of states in which one usually chooses a basis of position or momentum eigenvectors. In the case of GUP one usually chooses the momentum space, while in the case of GEUP Bergmann-Fock construction can be used [43]. becomes modified with appropriately chosen measure so that the observables (satisfying the modified commutation relations) stay symmetric on the dense domain of functions decaying faster than any power [11]. For the inner product modifications in the case of GUP see, e.g. [12,13] and for the relativistic case, see e.g. [66]. The inner product modifications in the case of EUP, i.e. in AdS and dS spaces were investigated e.g. in [41]. Subsequently the modified inner product can be used to study effects on solutions of Schrödinger equation, see e.g. [12,13,41]. Such modifications in the measure in the inner product have not been under investigation of the present paper.4Yang modelThe Yang model introduced in [55] is a Lorentz invariant model incorporating noncommutative space-time coordinates as well as noncommutative momenta, depending on the pair of dimensionful parameters α and β related with the curvatures of quantum space-time and momentum spaces (in similarity to SdS model). The defining relations are as follows:(46)[xˆμ,xˆν]=iħβ2Mˆμν,[pˆμ,pˆν]=iħα2Mˆμν(47)[Mˆμν,xˆρ]=iħ(ημρxˆν−ηνρxˆμ),(48)[Mˆμν,pˆρ]=iħ(ημρpˆν−ηνρpˆμ),(49)[Mˆμν,Mˆρτ]=iħ(ημρMˆντ−ημτMˆνρ+ηντMˆμρ−ηνρMˆμτ).However, the quantum phase space relation is described by an additional generator rˆ (central charge):(50)[xˆμ,pˆν]=iħημνrˆ,hence to obtain the full Yang algebra we need the additional relations:(51)[rˆ,xˆμ]=iħβ2pˆμ,[rˆ,pˆμ]=−iħα2xˆμ,[Mˆμν,rˆ]=0.The uncertainty principle corresponding to the Yang model, in the non-relativistic case, can be written in general as:(52)ΔxˆiΔpˆj≥ħδij2|〈rˆ〉|,where the generator rˆ can be realized in terms of the phase space variables rˆ=rˆ(xˆ,pˆ). It is also worth to mention that the Yang model is covariant (self-dual) under the Born reciprocity:(53)B:xˆμ→pˆμ,pˆμ→−xˆμ,Mˆμν↔Mˆμν,rˆ↔rˆ,α↔β.The similarity between the Yang model and the SdS model considered in the previous section is not coincidental. It has been shown [67] that the SdS algebra (17)-(21) can be viewed as a nonlinear realization of the Yang model (46)-(51).The classical limit of the Yang model (cf. [68], see also [69]) is:(54){xi,xj}=β2(xipj−xjpi),{pi,pj}=α2(xipj−xjpi)and one of the possible realizations for the rˆ generator on the canonical phase space [68], for example, gives:(55){xi,pj}=δij1−α2x2−β2p2−α2β2(x2p2−(xp)2).The remaining relations are obtained in a straightforward way. In D=1 this would simplify to:(56){x,p}=1−α2x2−β2p2,with the corresponding GEUP1414In higher dimensions additional mixed terms would be present in the GEUP relation.:(57)ΔxΔp≥ħ21−α2(Δx)2−β2(Δp)2. We see that in such 1-dimensional case higher order terms would appear, which is not uncommon for GUPs see e.g. [70–72] but this would be the first such example considering higher order GEUPs, up to our knowledge.Expanding the RHS of (56) in deformation parameters, up to α2 and β2, we can apply the results of the previous section (27) and obtain the invariant weighted phase space volume element as:(58)dxdp1−12(α2x2+β2p2). From the Yang model (46)-(51), in the realization (55) for rˆ generator we can obtain two special cases. Namely:•In the limit when α→0 we obtain the so-called “square-root modified” or “Maggiore algebra” (see e.g. [73]):(59)[xˆμ,xˆν]=iħβ2Mˆμν,[pˆμ,pˆν]=0,[xˆμ,pˆν]=iħημν1−β2pˆ2,supplemented by the Lorentz covariance conditions (47)-(49). Such modification of quantum mechanical phase space relations leads, in the non-relativistic case, to the higher order type of GUP1515We call this type of GUP “higher order” due to the expansion of the square root function in powers of p thanks to which, through series of inequalities, one obtains the formula (60), see e.g. [72]. of the form:(60)ΔxˆiΔpˆi≥ħ21−β2(Δpˆ)2. This version of GUP no longer produces a minimum observable length [48,72]. But it would still result in the weighted phase space volume element:(61)dxdp1−12β2p2 affecting the density of states.