Eur. Phys. J. C (2019) 79:300 https://doi.org/10.1140/epjc/s10052-019-6794-4

Regular Article - Theoretical Physics

Energy-dependent noncommutative quantum mechanics

Tiberiu Harko1,2,3,a, Shi-Dong Liang1,4,b 1 School of Physics, Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China 2 Department of Physics, Babes-Bolyai University, Kogalniceanu Street, 400084 Cluj-Napoca, Romania 3 Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK 4 State Key Laboratory of Optoelectronic Material and Technology, and Guangdong Province Key Laboratory of Display Material and Technology, Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China

Received: 15 January 2019 / Accepted: 15 March 2019 / Published online: 3 April 2019 © The Author(s) 2019

Abstract We propose a model of dynamical noncommu- 3.1 Probability current and density in the energy- tative quantum mechanics in which the noncommutative dependent potential ...... 7 strengths, describing the properties of the commutation rela- 3.2 The free particle ...... 8 tions of the coordinate and momenta, respectively, are arbi- 3.3 The harmonic oscillator ...... 8 trary energy-dependent functions. The Schrödinger equation 4 Physical mechanisms generating energy-dependent in the energy-dependent noncommutative algebra is derived noncommutative algebras, and their implications .. 9 for a two-dimensional system for an arbitrary potential. The 4.1 The noncommutative algebra of the spacetime resulting equation reduces in the small energy limit to the quantum fluctuations (SQF) model ...... 9 standard quantum mechanical one, while for large energies 4.2 The noncommutative algebra of the energy cou- the effects of the noncommutativity become important. We pling (EC) model ...... 10 investigate in detail three cases, in which the noncommu- 4.3 The noncommutative algebra of the energy oper- tative strengths are determined by an independent energy ator (EO) model ...... 10 scale, related to the vacuum quantum fluctuations, by the 5 Quantum evolution in the spacetime quantum fluctuation particle energy, and by a quantum operator representation, (SQF) energy-dependent noncommutative model ..... 11 respectively. Specifically, in our study we assume an arbi- 5.1 Quantum mechanics of the free particle in the trary power-law energy dependence of the noncommutative SQF model ...... 11 strength parameters, and of their algebra. In this case, in the 5.2 The harmonic oscillator ...... 12 quantum operator representation, the Schrö dinger equation 6 Quantum dynamics in the energy coupling model .. 12 can be formulated mathematically as a fractional differential 6.1 Quantum evolution: the Schrödinger equation .12 equation. For all our three models we analyze the quantum 6.2 Quantum evolution: wave function and energy levels .... 13 evolution of the free particle, and of the harmonic oscillator, 6.3 The case of the free particle ...... 14 respectively. The general solutions of the noncommutative 6.4 The harmonic oscillator ...... 15 Schrödinger equation as well as the expressions of the energy 7 Quantum evolution in the energy operator (EO) levels are explicitly obtained. energy-dependent noncommutative geometry .... 16 7.1 Fractional calculus: a brief review ...... 16 7.2 The fractional Schrödinger equation ...... 17 α = Contents 7.3 The free particle: the case 1 ...... 18 8 Discussions and final remarks ...... 18 1 Introduction ...... 1 References ...... 21 2 Energy-dependent noncommutative geometry and algebra ... 5 3 The Schrödinger equation in the energy-dependent noncommutative geometry ...... 7 1 Introduction

It is generally believed today that the description of the space- a e-mail: [email protected] time as a manifold M, locally modeled as a flat Minkowski b 3 e-mail: [email protected] space M0 = R × R , may break down at very short dis- 123 300 Page 2 of 22 Eur. Phys. J. C (2019) 79 :300  3   2 tances of the order of the Planck length lP = Gh¯ /c ≈ a − xμ,xν = i Lμν, (2) 1.6 × 10 33 cm, where G, h¯ and c are the gravitational con- h¯ stant, Planck’s constant, and the speed of light, respectively [1]. This assumption is substantiated by a number of argu- where a is a basic length unit, and Lμν are the generators ments, following from quantum mechanical and general rela- of the Lorentz group. In this approach, Lorentz covariance tivistic considerations, which point towards the impossibility is maintained, but the translational invariance is lost. A rig- of an arbitrarily precise location of a physical particle in terms orous mathematical approach to noncommutative geometry of points in spacetime. was introduced in [8–11], by generalizing the notion of a ∗ One of the basic principles of quantum mechanics, Heisen- differential structure to arbitrary C algebras, as well as to berg’s uncertainty principle, requires that a localization x quantum groups and matrix pseudo-groups. This approach in spacetime can be reached by a momentum transfer of the led to an operator algebraic description of noncommutative order of p = h¯ /x, and an energy of the order of E = spacetimes, based entirely on algebras of functions. hc¯ /x [2–4]. On the other hand, the energy E must contain Since at the quantum level noncommutative spacetimes amassm, which, according to Einstein’s general theory do appear naturally when gravitational effects are taken into of relativity, generates a gravitational field. If this gravita- account, their existence must also follow from string theory. tional field is so strong that it can completely screen out from In [12] it was shown that if open strings have allowed end- observations some regions of spacetime, then its size must points on D-branes in a constant B-field background, then be of the order of its Schwarzschild radius R ≈ Gm/c2. the endpoints live on a noncommutative space with the com- Hence we easily find R ≈ GE/c4 = Gh¯ /c3x,giving mutation relations Rx ≈ Gh¯ /c3. Thus the Planck length appears to give the [μ,ν]= θ μν, lower quantum mechanically limit of the accuracy of position x x i (3) measurements [5]. Therefore the combination of the Heisen- θ μν berg uncertainty principle with Einstein’s theory of general where is an antisymmetric constant matrix, with com- −2 relativity leads to the conclusion that at short distances the ponents c-numbers with the dimensionality (length) .More standard concept of space and time may lose any operational generally, a similar relation can also be imposed on the par- meaning. ticle momenta, which generates a noncommutative algebra On the other hand, the very existence of the Planck length in the momentum space of the form requires that the mathematical concepts for high-energy    ,  = η , (short distance) physics have to be modified. This follows pμ pν i μν (4) from the fact that classical geometrical notions and con- η cepts may not be well suited for the description of physical where μν are constants. In [13] noncommutative field phenomena at very short distances. Moreover, some drastic theories with commutator of the coordinates of the form μ, ν = μν ω  changes are expected in the physics near the Planck scale, [x x ] i ωx have been studied. By considering a with one important and intriguing effect being the emer- Lorentz tensor, explicit Lorentz invariance is maintained, a gence of the noncommutative structure of the spacetime. The free quantum field theory is not affected. On the other hand, basic idea behind spacetime noncommutativity is very much since invariance under translations is broken, the conserva- inspired by quantum mechanics. A quantum phase space is tion of energy–momentum tensor is violated, and a new law defined by replacing canonical position and momentum vari- expressed by a Poincaré-invariant equation is obtained. The λφ4 ables xμ, pν with Hermitian operators that obey the Heisen- quantum field theory was also considered. It turns out berg commutation relations, that the usual UV divergent terms are still present in this model. Moreover, new type of terms also emerge that are μ ν μν [x , p ]=ih¯ δ . (1) IR divergent, violates momentum conservation and lead to corrections to the dispersion relations. Hence the phase space becomes smeared out, and the The physical implications and the mathematical proper- notion of a point is replaced with that of a Planck cell. The ties of the noncommutative geometry have been extensively  ,  = generalization of commutation relations for the canonical investigated in [14–42]. In the case when pi p j 0, the operators (coordinate-momentum or creation-annihilation noncommutative quantum mechanics goes into the usual one, operators) to non-trivial commutation relations for the coor- described by the non-relativistic Schrödinger equation, dinate operators was performed in [6,7], where it was first ( ˜, ) ψ ( ˜) = ψ ( ˜) , suggested that the coordinates xμ may be noncommutating H x p x E x (5) operators, with the six commutators being given by