•By taking β→0 in (46)-(51), we obtain(62)[xˆμ,xˆν]=0,[pˆμ,pˆν]=iħα2Mˆμν,[xˆμ,pˆν]=iħημν1−α2xˆ2, leading to the higher order type of EUP1616By similar expansion of the square-root function in powers of x.:(63)ΔxˆiΔpˆi≥ħ21−α2(Δxˆ)2 with the weighted phase space volume element as:(64)dxdp1−12α2x2 affecting the density of states.We postpone the investigation of the full D-dimensional case of Yang model (46)-(51) in the context of density of states to the future work. It is also worth to mention that in [74] generalizations of the Snyder algebra to a curved space-time background with de Sitter symmetry were considered where the SdS model and Yang model were obtained as special cases. The realizations of these algebras were considered in terms of canonical phase space coordinates, up to the fourth order in the deformation parameters. Therefore the results of the present paper could be generalized to these type of models and realizations as well.5Lie-algebraic case with commuting momenta: fuzzy sphereMany noncommutative (quantum) space-times proposed in the quantum gravity motivated literature have the Lie-algebraic form for the noncommutativity of coordinates and include commuting momenta. Hence, for the sake of completeness we discuss here one example of such model defined by the following commutation relations:(65)[xˆi,xˆj]=iħϵijkxˆk,[xˆi,pˆj]=iħϵijkpˆk,[pˆi,pˆj]=0 where ϵijk is totally skew-symmetric tensor, ϵ123=1. The subalgebra [xˆi,xˆj]=iħϵijkxˆk supplemented by the relation xˆixˆi=r corresponds to the fuzzy sphere, with r the constant radius of the sphere.Since we are interested in the effects on the density of states in this Lie algebraic case we follow the steps outlined in Sec. 2. In the classical limit, we obtain:(66){xi,xj}=ϵijkxk≡aij,{xi,pj}=ϵijkpk≡cij,{pi,pj}=0≡bij, where we identified RHSs with the notation used in Sec. 2. Directly plugging these relations into (15) we obtain:(67)dDx′dDp′=dDxdDp(1+[(∂∂xi(ϵijkxk)−∂∂pi(ϵjikpk))∂H∂xj+(∂∂xi(ϵijkpk))∂H∂pj]δt+O(δt2))=dDxdDp(1+[(ϵijkδik−ϵjikδik)∂H∂xj+0]δt+O(δt2))=dDxdDp. Hence we see that the infinitesimal phase space volume element stays invariant under the time evolution without the need to introduce any additional factor and there will be no change in the density of states for the case of fuzzy sphere.6Final remarksIn this paper we have focused on models exhibiting noncommutativity in both space-time coordinates and in momenta (i.e. models with curved space-time and curved momentum space), such that in the quantum phase space relations both the Planck length lP and the cosmological constant Λ appear as fundamental parameters on equal footing. The modified Heisenberg commutation relations lead to GEUPs where the symmetry between position and momentum is preserved (which is not the case in the usual GUPs or EUPs). Such symmetric GEUPs may be seen as an indication of quantum gravitational corrections to the classical space-time and standard quantum mechanics, at both very small and very large scales. We point out that, on the contrary to many works which have proposed multiple variants of the GUPs, EUPs and GEUPs by arbitrarily choosing specific forms of the commutation relations between space-time coordinates and momenta [x,p] with supposedly desirable properties, we have focused here on studying the consequences of known models which arise in the noncommutative geometry approach to quantum gravity.In general since the canonical commutation relations are modified, one expects that thermodynamics and statistical mechanics will be affected by the introduced modifications, possibly leading to some new effects. As a consequence of the GEUP arising from the