123 Eur. Phys. J. C (2019) 79 :300 Page 3 of 22 300

μ μ μν where x˜ = x − (1/2)θ pν [43]. In the presence of a properties of systems with dynamic noncommutativity were constant magnetic field B and an arbitrary central potential considered in [51–58]. V (r), with Hamiltonian A quantum mechanical system on a noncommutative space for which the structure constant is explicitly time- p2 dependent was investigated in [59], in a two-dimensional H = + V (r), (6) 2m space with nonvanishing commutators for the coordinates X, Y and momenta Px , Py given by the operators p, x obey the commutation relations [43]   [ , ]= θ( ), , = ( ),     X Y i t Px Py i t (11a) 1 2 μ μ e   x ,x = iθ, x , pν = ih¯ δν , [p1, p2] = i B. (7) θ(t)(t) c [X, Px ] = Y, Py = ih¯ + i . (11b) 4h¯ Several other types of noncommutativity, extending the Any autonomous Hamiltonian on such a space acquires a canonical one, have also been proposed. For example, in [44], time-dependent form in terms of the conventional canoni- a three-dimensional noncommutative quantum mechanical cal variables. A generalized version of Heisenberg’s uncer- system with mixing spatial and spin degrees of freedom was tainty relations for which the lower bound becomes a investigated. In this study it was assumed that the noncom- time-dependent function of the background fields was also mutative spatial coordinates xi , the conjugate momenta pi , obtained. For a two-dimensional harmonic oscillator, after and the spin variables si obey the nonstandard Heisenberg performing the Bopp shift, the Hamiltonian becomes times algebra     dependent, and t is given by [59]    xi ,x j = iθ 2 ijksk, xi , p = iδi , p , p = 0, (8) j j i j ( ) = 1 2 + 2 H t px py 2me(t) and ( )      ke t 2 2 + x + y + Be(t)Lz, (12) xi ,s j = iθ ijksk, si ,s j = i ijksk, (9) 2 where respectively, where θ ∈ R is the parameter of the noncom- 1 1 mω2 2(t) mutativity. A classical model of spin noncommutativity was = + θ 2(t), k (t) = mω2 + , (13a) ( ) 2 e 2 investigated in [45]. In the nonrelativistic case, the Poisson me t m 4h¯ 4mh¯ brackets between the coordinates are proportional to the spin mω2θ(t) (t) Be = + , Lz = px y − xpy . (13b) angular momentum. The quantization of the model leads to 2h¯ 2hm¯ the noncommutativity with mixed spatial and spin degrees From a general physical point of view we can interpret the μν of freedom. A modified Pauli equation, describing a spin noncommutativity parameters θ and ημν as describing the half particle in an external electromagnetic field was also strength of the noncommutative effects exerted in an inter- obtained, and it was shown that in spite of the presence action. In this sense they are the analogues of the coupling of noncommutativity and nonlocality, the model is Lorentz constants in standard quantum field theory. invariant. Other physical and mathematical implications of It is a fundamental assumption in quantum field theory that spin noncommutativity were investigated in [46–48] the properties of a physical system (including the underlying A model of dynamic position-dependent noncommutativ- force laws) change when viewed at different distance scales, ity, involving the complete algebra of noncommutative coor- and these changes are energy dependent. This is the fun- dinates damental idea of the renormalization group method, which   has found fundamental applications in quantum field theory, μ ν μν x ,x = iω (x) , (10) elementary particle physics, condensed matter etc. [60]. It is the main goal of the present paper to introduce was proposed in [49], and further investigated in [50]. In [50] and analyze a dynamic noncommutative model of quan- a system consisting of two interrelated parts was analyzed. tum mechanics, in which the noncommutative strengths θ μν The first describes the physical degrees of freedom with the and ημν are energy-dependent quantities. This would imply coordinates x1 and x2, while the second corresponds to the the existence of several noncommutative scales that range noncommutativity η, which has a proper dynamics. It turns from the energy level of the standard model, where the low- out that after quantization, the commutator of two physical energy scales of the physical systems reduce the general coordinates is proportional to an arbitrary function of η.An noncommutative algebra to the standard Heisenberg alge- interesting feature of this model is the dependence of non- bra, and ordinary quantum mechanics, to the Planck energy locality on the energy of the system, so that the increase of scale. On energy scales of the order of the Planck energy, 5 19 the energy leads to the increase in nonlocality. The physical E P = hc¯ /G ≈ 1.22 × 10 GeV, the noncommutative 123 300 Page 4 of 22 Eur. Phys. J. C (2019) 79 :300 effects become maximal. Under the assumption of the energy functions are determined. The particle oscillation frequen- dependence of the noncommutative parameters, with the help cies are either dependent on the vacuum fluctuation energy of the generalized Seiberg–Witten map, we obtain the gen- scale, or they have an explicit dependence on the particle eral form of the Schrödinger equation describing the quantum energy. In the limiting case of small energies we recover the evolution in an energy-dependent geometry. The noncommu- standard results of quantum mechanics. tative effects can be included in the equation via a generalized As a simple application of the developed general for- quantum potential, which contains an effective (analogue) malism we consider the case in which the noncommuta- magnetic field, and an effective elastic constant, whose func- tive strengths η and θ are power-law functions of energy, tional forms are determined by the energy-dependent non- with arbitrary exponents. However, the quantization of such commutative strengths. systems, in which we associate an operator to the energy, The possibility of an energy-dependent Schrödinger equa- requires the mathematical/physical interpretation of opera- tion was first suggested by Pauli [61], and it was further tors of the form ∂α/∂tα, where α can have arbitrary real) considered and investigated extensively (see [62–67] and the values, like, for example, α = 1/2, α = 5/4 etc. These types references therein). Generally, the nonlinearity induced by of problems belong to the field of fractional calculus [71–73], the energy dependence requires modifications of the stan- whose physical applications have been intensively investi- dard rules of quantum mechanics [62]. In the case of a linear gated. In particular, the mathematical and physical properties energy dependence of the potential for confining potentials of the fractional Schrö dinger equation, whose introduction the saturation of the spectrum is observed, which implies was based on a purely phenomenological or abstract mathe- that with the increase of the quantum numbers the eigen- matical approach, have been considered in detail in [74–93]. values reach an upper limit [65]. The energy-dependent It is interesting to note that the present approach gives a phys- Schrödinger equation was applied to the description of heavy ical foundation for the mathematical use of fractional deriva- quark systems in [63], where for a linear energy dependence tives in quantum mechanics, as resulting from the noncom- the harmonic oscillator was studied as an example of a sys- mutative and energy-dependent structure of the spacetime. tem admitting analytical solutions. A new quark interaction We present in full detail the fractional Schrödinger equations was derived in [64], by means of a Tamm–Dancoff reduc- obtained by using two distinct quantization of the energy (the tion, from an effective field theory constituent quark model. time operator and the Hamiltonian operator approach, respec- The obtained interaction is nonlocal and energy dependent. tively), and we investigate the quantum evolution of the free Moreover, it becomes positive and rises up to a maximum particle and of the harmonic oscillator in the time operator value when the interquark distance increases. The quan- formalism for a particular simple choice of the energy depen- tum mechanical formalism for systems featuring energy- dent noncommutativity strength parameters. dependent potentials was extended to systems described by The present paper is organized as follows. We introduce generalized Schrödinger equations that include a position- the energy-dependent noncommutative quantum geometry, dependent mass in [67]. Modifications of the probability its corresponding algebra, and the Seiberg–Witten map that density and of the probability current need the adjustments allows one to construct the noncommutative set of variables in the scalar product and the norm. The obtained results from the commutative ones in Sect. 2. The Schrö dinger have been applied to the energy-dependent modifications of equation describing the quantum evolution in the energy- the Mathews–Lakshmanan oscillator, and to the generalized dependent noncommutative geometry is obtained in Sect. 3, Swanson system. where the form of the effective potential induced by the non- From a physical point of view we can assume that the commutative effects is also obtained. Three relevant physi- energy dependent noncommutative effects can be described cal and mathematical mechanisms that could induce energy- by two distinct energy scales. One is the energy scale of dependent quantum behaviors in noncommutative geometry the spacetime quantum fluctuations, generated by the vac- are discussed in Sect. 4, and their properties are explored uum background and the zero point energy of the quantum in the framework of a particular model in which the non- fields. The noncommutativity is then essentially determined commutativity parameters have a power-law dependence on by this energy scale, which is independent of the particle energy. The quantum dynamics of a free particle and of the energy. This is the first explicit model we are considering, a harmonic oscillator in the spacetime quantum fluctuations two energy scales model, in which the energy of the quan- model is analyzed in Sect. 5, while the same physical sys- tum fluctuations and the particle energy evolve in different tems are analyzed in the energy coupling model in Sect. 6. and independent ways. The alternative possibility, in which The fractional Schrödinger equations for the quantum evo- the noncommutative strengths are dependent on the particle lution of general quantum systems in the energy operator energy only, is also investigated. We consider the quantum approach are presented in Sect. 7, where the dynamics of the evolution of the free particle and of the harmonic oscillator free particles and of the harmonic oscillator are analyzed in in these cases, and the resulting energy spectrum and wave 123 Eur. Phys. J. C (2019) 79 :300 Page 5 of 22 300 detail. A brief review of the fractional calculus is also pre- In the following we will not consider time-like noncom- sented. We discuss and conclude our results in Sect. 8. mutative relations, that is, we take θ 0i = 0 and η0i = 0, since otherwise the corresponding quantum field theory is not unitary. 2 Energy-dependent noncommutative geometry and The parameters θ and η from Eqs. (14a)–(14c) can be algebra represented generally as  0 θ(E) In the present section we will introduce the basic definitions, θij = , (17a) −θ(E) 0 conventions and relations for an energy-dependent general-  0 η(E) ization of the noncommutative geometry and algebra of phys- ηij = , (17b) −η(E) 0 ical variables, valid in the high-energy/small distance regime.   In high-energy physics theoretical models where both the  γ 2 + θ(E)η(E) θ(E)η(E) γ 2 2 γ 2 2 ij = 2 h¯ 4 h¯ , (17c) coordinate and momentum space noncommutativity is taken θ(E)η(E) γ 2 + θ(E)η(E) into account, in a four-dimensional space the coordinates and 4γ 2h¯ 2 2γ 2h¯ 2 momenta satisfy the following algebra: where i, j correspond to x, y and z. It can be seen that θ and η μ ν μν  [x ,x ]=iθ , (14a) are antisymmetric, but is symmetric. They are assumed to [μ, ν]= ημν, be energy-dependent, and we take them as independent of the p p i (14b) γ = μ ν μν spacetime coordinates. We may set 1 for convenience [x , p ]=ih¯  , (14c) without losing the basic physics in the following section. − where the noncommutativity strength parameters θ μν and Moreover, we also limit our analysis to the x y ημν are antisymmetric. Due to the commutation relation plane, where the two-dimensional noncommutative energy- given by Eq. (14c), this algebra is consistent with ordinary dependent algebra can be formulated as     Quantum Mechanics. We shall assume in the following that x,y = iθ(E), p , p = iη(E), x , p = ih¯ δ , μν μν [ ] x y i j ij the two matrices θ and η are invertible, and moreover μν μν 2 μα ν (18a) the matrix  = δ + 1/h¯ θ ηα is also invertible [26]. , = , ,  = , = , , Under a linear transformation of the form [E xi ] 0 [E pi ] 0 i 1 2 (18b)