cases of Snyder-de Sitter and Yang models we have shown that the analogue of the Liouville theorem in statistical physics requires considering the weighted phase space volume and introduces modification in the density of states, with the weight factor depending on both coordinates and momenta. Since such modification of the density states is required this will influence the statistical and thermodynamical properties of physical systems. Various applications can now be studied and the effects of both noncommutativity in coordinates and momenta (or the presence of the Planck length and the cosmological constant in modified UPs) on atomic physics, condensed matter physics, preheating phase of the universe and black holes etc. can now be investigated. Declaration of Competing InterestThe authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Anna Pachol reports financial support was provided by NCN grant number 2022/45/B/ST2/01067. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.AcknowledgementsAP thanks A. Wojnar and S. Zonetti for interesting discussions and acknowledges the support of the Polish NCN grant 2022/45/B/ST2/01067 and COST Action CA21109 - CaLISTA. References[1]NathanSeibergEdwardWittenString theory and noncommutative geometryJ. High Energy Phys.091999032arXiv:hep-th/9908142Nathan Seiberg and Edward Witten. String theory and noncommutative geometry. JHEP, 09:032, 1999, arXiv:hep-th/9908142.[2]David J.GrossPaul F.MendeString theory beyond the Planck scaleNucl. Phys. B3031988407454David J. Gross and Paul F. Mende. String Theory Beyond the Planck Scale. Nucl. Phys. B, 303:407–454, 1988.[3]CarloRovelliLeeSmolinDiscreteness of area and volume in quantum gravityNucl. Phys. B4421995593622arXiv:gr-qc/9411005Erratum:Nucl. Phys. B4561995753754Carlo Rovelli and Lee Smolin. Discreteness of area and volume in quantum gravity. Nucl. Phys. B, 442:593–622, 1995, arXiv:gr-qc/9411005. [Erratum: Nucl.Phys.B 456, 753–754 (1995)].[4]J.AmbjornJ.JurkiewiczR.LollSpectral dimension of the universePhys. Rev. Lett.952005171301arXiv:hep-th/0505113J. Ambjorn, J. Jurkiewicz, and R. Loll. Spectral dimension of the universe. Phys. Rev. Lett., 95:171301, 2005, arXiv:hep-th/0505113.[5]O.LauscherM.ReuterFractal spacetime structure in asymptotically safe gravityJ. High Energy Phys.102005050arXiv:hep-th/0508202O. Lauscher and M. Reuter. Fractal spacetime structure in asymptotically safe gravity. JHEP, 10:050, 2005, arXiv:hep-th/0508202.[6]PetrHoravaSpectral dimension of the universe in quantum gravity at a Lifshitz pointPhys. Rev. Lett.1022009161301arXiv:0902.3657Petr Horava. Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point. Phys. Rev. Lett., 102:161301, 2009, arXiv:0902.3657.[7]Luis J.GarayQuantum gravity and minimum lengthInt. J. Mod. Phys. A101995145166arXiv:gr-qc/9403008Luis J. Garay. Quantum gravity and minimum length. Int. J. Mod. Phys. A, 10:145–166, 1995, arXiv:gr-qc/9403008.[8]MicheleMaggioreA generalized uncertainty principle in quantum gravityPhys. Lett. B30419936569arXiv:hep-th/9301067Michele Maggiore. A Generalized uncertainty principle in quantum gravity. Phys. Lett. B, 304:65–69, 1993, arXiv:hep-th/9301067.[9]MicheleMaggioreQuantum groups, gravity and the generalized uncertainty principlePhys. Rev. D49199451825187arXiv:hep-th/9305163Michele Maggiore. Quantum groups, gravity and the generalized uncertainty principle. Phys. Rev. D, 49:5182–5187, 1994, arXiv:hep-th/9305163.[10]MicheleMaggioreThe algebraic structure of the generalized uncertainty principlePhys. Lett. B31919938386arXiv:hep-th/9309034Michele Maggiore. The Algebraic structure of the generalized uncertainty principle. Phys. Lett. B, 319:83–86, 1993, arXiv:hep-th/9309034.[11]AchimKempfGianpieroManganoRobert B.MannHilbert space representation of the minimal length uncertainty relationPhys. Rev. D52199511081118arXiv:hep-th/9412167Achim Kempf, Gianpiero Mangano, and Robert B. Mann. Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D, 52:1108–1118, 1995, arXiv:hep-th/9412167.[12]Lay NamChangDjordjeMinicNaotoshiOkamuraTatsuTakeuchiThe effect of the minimal length uncertainty relation on the density of states and the cosmological constant problemPhys. Rev. D652002125028arXiv:hep-th/0201017Lay Nam Chang, Djordje Minic, Naotoshi Okamura, and Tatsu Takeuchi. The Effect of the minimal length uncertainty relation on the density of states and the cosmological constant problem. Phys. Rev. D, 65:125028, 2002, arXiv:hep-th/0201017.[13]Lay NamChangDjordjeMinicNaotoshiOkamuraTatsuTakeuchiExact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relationsPhys. Rev. D652002125027arXiv:hep-th/0111181Lay Nam Chang, Djordje Minic, Naotoshi Okamura, and Tatsu Takeuchi. Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations. Phys. Rev. D, 65:125027, 2002, arXiv:hep-th/0111181.[14]Elias C.VagenasAhmed FaragAliMohammedHemedaHassanAlshalLinear and quadratic GUP, Liouville theorem, cosmological constant, and Brick Wall entropyEur. Phys. J. C7952019398arXiv:1903.08494Elias C. Vagenas, Ahmed Farag Ali, Mohammed Hemeda, and Hassan Alshal. Linear and Quadratic GUP, Liouville Theorem, Cosmological Constant, and Brick Wall Entropy. Eur. Phys. J. C, 79(5):398, 2019, arXiv:1903.08494.[15]JohnPaulR.BernaldezAdrian G.AbacRolandEmeritoS.OtadoyRadial oscillations and dynamical instability analysis for linear-quadratic gup-modified white dwarfsAnn. Phys.1694022023John Paul R Bernaldez, Adrian G Abac, and Roland Emerito S Otadoy. Radial oscillations and dynamical instability analysis for linear-quadratic gup-modified white dwarfs. Annals of Physics, 169402, 2023.[16]AnetaWojnarUnveiling phase space modifications: a clash of modified gravity and the generalized uncertainty principlePhys. Rev. D10922024024011arXiv:2311.14066Aneta Wojnar. Unveiling phase space modifications: A clash of modified gravity and the generalized uncertainty principle. Phys. Rev. D, 109(2):024011, 2024, arXiv:2311.14066.[17]AnetaWojnarDébora AguiarGomesBose and Fermi gases in metric-affine gravity and linear generalized uncertainty principleUniverse1052024217arXiv:2404.03345Aneta Wojnar and Débora Aguiar Gomes. Bose and Fermi Gases in Metric-Affine Gravity and Linear Generalized Uncertainty Principle. Universe, 10(5):217, 2024, arXiv:2404.03345.[18]Ahmed FaragAliAnetaWojnarA covariant tapestry of linear GUP, metric-affine gravity, their Poincaré algebra and entropy boundClass. Quantum Gravity41102024105001arXiv:2401.05941Ahmed Farag Ali and Aneta Wojnar. A covariant tapestry of linear GUP, metric-affine gravity, their Poincaré algebra and entropy bound. Class. Quant. Grav., 41(10):105001, 2024, arXiv:2401.05941.[19]FabioScardigliGeneralized uncertainty principle in quantum gravity from micro - black hole Gedanken experimentPhys. Lett. B45219993944arXiv:hep-th/9904025Fabio Scardigli. Generalized uncertainty principle in quantum gravity from micro - black hole Gedanken experiment. Phys. Lett. B, 452:39–44, 1999, arXiv:hep-th/9904025.[20]S.CapozzielloG.LambiaseG.ScarpettaGeneralized uncertainty principle from quantum geometryInt. J. Theor. Phys.3920001522arXiv:gr-qc/9910017S. Capozziello, G. Lambiase, and G. Scarpetta. Generalized uncertainty principle from quantum geometry. Int. J. Theor. Phys., 39:15–22, 2000, arXiv:gr-qc/9910017.[21]FabianBrauFabienBuisseretMinimal length uncertainty relation and gravitational quantum wellPhys. Rev. D742006036002arXiv:hep-th/0605183Fabian Brau and Fabien Buisseret. Minimal Length Uncertainty Relation and gravitational quantum well. Phys. Rev. D, 74:036002, 2006, arXiv:hep-th/0605183.[22]SauryaDasElias C.VagenasUniversality of quantum gravity correctionsPhys. Rev. Lett.1012008221301arXiv:0810.5333Saurya Das and Elias C. Vagenas. Universality of Quantum Gravity Corrections. Phys. Rev. Lett., 101:221301, 2008, arXiv:0810.5333.