μ μ ν μ ν μ μ ν μ ν where, in the last three commutation relations, we have x = Aν x + Bν p , p = Cν x + Dν p , (14d) denoted x1 ≡ x, x2 ≡ y, p1 ≡ px and p2 ≡ py. Starting from the canonical quantum mechanical Heisen- D also called the map, where A, B, C, and D are real con- berg commutation relations one can easily verify that the stant matrices, the noncommutative algebra (14a)–(14c) can commutation relations Eq. (18a) can be obtained through the xμ, xν = be mapped to the usual Heisenberg algebra, [ ] 0, linear transformations [12,21] [pμ, pν] = 0, and [xμ, pν] = ih¯ δμν, respectively [26]. The θ( ) T − T = x x − E p matrices A, B, C, and D satisfy the equations AD BC = h¯ y , (19a) I × ,ABT − BAT = /h¯ , and CDT − DCT = N/h¯ [26], y y d d   where  and N are matrices with the entries θ μν and ημν, px px respectively [26]. Due to the linear transformations (14d), the = η( ) , (19b) p p − E x noncommutative algebra (14a)–(14c) admits a Hilbert space y y h¯ representation of ordinary quantum mechanics. However, it or, equivalently, through the alternative set of linear transfor- is important to note that the D map is not unique. mations In the present paper we generalize the algebra given by μν x x Eqs. (14a)–(14c) to the case when the parameters θ and = θ( ) , (20a) y + E ημν y h¯ px are energy-dependent functions, so that    + η(E) px = px h¯ y . θ μν = θ μν( )  (20b) E (15) py py and These two types of linear transformations can be combined into a single one, which simultaneously modifies all coordi- μν μν η = η (E), (16) nates and momenta, and not just x and py or y and px ,as given in Eqs. (19a), (19b) and (20a), (20b), respectively. respectively, where E is a general energy parameter whose One possible way of implementing the algebra defined by physical interpretation depends on the concrete physical Eqs. (18a) and (18b) is to construct the noncommutative set problem under consideration. of variables x,y, px , py from the commutative variables 123 300 Page 6 of 22 Eur. Phys. J. C (2019) 79 :300

x, y, px , py by means of linear transformations. This can Moreover, it also turns out that the commutator of the coor- be generally done by using the Seiberg–Witten map,given dinate and momentum operators, [xμ, pν], is not diagonal by [12,21] any longer, with the off-diagonal elements obtained as the   μν μν θ( ) algebraic products of the components of θ and η .  x − E p x = 2h¯ y , The linear transformations (26) can be further generalized  θ(E) (21a) y y + px  2h¯  to the form [23] η( )  p + E y px = x 2h¯ , μ μ η(E) (21b) μ μ θν ν μ μ ην ν py p − x x = ξ x − p , p = ξ p + x , (29) y 2h¯ ¯ ¯ 2h 2h where the canonical variables x, y, px , py satisfy Heisen- ξ berg commutation relations [12,21], where is a scaling factor. It corresponds to a scale transfor- mation of the coordinates and of the momenta [23]. Such a [x, y]=[p , p ]=0, (22) x y scaling can be used to make the Planck constant a true con- 2 −1/2 [xi , p j ]=ih¯ δij, i = 1, 2, (23) stant. Indeed, by choosing ξ = 1 + θη/4h¯ , we obtain the two-dimensional noncommutative algebra given by [23] With the help of transformations (21a) and (21b) we can   immediately recover the two first commutation relations in [x,y] = iξ 2θ = iθ , p , p = iξ 2η = iη , eff x y eff Eq. (18a). However, the last one takes the form   θη x , p = h¯ ξ 2 1 + δ = ih¯ δ = ih¯ δ ,   i j ¯ 2 ij eff ij ij θ(E)η(E) 4h [xi , p j ]=ih¯ 1 + δij, i = 1, 2 . (24) 4h¯ 2 where we have denoted θ η θ = ,η = , Comparing Eqs. (18a) and (24), we find that the linear eff θη eff θη 1 + 1 + transformations given by Eqs. (21a) and (21b) generate an 4h¯ 2 4h¯ 2 θη effective energy-dependent Planck constant, which is a func- 2 h¯ eff = h¯ ξ 1 + = h¯ . tion of the noncommutativity parameters θ(E) and η(E), and 4h¯ 2 it is given by [21] Hence by a simple rescaling of the noncommutativity param- eters one can ensure the constancy of the Planck constant. h¯ = h¯ [1 + ζ(E)] , (25) eff On the√ other hand, in [21] it was shown that by assuming that θ, giving the fundamental length scale in noncom- ζ ≡ θ( )η( )/ ¯ 2 where E E 4h . This approach is consistent with mutative geometry, is smaller than the average neutron size, the usual commutative spacetime quantum mechanics if we having an order of magnitude of around 1 fm, it follows that ξ  impose the condition 1, expected to be generally satis- (h¯ h¯ ) /h¯ ≤ O (10). Hence, in practical calculations one θ η ζ eff fied, since the small noncommutative parameters and , can ignore the deviations between the numerical values of are of second order. the effective Planck constant and the usual Planck constant. For the sake of completeness we also present the general In the two-dimensional case, which we will investigate case. In the four-dimensional spacetime equations (21a) and next, in order to convert a commutative Hamiltonian into a (21b) can be written as [21] noncommutative one, we first find the inverse of the trans-  μ  formations given by Eqs. (21a) and (21b). This set is given μ μ − θ ν (E) ν x x 2h¯ p by [21] μ = μ . (26) p μ + η ν (E) ν   p 2h¯ x θ( ) x x + E p = k(E) 2h¯ y , (30a) y − θ(E)  Therefore we obtain the following four-dimensional com- y 2h¯ px   mutation relations:  − η(E) px px 2h¯ y μ ν μν μ ν μν = k(E) , (30b) [x ,x ]=iθ (E), [p , p ]=iη (E), (27a) p  + η(E)   y py 2h¯ x μ ν μν 1 μα ν [x , p ]=ih¯ δ + θ (E)η α(E) . (27b) where k(E) is obtained: 4h¯ 2 Hence it follows that in the four-dimensional case the effec- 1 θ(E)η(E) k(E) = ≈ 1 + . (31) tive energy-dependent Planck constant is given by [21] 1 − θ(E)η(E)/4h¯ 2 4h¯ 2    1 In the following we will approximate k(E) as being one, h¯eff = h¯ 1 + Tr θ(E)η(E) . (28) 4h¯ 2 k(E) ≈ 1. 123 Eur. Phys. J. C (2019) 79 :300 Page 7 of 22 300