[23]FabioScardigliGaetanoLambiaseEliasVagenasGUP parameter from quantum corrections to the Newtonian potentialPhys. Lett. B7672017242246arXiv:1611.01469Fabio Scardigli, Gaetano Lambiase, and Elias Vagenas. GUP parameter from quantum corrections to the Newtonian potential. Phys. Lett. B, 767:242–246, 2017, arXiv:1611.01469.[24]E.HarikumarZuhair NambipunnilathSiddiqueCompact stars in quantum spacetimeJ. Phys. Commun.232018035016arXiv:1707.00422E. Harikumar and Zuhair Nambipunnilath Siddique. Compact stars in quantum spacetime. J. Phys. Comm., 2(3):035016, 2018, arXiv:1707.00422.[25]PengWangHaitangYangXiumingZhangQuantum gravity effects on statistics and compact star configurationsJ. High Energy Phys.082010043arXiv:1006.5362Peng Wang, Haitang Yang, and Xiuming Zhang. Quantum gravity effects on statistics and compact star configurations. JHEP, 08:043, 2010, arXiv:1006.5362.[26]Ahmed FaragAliMinimal length in quantum gravity, equivalence principle and holographic entropy boundClass. Quantum Gravity282011065013arXiv:1101.4181Ahmed Farag Ali. Minimal Length in Quantum Gravity, Equivalence Principle and Holographic Entropy Bound. Class. Quant. Grav., 28:065013, 2011, arXiv:1101.4181.[27]Ahmed FaragAliAbdel NasserTawfikEffects of the generalized uncertainty principle on compact starsInt. J. Mod. Phys. D2220131350020arXiv:1301.6133Ahmed Farag Ali and Abdel Nasser Tawfik. Effects of the Generalized Uncertainty Principle on Compact Stars. Int. J. Mod. Phys. D, 22:1350020, 2013, arXiv:1301.6133.[28]MateuszBawajProbing deformed commutators with macroscopic harmonic oscillatorsNat. Commun.620157503arXiv:1411.6410Mateusz Bawaj et al. Probing deformed commutators with macroscopic harmonic oscillators. Nature Commun., 6:7503, 2015, arXiv:1411.6410.[29]ArunMathewMalay K.NandyEffect of minimal length uncertainty on the mass–radius relation of white dwarfsAnn. Phys.3932018184205arXiv:1712.03953Arun Mathew and Malay K. Nandy. Effect of minimal length uncertainty on the mass–radius relation of white dwarfs. Annals Phys., 393:184–205, 2018, arXiv:1712.03953.[30]FabrizioTamburiniFabianoFeleppaBoThidéConstraining the generalized uncertainty principle with the light twisted by rotating black holes and M87*Phys. Lett. B8262022136894arXiv:2103.13750Fabrizio Tamburini, Fabiano Feleppa, and Bo Thidé. Constraining the Generalized Uncertainty Principle with the light twisted by rotating black holes and M87*. Phys. Lett. B, 826:136894, 2022, arXiv:2103.13750.[31]AshmitaDasSauryaDasNoor R.MansourElias C.VagenasBounds on GUP parameters from GW150914 and GW190521Phys. Lett. B8192021136429arXiv:2101.03746Ashmita Das, Saurya Das, Noor R. Mansour, and Elias C. Vagenas. Bounds on GUP parameters from GW150914 and GW190521. Phys. Lett. B, 819:136429, 2021, arXiv: 2101.03746.[32]JamesDavidM.TuñacaoAdrian G.AbacRolandEmeritoS.OtadoyFinite temperature considerations in the structure of quadratic gup-modified white dwarfsInt. J. Mod. Phys. D23500492023James David M Tuñacao, Adrian G Abac, and Roland Emerito S Otadoy. Finite temperature considerations in the structure of quadratic gup-modified white dwarfs. International Journal of Modern Physics D, 2350049, 2023.[33]William M.CampbellMichael E.TobarSergeGalliouMaximGoryachevImproved constraints on the minimum length with a macroscopic low loss phonon cavityarXiv:2304.006884 2023William M. Campbell, Michael E. Tobar, Serge Galliou, and Maxim Goryachev. Improved Constraints on the Minimum Length with a Macroscopic Low Loss Phonon Cavity. 4 2023, arXiv:2304.00688.[34]AnnaPachołAnetaWojnarFermi equation of state with finite temperature corrections in quantum space-times approach: Snyder model vs GUP caseClass. Quantum Gravity40192023195021arXiv:2304.08215Anna Pachoł and Aneta Wojnar. Fermi equation of state with finite temperature corrections in quantum space-times approach: Snyder model vs GUP case. Class. Quant. Grav., 40(19):195021, 2023, arXiv:2304.08215.[35]AnnaPachołAnetaWojnarConstraining Snyder and GUP models with low-mass starsEur. Phys. J. C831220231097arXiv:2307.03520Anna Pachoł and Aneta Wojnar. Constraining Snyder and GUP models with low-mass stars. Eur. Phys. J. C, 83(12):1097, 2023, arXiv:2307.03520.[36]PasqualeBossoGiuseppe GaetanoLucianoLucianoPetruzzielloFabianWagner30 years in: quo vadis generalized uncertainty principle?arXiv:2305.161935 2023Pasquale Bosso, Giuseppe Gaetano Luciano, Luciano Petruzziello, and Fabian Wagner. 30 years in: Quo vadis generalized uncertainty principle? 5 2023, arXiv:2305.16193.[37]MaxBornReciprocity theory of elementary particlesRev. Mod. Phys.2131949463473Max Born. Reciprocity Theory of Elementary Particles. Rev. Mod. Phys., 21(3):463–473, 1949.[38]BrettBolenMarcoCavaglia(Anti-)de Sitter black hole thermodynamics and the generalized uncertainty principleGen. Relativ. Gravit.37200512551262arXiv:gr-qc/0411086Brett Bolen and Marco Cavaglia. (Anti-)de Sitter black hole thermodynamics and the generalized uncertainty principle. Gen. Rel. Grav., 37:1255–1262, 2005, arXiv:gr-qc/0411086.[39]Mu-inParkThe generalized uncertainty principle in (A)dS space and the modification of Hawking temperature from the minimal lengthPhys. Lett. B6592008698702arXiv:0709.2307Mu-in Park. The Generalized Uncertainty Principle in (A)dS Space and the Modification of Hawking Temperature from the Minimal Length. Phys. Lett. B, 659:698–702, 2008, arXiv:0709.2307.[40]S.MignemiExtended uncertainty principle and the geometry of (anti)-de Sitter spaceMod. Phys. Lett. A25201016971703arXiv:0909.1202S. Mignemi. Extended uncertainty principle and the geometry of (anti)-de Sitter space. Mod. Phys. Lett. A, 25:1697–1703, 2010, arXiv:0909.1202.[41]B.HamilM.MeradT.BirkandanApplications of the extended uncertainty principle in AdS and dS spacesEur. Phys. J. Plus13462019278B. Hamil, M. Merad, and T. Birkandan. Applications of the extended uncertainty principle in AdS and dS spaces. Eur. Phys. J. Plus, 134(6):278, 2019.[42]KouroshNozariSaraSaghafiMiladHajebrahimiExtended uncertainty principle: a deeper insight into the Hubble tension?Phys. Dark Universe462024101571arXiv:2407.01961Kourosh Nozari, Sara Saghafi, and Milad Hajebrahimi. Extended uncertainty principle: A deeper insight into the Hubble tension? Phys. Dark Univ., 46:101571, 2024, arXiv:2407.01961.[43]AchimKempfUncertainty relation in quantum mechanics with quantum group symmetryJ. Math. Phys.35199444834496arXiv:hep-th/9311147Achim Kempf. Uncertainty relation in quantum mechanics with quantum group symmetry. J. Math. Phys., 35:4483–4496, 1994, arXiv:hep-th/9311147.[44]CosimoBambiF.R.UrbanNatural extension of the generalised uncertainty principleClass. Quantum Gravity252008095006arXiv:0709.1965Cosimo Bambi and F. R. Urban. Natural extension of the Generalised Uncertainty Principle. Class. Quant. Grav., 25:095006, 2008, arXiv:0709.1965.[45]FabianWagnerReinterpreting deformed Heisenberg algebrasEur. Phys. J. C8322023154arXiv:2206.04601Fabian Wagner. Reinterpreting deformed Heisenberg algebras. Eur. Phys. J. C, 83(2):154, 2023, arXiv:2206.04601.[46]HafidaMoullaDjamilBouazizHabibAissaouiNoureddineMebarkiWeakly q-deformed Heisenberg algebra and non-Hermitian Hamiltonians: application in statistical physicsNucl. Phys. B10062024116626Hafida Moulla, Djamil Bouaziz, Habib Aissaoui, and Noureddine Mebarki. Weakly q-deformed Heisenberg algebra and non-Hermitian Hamiltonians: Application in statistical physics. Nucl. Phys. B, 1006:116626, 2024.[47]AfzalRaghaviRamazan AliMohammadianSaeedMohammadiA unified approach to the generalized uncertainty principleRep. Math. Phys.93120245769Afzal Raghavi, Ramazan Ali Mohammadian, and Saeed Mohammadi. A unified approach to the generalized uncertainty principle. Rept. Math. Phys., 93(1):57–69, 2024.