3 The Schrödinger equation in the energy-dependent 3.1 Probability current and density in the energy-dependent noncommutative geometry potential

In order to develop some basic physics applications in the One of the interesting properties of the energy-dependent energy-dependent noncommutative quantum mechanics, as Schrödinger equation is that it leads to modified versions a first step we investigate a 2-D noncommutative quantum of the probability density and of the probability current [62– system by using the map between the energy-dependent non- 64]. This also implies modifications in the scalar product and commutative algebra and the Heisenberg algebra (21a) and the norm of the vectors in the Hilbert space. To investigate (21b). In this approach the Hamiltonian H of a particle in an the nature of these modifications we will consider the one- exterior potential V can be obtained: dimensional Schrödinger equation in an energy-dependent   potential V = V (x, y, E), which is given by  2  2  1 η(E) η(E) H = px − y + py + x 2m 2h¯ 2h¯     ∂(x, y, t) h¯ 2 θ( ) θ( ) ih¯ = − 2 + V (x, y, E) (x, y, t), E E ∂t 2m +V x + py , y − px . (32) 2h¯ 2h¯ (38) Equivalently, the two-dimensional Hamiltonian (32) can  2 2 be written as  =∇ ·∇ = ∂ + ∂ ∇ = ∂ i + ∂ j where 2 2 2 ∂x2 ∂y2 , with 2 ∂x ∂y .  E E 1 η(E) η2(E) Let us now consider two solutions of energy and of H = p2 + p2 + xp − yp + x y ¯ y x 2 the above Schrödinger equation, given by 2m  2mh 8mh¯   θ( ) θ( ) 2 2 E E − i (E−i )t × x + y + V x + py , y − px .  ( , , ) = h¯ ( , ), 2h¯ 2h¯ x y t e x y (39) − i (E −i )t (33)  (x, y, t) = e h¯ (x, y), (40)

Consequently, we obtain the generalized Schrödinger where is a small parameter, → 0. Then from the equation in the noncommutative geometry with energy- Schrödinger equation (38) we obtain the continuity equation: dependent strengths: ∂ρ +∇·J = 0, (41) ∂ ∂t ih¯ (x, y, t) = H(x, y, t), (34) ∂t where where the total Hamiltonian H can be written as ∗ ρ =  (x, y, t)  (x, y, t) + ρa (x, y, t) , (42)  2  H = H0 + Veff, (35) h¯ ∗ J =−  (x, y, t) ∇2 (x, y, t) 2im  with ∗ − (x, t) ∇2 (x, t) , (43)  1 2 2 H0 = p + p , (36) ρ 2m x y and a is obtained as a solution of the equation

  is the standard quantum mechanical kinetic energy in the ∂ i ∗ ρa =  (x, y, t) V (x, y, E) − V x, y, E  (x, y, t) . Heisenberg representation. In general, the effective potential ∂t h¯ Veff in Eq. (35) is given by (44)

η( ) η2( )  E E 2 2 V (x, y, px , py) = xpy − ypx + x + y By taking into account the explicit form of the wave func- eff 2mh¯ ¯ 2  8 mh  tions as given in Eqs. (39) and (40), after integration and θ(E) θ(E) +V x + py , y − px ; taking the limit → 0, we obtain for ρa the expression 2h¯ 2h¯ (37)   V x, y, E − V (x, y, E) ρ ( , ) =−∗( , ) ( , ). a x y x y x y it comes from both of the kinetic energy and the potential in E − E the noncommutative algebra. (45) 123 300 Page 8 of 22 Eur. Phys. J. C (2019) 79 :300

By considering limit E → E it follows that for the energy- can be interpreted as an effective magnetic field, while dependent wave function its norm (scalar product) in the Hilbert space is defined as [62] η2( ) ( ) = E , ke E 2 (52)  +∞   8mh¯ ∗ ∂V (x, y, E) N =  (x, y) 1 − (x, y)dxdy > 0. −∞ ∂ E is the effective elastic constant corresponding to a harmonic (46) oscillator. Hence the effective potential for free particles induced by the noncommutative algebra can be interpreted as If we specify the stationary states by their quantum num- generating two distinct physical processes: an effective mag- bers n, it follows that the orthogonality relation between two netic field and an effective harmonic oscillator, respectively. states n and n , n = n , is given by  3.3 The harmonic oscillator ∗  ( , ) − ϕ ( , )  ( , ) = , n x y [1 n n x y ] n x y dxdy 0 (47) As a second example of quantum evolution in the noncom- where mutative geometry with energy-dependent noncommutative strengths let us consider the case of the two-dimensional V (x, y, En ) − V (x, y, En) quantum harmonic oscillator. The potential energy is writ- ϕn n (x, y) = . (48) (En − En) ten as  In the case of the energy-dependent quantum mechanical (,) = 1 2 + 2 ,  , V x y k x y (53) systems the standard completeness relation n n x y 2 ∗ ( , ) = δ − δ − n x y x x y y does not hold gener- ally. This is a consequence of the fact that the func- where k is a constant. By using the 2D Seiberg–Witten map tions n(x, y) do not represent eigenfunctions of the same as given by Eq. (21a), the potential can be expressed as (linear self-adjoint) operator on L2(−∞, +∞). An alter-   θ(E) θ(E) native procedure was proposed in [62], and it is given V x + p , y − p ∗ y x  , − φ ( , )  ( , ) = δ − δ 2h¯ 2h¯ by n n x y [1 nn x y ] n x y x x   − θ( ) θ 2( )  y y . Finally, we would like to mention that due to k E E 2 2 = V (x, y) + − Lz + p + p , the presence of the energy eigenvalue in the Hamiltonian 2 h¯ 4h¯ 2 x y the commutator [H, x] is obtained as [H, x] =−ihp¯ + (54) [∂V (H, x) /∂ H] (∂ H/∂p), and similarly for the coordinate ( , ) = 1 2 + 2 y. where V x y 2 k x y is the potential energy of the harmonic oscillator in Heisenberg’s representation. The 3.2 The free particle Hamiltonian can be written as  For free particles, V (x,y) = 0, and the effective Hamiltonian  = 1 2 + 2 − + 1 ( 2 + 2), H ∗ px py Bh Lz Kh x y (55) takes the form 2m 2   1 2 2 where H = p + p + V x, y, px , py , (49) 2m x y eff 1 = 1 + k θ 2( ), ∗ E (56a) where the effective potential comes from the kinetic energy m m 4h¯ 2 term only via the Seiberg–Witten map, and it is given by kθ(E) η(E) kθ(E) Bh(E) = Be(E) + = + , (56b) 2h¯ 2mh¯ 2h¯ η( ) η2( ) E E 2 2 η2( ) Veff x, y, px , py =− Lz + (x + y ) E 2mh¯ ¯ 2 Kh(E) = k + ke(E) = k + . (56c) 8mh ¯ 2 ( ) 8mh ke E 2 2 ≡−Be(E)Lz + (x + y ), (50) ∗ 2 In the above equations m is the effective mass of the oscil- lator, including the modifications of the harmonic potential where Lz = (xpy − ypx ) is the z-component of the angular momentum, due to the noncommutative algebra, Bh is the effective mag- netic field, in which the first term comes from the kinetic η(E) energy and the second term comes from the harmonic poten- Be(E) = , (51) 2mh¯ tial energy in the noncommutative algebra, while Kh is the 123 Eur. Phys. J. C (2019) 79 :300 Page 9 of 22 300 effective elastic constant, in which the second term comes 4.1 The noncommutative algebra of the spacetime quantum from the noncommutative algebra. For free particle, namely fluctuations (SQF) model ∗ V (x,y) = 0, m = m, Bh = Be(E), and Kh = ke(E).The Hamiltonian (55)givesa unified description of the quan- Let us consider first there exists an intrinsic and universal tum evolution for both the free particle and for the harmonic energy scale ε inducing the noncommutative algebra. This oscillator in the energy-dependent noncommutative geome- energy scale is different and independent from the particle try. By taking k = 0 we immediately obtain the case of the energy E. Hence in this approach we are dealing with a model free particle. with two distinct energy scales. For the sake of concreteness we assume that the noncommutativity parameters η and θ have a power-law dependence on the intrinsic energy scale ε 4 Physical mechanisms generating energy-dependent , so that noncommutative algebras, and their implications

ε α ε β η(ε) = η ,θ(ε)= θ , The idea of the energy-dependent noncommutative geome- 0 0 (57) ε0 ε0 try and its underlying algebra must be supplemented by the description of different physical processes that could lead to η θ α β such mathematical structures. In the following we propose with 0, 0, , are parameters describing the strength of several possible mechanisms that could generate quantum the energy-dependent noncommutative effects. The energy ε energy-dependent behaviors described by the corresponding parameter 0 describes the basic energy scale, which is noncommutative algebra. related to the spacetime quantum fluctuation or Planck scale. This energy-dependent noncommutative mechanism is called the spacetime quantum fluctuation (SQF) process. • We assume first that there exists an intrinsic and univer- When ε  ε , both η(ε) and θ(ε) tend to zero, and thus ε 0 sal energy scale , different of the particle energy scale we recover the canonical quantum mechanics. When ε ≈ E, which induces the noncommutative effects, and the ε0, we reach the opposite limit of noncommutative quan- corresponding algebra. This intrinsic universal energy tum mechanics with constant noncommutative parameters. scale could be related to the spacetime quantum fluc- Thus, the basic physical parameters describing of effects tuations (SQF), and to the Planck energy scale, respec- of the noncommutativity in the effective potential of the tively. Therefore the energy-dependence in the commu- Schrödinger equation in the framework of the power-law

tation relations is determined by the energy scale, or energy-dependent strength noncommutative algebra Eq. (50) by the magnitude of the quantum fluctuations. Hence in become this approach the dynamics of the quantum particle is α α determined by two independent energy scales. η0 ε ε • Be(E) → Bε = ≡ B0 , (58a) For the second mechanism we assume that there is an 2mh¯ ε ε 0 0 energy coupling (EC) between the noncommutative evo- η2 ε 2α ε 2α 0 k0 lution, and the dynamical energy E of the quantum sys- k (E) → kε = ≡ , (58b) e 2 ε ε tems. Hence in this approach the interaction between the 8mh¯ 0 2 0 particle dynamics and the spacetime fluctuations is fully η η2 determined by the particle energy, and all physical pro- where B = 0 and k = 0 . For the harmonic oscillator 0 2mh¯ 0 4mh¯ 2 cesses related to the energy-dependent noncommutativity we obtain are described in terms of the particle energy scale E. • 2β Finally, the third mechanism we are going to consider 1 = 1 + k θ 2 , ∗ 2 0 (58c) follows from the possibility that the energy of a quan- m m 4h¯ 0 tum system can be mapped to an energy operator, which α β η0 k modifies the Hamiltonian of the system, and the corre- Bh ( ) = + θ , (58d) 2mh¯ 2h¯ 0 sponding Schrödinger equation. This approach we call 0 0 the EO (energy operator) approach; it assumes again that and the dominant energy scale describing noncommutative effects is the particle energy scale, E. η2 2α K ( ) = k + 0 , (58e) h 2 In the following we will consider in detail the mathemati- 8mh¯ 0 cal formulations of the above physical mechanisms, and their physical implications. respectively. 123 300 Page 10 of 22 Eur. Phys. J. C (2019) 79 :300

4.2 The noncommutative algebra of the energy coupling where

(EC) model α β I ih¯ I ih¯ ηα = η0 ,θβ = θ0 . (62) In our second model we assume that there is a coupling ε0 ε0 between the energy-dependent noncommutative geometry, and the energy of the quantum dynamical systems, and that The effective magnetic field and elastic constant in Eq. (61) this coupling can be described in terms of the particle energy for Case I are represented by operators that can be expressed in terms of fractional derivative as [71–73] E only. By adopting again a power-law dependence of the η θ α ∂ α noncommutativity parameters and on the particle energy ih¯ I α B (E) → B ≡ Bα D , (63a) E we have e 0 ε ∂t t 0 2α 2α I α β k ih¯ ∂ kα E E k (E) → 0 ≡ Dα, η(E) = η ,θ(E) = θ , (59) e ε ∂ t (63b) 0 E 0 E 2 0 t 2 0 0   α 2α α I ih¯ I ih¯ α ∂ where Bα ≡ B0 ε , kα ≡ k0 ε and D ≡ ∂ . with η0, θ0, α, β are parameters describing the strength of the 0 0 t t noncommutative effects. E0 is a critical energy, which can interpreted as the ground-state energy of the quantum sys- Case II. In the second case we assume that the energy can tem, or a critical energy in some phase transition. We call this be mapped to the Hamiltonian operator according to the rule energy-dependent noncommutativity generating mechanism 2  h¯ the energy coupling (EC) mechanism. When E E0, both ε → H =− 2. (64) η(E) and θ(E) tend to zero, and the noncommutative algebra 2m ≈ reduces to the standard Heisenberg algebra. When E E0, In this case for the power-law dependent noncommutativity we reach the opposite limit of noncommutative quantum parameters we obtain mechanics with constant parameters. Similarly, the physical α α ε −h¯ 2 parameters of the power-law energy-dependent noncommu- η(ε) = η → η ( )α ≡ ηIIα, 0 0 2 α 2 (65) tative algebra in the effective potential given by Eq. (50)take ε0 2mε0 β the form ε β −¯ 2 h β II β α α θ(ε) = θ0 → θ0 (2) ≡ θβ  , (66) η0 E E ε 2mε 2 B (E) = ≡ B , (60a) 0 0 e ¯ 0 2mh E0 E0 where α α η2 E 2 k E 2 k (E) = 0 ≡ 0 , α β e 2 (60b) −¯ 2 −¯ 2 8mh¯ E0 2 E0 II h II h ηα = η ,θβ = θ . (67) 0 2mε 0 2mε η η2 0 0 where B = 0 and k = 0 . 0 2mh¯ 0 4mh¯ 2 The effective magnetic field and the elastic constant in 4.3 The noncommutative algebra of the energy operator Eq. (65) are represented by the fractional derivatives,

(EO) model − 2 α h¯ α II α Be(E) → B0 (2) ≡ Bα (2) , (68a) 2mε0 Finally, we consider the model in which the energy-dependent − 2 2α II noncommutative geometry can be mapped to a quantum k0 h¯ 2α kα 2α ke(E) → (2) ≡ (2) , (68b) mechanical representation . This can be realized by asso- 2 2mε0 2   ciating a quantum operator to the considered energy scales α 2α II −h¯ 2 II −h¯ 2 where Bα ≡ B , kα ≡ k , and or E. There are two possibilities to construct such a map- 0 2mε0 0 2mε0  α ping between energy and operators. ( )α ≡ ∂2 + ∂2 2 ∂x2 ∂y2 , respectively. ε → ¯ ∂ Since α, β are real variables, the energy-dependent non- Case I. In the first case we consider the mapping ih ∂t , that is, we map the energy to the standard quantum mechani- commutative geometry in the EO model now involves frac- cal representation. Hence we obtain for the noncommutativ- tional derivative differential equations. ity parameters the representation Hence the energy operator (EO) representation of the energy-dependent noncommutative quantum mechanics leads ε α α ∂ α ih¯ I α η(ε) = η → η ≡ ηα D , (61a) to the emergence of fractional calculus for the physical 0 ε 0 ε ∂t t 0 0 description of the high-energy scale quantum processes. In ε β ¯ β ∂ β general there are several definitions of the fractional deriva- θ(ε) = θ → θ ih ≡ θ I β , 0 0 β Dt (61b) ε0 ε0 ∂t tives, which we will discuss briefly in Sect. 7. 123 Eur. Phys. J. C (2019) 79 :300 Page 11 of 22 300   2 For convenience we rewrite the notations of Cases I and h¯ kε 2 2 −  − Bε Lz + (x + y ) ψ(x, y) = Eψ(x, y). II in a unified form, 2m 2 2