[48]SebastianoSegretoGiovanniMontaniExtended GUP formulation and the role of momentum cut-offEur. Phys. J. C8352023385arXiv:2208.03101Sebastiano Segreto and Giovanni Montani. Extended GUP formulation and the role of momentum cut-off. Eur. Phys. J. C, 83(5):385, 2023, arXiv:2208.03101.[49]MichaelBishopJaeyeongLeeDouglasSingletonModified commutators are not sufficient to determine a quantum gravity minimal length scalePhys. Lett. B8022020135209arXiv:1911.04248Michael Bishop, Jaeyeong Lee, and Douglas Singleton. Modified commutators are not sufficient to determine a quantum gravity minimal length scale. Phys. Lett. B, 802:135209, 2020, arXiv:1911.04248.[50]MichaelBishopJoeyContrerasDouglasSingletonA subtle aspect of minimal lengths in the generalized uncertainty principleUniverse832022192arXiv:2203.10628Michael Bishop, Joey Contreras, and Douglas Singleton. A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle. Universe, 8(3):192, 2022, arXiv:2203.10628.[51]PasqualeBossoLucianoPetruzzielloFabianWagnerMinimal length: a cut-off in disguise?Phys. Rev. D107122023126009arXiv:2302.04564Pasquale Bosso, Luciano Petruzziello, and Fabian Wagner. Minimal length: A cut-off in disguise? Phys. Rev. D, 107(12):126009, 2023, arXiv:2302.04564.[52]SergioDoplicherKlausFredenhagenJohn E.RobertsThe quantum structure of space-time at the Planck scale and quantum fieldsCommun. Math. Phys.1721995187220arXiv:hep-th/0303037Sergio Doplicher, Klaus Fredenhagen, and John E. Roberts. The Quantum structure of space-time at the Planck scale and quantum fields. Commun. Math. Phys., 172:187–220, 1995, arXiv:hep-th/0303037.[53]S.DoplicherK.FredenhagenJ.E.RobertsSpace-time quantization induced by classical gravityPhys. Lett. B33119943944S. Doplicher, K. Fredenhagen, and J. E. Roberts. Space-time quantization induced by classical gravity. Phys. Lett. B, 331:39–44, 1994.[54]J.Kowalski-GlikmanLeeSmolinTriply special relativityPhys. Rev. D702004065020arXiv:hep-th/0406276J. Kowalski-Glikman and Lee Smolin. Triply special relativity. Phys. Rev. D, 70:065020, 2004, arXiv:hep-th/0406276.[55]C.N.YangOn quantized space-timePhys. Rev.721947874C. N. Yang. On quantized space-time. Phys. Rev., 72:874, 1947.[56]JerzyKowalski-GlikmanSebastianNowakDoubly special relativity and de Sitter spaceClass. Quantum Gravity20200347994816arXiv:hep-th/0304101Jerzy Kowalski-Glikman and Sebastian Nowak. Doubly special relativity and de Sitter space. Class. Quant. Grav., 20:4799–4816, 2003, arXiv:hep-th/0304101.[57]JerzyLukierskiStjepanMeljanacSalvatoreMignemiAnnaPachołMariuszWoronowiczFrom Snyder space-times to doubly κ-dependent Yang quantum phase spaces and their generalizationsPhys. Lett. B8542024138729arXiv:2311.16994Jerzy Lukierski, Stjepan Meljanac, Salvatore Mignemi, Anna Pachoł, and Mariusz Woronowicz. From Snyder space-times to doubly κ-dependent Yang quantum phase spaces and their generalizations. Phys. Lett. B, 854:138729, 2024, arXiv:2311.16994.[58]PouriaPedramA higher order GUP with minimal length uncertainty and maximal momentum II: applicationsPhys. Lett. B7182012638645arXiv:1210.5334Pouria Pedram. A Higher Order GUP with Minimal Length Uncertainty and Maximal Momentum II: Applications. Phys. Lett. B, 718:638–645, 2012, arXiv:1210.5334.[59]RobertoCasadioFabioScardigliGeneralized uncertainty principle, classical mechanics, and general relativityPhys. Lett. B8072020135558arXiv:2004.04076Roberto Casadio and Fabio Scardigli. Generalized Uncertainty Principle, Classical Mechanics, and General Relativity. Phys. Lett. B, 807:135558, 2020, arXiv:2004.04076.[60]Hartland S.SnyderQuantized space-timePhys. Rev.711194738Hartland S Snyder. Quantized space-time. Physical Review, 71(1):38, 1947.[61]S.MignemiClassical and quantum mechanics of the nonrelativistic Snyder model in curved spaceClass. Quantum Gravity292012215019arXiv:1110.0201S. Mignemi. Classical and quantum mechanics of the nonrelativistic Snyder model in curved space. Class. Quant. Grav., 29:215019, 2012, arXiv:1110.0201.