 α (75) Be(E) → BαD , (69)  kα α We now introduce the particle representation, ke(E) → D , (70) 2    1 ∂  ωε x − α α α α a m ihm¯ ωε ∂x  = , D = D = = ∂ , (76a) where I II, I Dt and II 2 for Cases I and a† ¯ + 1 2h x ihm¯ ωε ∂x II.     1 ∂ b mωε y − ω ∂ = ihm¯ ε y , (76b) b† ¯ + 1 ∂ 2h y ihm¯ ωε ∂y 5 Quantum evolution in the spacetime quantum  † fluctuation (SQF) energy-dependent noncommutative where a and b are the particle annihilation and creation ω = kε model operators, and we have denoted ε m . Then it is easy † to show that the operators a and b satisfy the Bose algebra, In the present section we explore the physical implications of namely a,a† = 1 and b,b† = 1, respectively. The other the SQF noncommutative algebra and the underlying quan- operators are commutative. Hence the Hamiltonian can be tum evolution. To gain some insights into the effects of represented as the energy-dependent noncommutativity on the dynamics of quantum particles we analyze two basic models of quantum ω   †  h¯ ε ihB¯ ε a H0 = a b  . (77) mechanics – the free particle and the harmonic oscillator, −ihB¯ ε h¯ ωε b† respectively. By using the Bogoliubov transformation, 5.1 Quantum mechanics of the free particle in the SQF model α† u v a = , (78) β v u b† Let first us consider a free particle whose quantum mechani- cal evolution is described by the SQF noncommutative alge- to diagonalize the Hamiltonian, we obtain bra with V (x,y) = 0. Hence the effective potential becomes   † † H0 = h¯ α α α + β β + 1 , (79) kε 2 2 Veff =−Bε Lz + (x + y ). (71) 2 where The generalized Schrödinger equation reduces to  α kε η0 1 ε α = ωε + Bε = + Bε = 1 + √ ∂ m 2mh¯ 2 ε0 ih¯ (x, y, t) = H(x, y, t), (72) ∂t (80) where is the effective frequency of the effective “harmonic oscilla- tor” associated with the quantum evolution of the free parti- ¯ 2   h kε 2 2 cle. The eigenvalues of H0 can be written as H =− 2 − Bε Lz + (x + y ), (73) 2m 2 E = h¯ α nα + nβ + 1 , (81) and with Bε and kε given by Eqs. (58a) and (58b), respec- tively. Since H0 is independent of time, the wave function is where nα, nβ = 0, 1,...The corresponding eigenstates can of the form be expressed as

− i Et (t, x, y) = e h¯ ψ(x, y), (74) 1 |ψnα,nβ = nαnβ |0, 0 (82) nα!nβ !

  where E is the energy of the free particle. By substituting the where nα = α†α and nβ = β†β are the quasi- particle wave function (74) into the Schrödinger equation (72), the operators, and |0, 0 is the ground state of the associated stationary Schrödinger equation is obtained: two-dimensional harmonic oscillator. 123 300 Page 12 of 22 Eur. Phys. J. C (2019) 79 :300

5.2 The harmonic oscillator where E is the energy of the oscillator. By substituting the wave function (88) into the Schrödinger equation (85), we For the two-dimensional quantum harmonic oscillator in obtain the stationary Schrödinger equation as given by the SQF noncommutative geometry, the potential energy by   using the 2D Seiberg–Witten map as given by Eq. (21a) can 2 h¯ Kh 2 2 −  − Bh Lz + (x + y ) ψ(x, y)=Eψ(x, y). be expressed as 2m∗ 2 2  (89) 1 V (x,y) = k x2 + y2 , (83) 2 By using the same procedure as in the case of the free particle, we diagonalize the Hamiltonian, thus obtaining giving     † † θ(E) θ(E) H = h¯ α α α + β β + 1 , (90) V x + py , y − px 2h¯ 2h¯  α where k θ ε  = V (x, y) + − 0 L ε z 2α 2 h¯ 0 k k0 ε  α = ω + B = + 2β  h h ∗ ∗ θ 2 ε m m ε0 + 0 2 + 2 ,   2 px py (84) α β−α 4h¯ ε0 η kmθ + 0 1 + 0 , (91) 2mh¯ η where V (x, y) = 1 k x2 + y2 is the potential energy of the 0 0 0 2 √ ∗ harmonic oscillator in the Heisenberg representation. The where ωh = Kh/m is the generalized effective frequency generalized Schrödinger equation takes the form of the two-dimensional harmonic oscillator in the SQF non- commutative algebra. The eigenvalues of H can be written ∂ as ih¯ (x, y, t) = H(x, y, t), (85) ∂t E = h¯ α nα + nβ + 1 , (92) where where nα, nβ = 0, 1,...The corresponding eigenstates can ¯ 2  h 1 2 2 be obtained: H =−  − Bh Lz + Kh(x + y ), (86) 2m∗ 2 2 1 |ψnα,nβ = nαnβ |0, 0, (93) and we have denoted nα!nβ !

κ θ 2 ε 2β 1 1 2 ε k 0  = α†α  = β†β = + ,κε = , (87a) where nα and nβ are the quasi-particle oper- m∗ m h¯ 2 8 ε | ,  0 ators, and 0 0 is the ground state of the two-dimensional β 2β kθ0 ε ε harmonic oscillator in the SQF noncommutative algebra. Bh = Bε + , Kh = k + k0 . (87b) 2h¯ ε0 ε0 ∗ In the above equations m is the effective mass of the oscil- 6 Quantum dynamics in the energy coupling model lator, including the modifications of the harmonic potential due to the SQF noncommutative algebra, Bh is the effective In the present section we investigate the two basic quantum magnetic field, in which the first term comes from the kinetic mechanical models, the free particle, and the harmonic oscil- energy and the second term comes from the potential energy lator, respectively, in the energy coupling (EC) noncommuta- in the SQF noncommutative algebra, while Kh is the effec- tive algebra, by assuming that the noncommutativity param- tive elastic constant, in which the second term comes from eters θ and η are functions of the particle energy E only, the SQF noncommutative algebra. For a free particle, namely and independent of energy scale of the quantum spacetime  α ∗ ε 2 V (x,y) = 0, m = m, B = Bε, and K = k . fluctuations. h h 0 ε0 Similarly to the free particle case, since H is independent of time, the wave function is of the form 6.1 Quantum evolution: the Schrödinger equation

− i For the EC noncommutative algebra, the effective potential ( , , ) = h¯ Etψ( , ), t x y e x y (88) can be written in a unified form for both the free particle and the harmonic potential: 123 Eur. Phys. J. C (2019) 79 :300 Page 13 of 22 300

( ) Kh E 2 2 where R (r) = dR(r)/dr. By introducing a new radial coor- Veff =−Bh(E)Lz + (x + y ). (94) 2 dinate ξ, defined as   Then the generalized Schrödinger equation can be obtained: ∗ 1/4 m Kh(E) ξ = √ r, (102) ∂ h¯ ih¯ (x, y, t) = H(x, y, t), (95) ∂t and by denoting √ ∗ where 2 m E + mφhB¯ (E) C = √ h , (103) h¯ K (E) ¯ 2 ( ) h  h Kh E 2 2 H =−  − Bh(E)Lz + (x + y ). (96) 2m∗ 2 2 Eq. (101) takes the form

H 2 (ξ) (ξ)  Since is independent of time, the wave function is of the 2 d R dR 2 4 2 ξ + ξ + Cξ − ξ − mφ R (ξ) = 0, form dξ 2 dξ (104) − i Et (t, x, y) = e h¯ ψ(x, y), (97) or, equivalently, By substituting the wave function (97) into the Schrödinger  d dR(ξ) equation (95), we obtain the stationary Schrödinger equation ξ ξ + Cξ 2 − ξ 4 − m2 R (ξ) = . ξ ξ φ 0 (105) as d d   2 ( ) ξ ξ 2 4 h¯ Kh E 2 2 In the range of values of so that C >> ξ , or, equiv- −  − Bh(E)Lz + (x + y ) ψ = Eψ. 2m∗ 2 2 alently, (98)   ∗ 1/4 1/2 (4m ) E + mφhB¯ h(E) ξ<< , (106) ¯ 1/2 1/4( ) 6.2 Quantum evolution: wave function and energy levels h Kh E

In the following we obtain the solutions (wave functions) Eq. (104) becomes and the energy levels of the Schrödinger equation (98)inthe 2 (ξ) (ξ)  EC noncommutative algebra. In the polar coordinate system 2 d R dR 2 2 ξ + ξ + Cξ − mφ R (ξ) = 0, (107) (r,φ) with x = r cos φ, y = r sin φ, the angular momentum dξ 2 dξ ∂ operator is represented by Lz =−ih¯ ∂φ. Then the general- ized stationary Schrödinger equation can be expressed as and it has the general solution given by  √ √ ¯ 2 ∂2 ∂ ∂2 ∂ h 1 1 Rmφ (ξ) = c1 Jmφ Cξ + c2Ymφ Cξ , (108) − + + + ihB¯ h(E) 2m∗ ∂r 2 r dr r 2 ∂φ2 ∂φ  ( ) where J (ξ) is the Bessel function of the first kind, while Kh E 2 mφ + r ψ(r,φ)= Eψ(r,φ). (99) (ξ) 2 Ymφ denotes the Bessel function of the second kind [68]. c1 and c2 denote two arbitrary integration constants. Since Due to the axial symmetry of Eq. (99), the wave function the function Ymφ (ξ) is singular at the origin, we must take can be represented as c2 = 0 in the solution (108). Therefore the general solution of Eq. (107) can be written as φ  ψ(r,φ)= R(r)eimφ , (100) √ Rmφ (ξ) = c1 Jmφ Cξ   ∞ √ 2l+mφ where mφ = 0, 1, 2,... By substituting Eq. (100) into Eq.  (−1)l Cξ = c1 (99), we obtain l! l + mφ + 1 2  l=0 ∗   π  √ 2 2m 2 c1 r R (r) + rR(r) + E + mφhB¯ h(E) r = cos mφτ − Cξ sin τ dτ, (109) h¯ 2 π  0 ∗ ( )  (z) − 2 − m Kh E 4 ( ) = , where is the gamma function. For small values of mφ 2 r R r 0 (101) (ξ) ≈ h¯ the argument, the wave function behaves like Rmφ 123 300 Page 14 of 22 Eur. Phys. J. C (2019) 79 :300    √ mφ 1/ mφ + 1 Cξ/2 , while for large values of the The radial wave function must satisfy the normalization  √ condition (ξ) ≈ /π ξ argument we have Rmφ c1 2 C cos  √ ∞  2 | (ξ)| (mφ ) Cξ − mφπ/2 − π/4 + e Im O (1/ |ξ|), a relation ( ) = ,   Rn r rdr 1 (117)  √  0 valid for arg Cξ <π[68]. or, equivalently, In the interval [0,ξs], the Bessel functions satisfy the ξ s x x = ( / ) ξ 2 condition Jmφ jmφl ξ Jmφ jmφ n ξ xdx 1 2 s  0 s s ∞   2   h¯ −ξ 2 (mφ ) 2 δ (ξ) c ξ 2mφ e L ξ 2 ξdξ = 1. Jmφ +1 jmφl ln, where jmφl is the lth zero of Jmφ ∗ 1/2 n [m K (E)] 0 [68]. There is a large literature on the zeros of the Bessel h (118) functions; for a review and some recent results see [69]. Now we consider the case in which in Eq. (105)thetermξ 4 By introducing a new variable ξ 2 = x,2ξdξ = dx,we cannot be neglected. To solve Eq. (105) we introduce a new obtain coordinate ζ defined as ζ = ξ 2. Then we obtain immediately 2    ξ d = 2ζ d , ξ d ξ dR = 4 ζ d R + ζ dR , and Eq. (105) c h¯ ∞ ( ) 2 dξ dζ dξ dξ dζ 2 dζ mφ −x mφ ( ) = . ∗ 1/2 x e Ln x dx 1 (119) takes the form 2 [m Kh(E)] 0   2 d2 R(ζ ) 1 dR(ζ ) C − ζ mφ By taking into account the mathematical identity [2,68] + + − = 0. (110) dζ 2 ζ dζ 4ζ 4ζ 2   ∞ 2 −x a (a) ( ) = ( + )!/ !, e x Ln x dx n a n (120) Next we introduce a new function L ζ,mφ by means of the 0 transformation we find for the integration constant c the expression

mφ /2 −ζ/2   R (ζ ) = ζ e L ζ,mφ . (111) ∗ 1/2 2n! m Kh(E) c = . (121) Hence Eq. (110) becomes n + mφ ! h¯

2 d L ζ,mφ dL ζ,mφ The quantized energy levels of the free particle, and of the ζ + mφ + 1 − ζ ζ 2 ζ d d harmonic oscillator, can be obtained in the noncommutative 1 C energy dependent quantum mechanics as solutions of the + − mφ − 1 L ζ,mφ = 0. (112) 2 2 algebraic equation For h¯ E + mφhB¯ h(E) √ + φ + = √ . ∗ 2n m 1 (122) 1 C m Kh(E) n = − mφ − 1 ≥ 0, n ∈ N, (113) 2 2 The commutative quantum mechanical limit for the har- that is, for n taking non-negative integer values, the well- monic oscillator is recovered, as one can see easily from << ( ) = behaved solution of Eq. (112) is given by Eqs. (56a)–(56c), when E E0,givingBh E 0 and Kh(E) = k, respectively. Then from Eq. (122 ) we immedi-

(mφ ) ately obtain L ζ,mφ = cLn (ζ ) , (114)

(mφ ) (mφ ) Ecom = En = h¯ ω 2n + mφ + 1 , (123) where c is an arbitrary integration constant, and Ln (ζ ) are the generalized Laguerre polynomials, defined as [68] where ω2 = k/m. This relation gives the energy spectrum of

mφ ζ n the harmonic oscillator in commutative quantum mechanics (mφ ) ζ e d −ζ + L (ζ ) = e ζ n mφ , (115) [70]. n n! dζ n

( ) 6.3 The case of the free particle mφ = −mφ d − n ζ n+mφ / ! or, alternatively, as Ln x dx 1 n Hence the physical solution of Eq. (104) can be obtained: Let us consider first the simple case of the free parti- α  cle, with Be(E) = Bh(E)| = = B0 (E/E0) , ke(E) = (mφ ) 2 (mφ ) k 0 (ξ) = −ξ /2ξ mφ ξ 2 . ( )| = / / 2α ∗ = Rn ce Ln (116) Kh E k=0 (k0 2)(E E0) , and m m, respectively. 123 Eur. Phys. J. C (2019) 79 :300 Page 15 of 22 300

Then from Eq. (122) it follows that the energy levels of the the ratio of the number N of particles scattered into direc- free particle are given by tion (θ,φ) per unit time per unit solid angle, divided by incident flux j,dσ/d = N/j [2,3]. Usually one con-  1/(α−1) (mφ ) 2m siders that the incident wave on the target corresponds to E = E n 2 0 h¯ k0 a free particle, and the scattering wave function is given ψ ( ) ∼ i K · r + (θ) i K · r / E0 by r e f e r. However, in energy- ×  √  ,α = 1. 1/(α−1) dependent quantum mechanics the wave function of the free 2n + 1 − B0 2m/k0 mφ + 1 particle at infinity cannot be described anymore as a simple (124) plane wave. Hence, at least in principle, energy-dependent For α = 1 the energy spectrum of the particle is continu- noncommutative effects could be determined and studied experimentally through their effects on the scattering cross ous, and the two quantum√ numbers n and m√φ must satisfy the sections in very high-energy particle collisions. A cross sec- condition E0 = h¯ k0/2m 2n + 1 − B0 2m/k0 mφ + 1 . The wave function of the two-dimensional free particle tion dependent on the particle energies may be an indicator of in energy dependent noncommutative quantum mechanics is the noncommutative quantum mechanical effects, and may given by provide an experimental method to detect the presence of the  quantum spacetime. mφ mφ α  α mk 4 2 − 0 E r2 (mφ ) mk0 E 8h¯ 2 E0 Rn (r) = c e 2h¯ 2 E  0  α 6.4 The harmonic oscillator ( ) mk E × mφ mφ 0 2 . r Ln 2 r (125) 2h¯ E0 Next we consider the harmonic oscillator problem in the In commutative quantum mechanics the wave function of energy coupling model of the energy-dependent noncom- a freely moving quantum particle with momentum p is given mutative quantum mechanics. By taking into account the i p · r/h¯ explicit forms of the effective mass, effective magnetic field by p = const. e . If we introduce the wave vector K , and effective elastic constant as given by Eqs. (56a)–(56c), it defined by K = p/h¯ , then the wave function of the free par- i K · r follows that the energy spectrum of the harmonic oscillator ticle is given by  = const. e . The energy spectrum of K is obtained as a solution of the nonlinear algebraic equation the particle is continuous, with E = p2/2m = h¯ 2 K 2/2m. given by The evolution of the free particle in the energy-dependent noncommutative quantum mechanics is qualitatively differ-  ent from the standard quantum mechanical case. The particle 2ω2θ m 0 β is not anymore “free”, but its dynamics is determined by the h¯ ω 1 + x 2n + mφ + 1 ¯ 2 presence of the effective potential generated by the noncom- 4h ω2  η  mutative effects. Moreover, the energy levels are quantized E x + mφ 0 xα + mθ xβ = 0  2 k 0 , in terms of two quantum numbers n and mφ. (128) η2 The ground state of the free particle corresponds to the 1 + 0 x2α 8m2ω2h¯ 2 choice n = mφ = 0. Then the ground-state energy is given by √ where we have denoted x = E/E0, and ω = k/m.In 1 α ( ) 2m 2(α−1) order to solve this equation we need to fix, from physical E 0 = E α−1 ,α = . 0 2 0 1 (126) considerations, the numerical values of the quantities α and h¯ k0 β. For arbitrary values of α and β the energy levels can be The radial wave function of the ground state of the free obtained generally only by using numerical methods. In the particle in energy-dependent quantum mechanics takes the first order approximation we obtain form