[62]RabinBanerjeeKuldeepKumarDibakarRoychowdhurySymmetries of Snyder-de Sitter space and relativistic particle dynamicsJ. High Energy Phys.032011060arXiv:1101.2021Rabin Banerjee, Kuldeep Kumar, and Dibakar Roychowdhury. Symmetries of Snyder-de Sitter space and relativistic particle dynamics. JHEP, 03:060, 2011, arXiv:1101.2021.[63]S.MignemiClassical and quantum mechanics of the nonrelativistic Snyder modelPhys. Rev. D842011025021arXiv:1104.0490S. Mignemi. Classical and quantum mechanics of the nonrelativistic Snyder model. Phys. Rev. D, 84:025021, 2011, arXiv:1104.0490.[64]SebastianFranchino-VinasSalvatoreMignemiIf the universe were curved and noncommutativePoSCORFU20232024277arXiv:2405.08934Sebastian Franchino-Vinas and Salvatore Mignemi. If the universe were curved and noncommutative. PoS, CORFU2023:277, 2024, arXiv:2405.08934.[65]Tea MartinićBilaćStjepanMeljanacSalvatoreMignemiGeneralized triply special relativity models and their classical limitarXiv:2407.106167 2024Tea Martinić Bilać, Stjepan Meljanac, and Salvatore Mignemi. Generalized Triply Special Relativity models and their classical limit. 7 2024, arXiv:2407.10616.[66]C.QuesneV.M.TkachukLorentz-covariant deformed algebra with minimal length and application to the 1+1-dimensional Dirac oscillatorJ. Phys. A3920061090910922arXiv:quant-ph/0604118C. Quesne and V. M. Tkachuk. Lorentz-covariant deformed algebra with minimal length and application to the 1+1-dimensional Dirac oscillator. J. Phys. A, 39:10909–10922, 2006, arXiv:quant-ph/0604118.[67]C.ChryssomalakosE.OkonLinear form of 3-scale special relativity algebra and the relevance of stabilityInt. J. Mod. Phys. D13200418171822arXiv:hep-th/0407080C. Chryssomalakos and E. Okon. Linear form of 3-scale special relativity algebra and the relevance of stability. Int. J. Mod. Phys. D, 13:1817–1822, 2004, arXiv:hep-th/0407080.[68]S.MeljanacS.MignemiRealizations of the Yang–Poisson model on canonical phase spaceInt. J. Mod. Phys. A3835n3620232350182arXiv:2307.09828S. Meljanac and S. Mignemi. Realizations of the Yang–Poisson model on canonical phase space. Int. J. Mod. Phys. A, 38(35n36):2350182, 2023, arXiv:2307.09828.[69]Tea MartinićBilaćStjepanMeljanacSalvatoreMignemiGeneralized Yang Poisson models on canonical phase spaceSIGMA202024049arXiv:2403.05464Tea Martinić Bilać, Stjepan Meljanac, and Salvatore Mignemi. Generalized Yang Poisson Models on Canonical Phase Space. SIGMA, 20:049, 2024, arXiv:2403.05464.[70]K.NouicerQuantum-corrected black hole thermodynamics to all orders in the Planck lengthPhys. Lett. B64620076371arXiv:0704.1261K. Nouicer. Quantum-corrected black hole thermodynamics to all orders in the Planck length. Phys. Lett. B, 646:63–71, 2007, arXiv:0704.1261.[71]Won SangChungHassanHassanabadiA new higher order GUP: one dimensional quantum systemEur. Phys. J. C7932019213Won Sang Chung and Hassan Hassanabadi. A new higher order GUP: one dimensional quantum system. Eur. Phys. J. C, 79(3):213, 2019.[72]MatteoFadelMicheleMaggioreRevisiting the algebraic structure of the generalized uncertainty principlePhys. Rev. D105102022106017arXiv:2112.09034Matteo Fadel and Michele Maggiore. Revisiting the algebraic structure of the generalized uncertainty principle. Phys. Rev. D, 105(10):106017, 2022, arXiv:2112.09034.[73]Marco ValerioBattistiStjepanMeljanacModification of Heisenberg uncertainty relations in noncommutative Snyder space-time geometryPhys. Rev. D792009067505arXiv:0812.3755Marco Valerio Battisti and Stjepan Meljanac. Modification of Heisenberg uncertainty relations in noncommutative Snyder space-time geometry. Phys. Rev. D, 79:067505, 2009, arXiv:0812.3755.[74]S.MeljanacS.MignemiGeneralizations of Snyder model to curved spacesPhys. Lett. B8332022137289arXiv:2206.04772S. Meljanac and S. Mignemi. Generalizations of Snyder model to curved spaces. Phys. Lett. B, 833:137289, 2022, arXiv:2206.04772.