  2 2 α m ω θ β mk E 2 0 ( ) − 0 r h¯ ω 1 + x 2n + mφ + 1 0 ( ) = 8h¯ 2 E0 . ¯ 2 R0 r ce (127) 8h ω2 η  η2 mφ 0 α β E0 0 2α+1 = E0x + x + mθ0x − x , A possibility to test the energy-dependent noncommu- 2 k 16m2ω2h¯ 2 tative quantum mechanics would be through the study of (129) the collision and scattering processes. Collisions are charac- terized by the differential cross section dσ/d, defined as 123 300 Page 16 of 22 Eur. Phys. J. C (2019) 79 :300 or, equivalently, Then the wave function of the ground state of the harmonic oscillator can be written as 2 2 2 m ω θ0 β mφω   1 + x Ecom = E0x + ω ( ) 2 ¯ 2 (0) m eff E r 8h  2  R (r) ∼ exp − . (135) η 0 h¯ 2 × 0 xα + mθ xβ k 0 2 Hence we have completely solved the problem of the quan- E0η α+ − 0 x2 1, (130) tum mechanical motion of the free particle, and of a parti- 16m2ω2h¯ 2 cle in a harmonic potential, in the noncommutative quantum mechanics with energy-dependent strengths. where Ecom denotes the energy levels of the harmonic oscil- lator in the commutative formulation of quantum mechanics. In the simple case α = β = 1, in the first approximation we obtain for the energy levels the algebraic equation 7 Quantum evolution in the energy operator (EO) energy-dependent noncommutative geometry   2 2 2  m ω θ0 mφω η0 1 + x Ecom = E0 + + mθ0 x 8h¯ 2 2 k Finally, we will consider in detail the third possibility of con- 2 structing quantum mechanics in the framework of energy- E0η − 0 x3. (131) dependent noncommutative geometry. This approach con- 2ω2 ¯ 2 16m h sists in mapping the energy in the noncommutative algebra to an operator. As we have already discussed, we have two By neglecting the term x3, we obtain the energy levels in possibilities to develop such an approach, by mapping the the energy-dependent noncommutative quantum mechanics energy to the time operator, or to the particle Hamiltonian. in the first order approximation: Under the assumption of a power-law dependence on energy of the noncommutative strengths, in the general case these (mφ ) Ecom maps lead to a fractional Schrödinger equation E ≈ . (132) .Inthefol- n ω2 η 2ω2θ + mφ 0 + θ − m 0 lowing we will first write down the basic fractional Schrö E0 2 k m 0 ¯ 2 Ecom 8h dinger equations for the free particle and the harmonic oscil- lator case, and after that we will proceed to a detailed study of The wave function of the ground state of the energy- their properties. We will concentrate on the models obtained dependent harmonic oscillator in noncommutative quantum by the time operator representation of the energy. But before = = mechanics, corresponding to n mφ 0, can be written as proceeding to discuss the physical implications of the gen- eralized Schrödinger equation with fractional operators, we ⎧ ⎫ ⎡  ⎤ / will present o very brief summary of the basic properties of ⎪ 2 2α 1 2 ⎪ ⎪ η E ⎪ ⎨ + 0 2 ⎬ the fractional calculus. ( ) mω ⎢1 2ω2 2 E ⎥ r 0 ( ) ∼ − ⎣ 8m h¯ 0 ⎦ . R0 r exp  β ⎪ h¯ 2ω2θ 2 ⎪ ⎩ 1 + m 0 E ⎭ 4h¯ 2 E0 7.1 Fractional calculus: a brief review (133) For α, β∈ / N, the quantum mechanical model introduced in the previous sections, based on the power-law energy depen- In the limit E << E0, we recover the standard com- dence of the noncommutative strengths, leads to the interest- mutative result for the eave function of the ground state ing question of the mathematical and physical interpretation ψ ( ) ∼ α of the- quantum. mechanical harmonic oscillator, r of the operators of the form [(ih/E0) ∂/∂t] . It turns out − mω r2 exp h¯ 2 [2,3]. that such operators can be written is terms of a fractional The wave function of the ground state of the harmonic derivative as oscillator in the energy-dependent noncommutative quantum α α ∂ ih¯ ∂α ih¯ mechanics can be written in a form analogous to the com- η ih¯  = η = η Dα, ∂ 0 ∂ α 0 t (136) mutative case by introducing the effective energy-depending t E0 t E0 frequency ω , defined as eff α = ∂α/∂ α  where Dt t is the fractional derivative of of the order α>0, which can be defined in terms of the fractional ⎡  ⎤ / η2 2α 1 2 −m( ) + 0 E integral Dt t as [71–73] ⎢1 2ω2 2 E ⎥ ω ( ) = ω 8m h¯ 0 . eff E ⎣  β ⎦ (134)   2ω2θ α −( −α) 1 + m 0 E D  (t) = Dm D m  (t) . (137) 4h¯ 2 E0 t t t 123 Eur. Phys. J. C (2019) 79 :300 Page 17 of 22 300

∂ Hence in fractional calculus a fractional derivative is defined ih¯ (x, y, t) = H(x, y, t), (143) via a fractional integral. By interpreting the time derivative ∂t operators as fractional derivatives we obtain the fractional Schrödinger equations as given by the equations presented where in the next section. 2  h¯ There are several definitions of the fractional derivative H =− ∗ 2 + Veff, (144) that have been intensively investigated in the mathematical 2m and physical literature. For example, the Caputo fractional derivative is defined as with

 ( ) 1 t  n (τ)dτ = ( , ) − α + 1 2α( 2 + 2), C α ( ) = . Veff V x y B D Lz K D x y (145) α Dt t α+1−n (138) 2  (n − α) a (t − τ)  and we have denoted ( ) = ∞ kα Given a function f x k=0 ak x , its fractional 2 Caputo derivative can be obtained according to 1 1 2κ kθα = + ,κ = , ∗ 2 (146a) m m h¯ 8 α ∞  + ( + )α θ d [1 k 1 ] kα  k α α f (x) = ak+1 x , (139) B = Bα + , K = k + k , (146b) dxα  (1 + kα) 2h¯ k=0 where  = I, II denote the two different operator represen- α where  (z) is the gamma function defined as  (z) = α ≡ ∂ ∞ − − tations of the EO noncommutative algebra. DI ∂t and t z 1e t dt, with the property  (1 + z) = z (z) [73]. α ≡ 2α + 2α 0 DII Dx Dy are the fractional derivatives with respect Consequently, we obtain the definition of the Caputo frac- to time and space when α is a rational integer. tional derivative of the exponential function: In the following we will concentrate only on the time oper- ∞ ∞ dα dα  xn  xn−α ator representation of the noncommutative quantum mechan- ex = = ics with energy-dependent strengths, namely  = I . Thus, dxα dxα n!  (1 + n − α) n=0 n=1 by representing the wave function as 1−α = x E1,2−α (x) , (140) − i Et ( )  (t, x, y) = e h¯ ψ (x, y) , (147) where E1,2−α x is the generalized Mittag-Leffler function, ( ) = ∞ n/( α + β) defined as Eα,β z n=0 z n [73]. Another definition of the fractional derivative is the Liou- the Schrödinger equation becomes a fractional differential ville definition, which implies [73] equation given by  α ¯ 2 d − h  + ( , ) − D ( ) kx = α kx, ≥ . ∗ 2 V x y B α t Lz α e k e k 0 (141) 2m dx   1 2 2 Finally, we also point out the definition of the left-sided + KD2α(t)(x + y ) ψ (x, y) = Eψ(x, y), (148) 2 Riemann–Liouville fractional integral of order ν of the func- tion f (t), which is defined as where   n x (τ ν+1−n i α − i i α − i ν 1 d f D ( ) = h¯ Et h¯ Et, D ( ) = h¯ Et 2 h¯ Et. D f (x) := dτ , (142) α t e Dt e 2α t e Dt e a ( − ν) n − τ n dx 0 x (149) − <ν< ∈ N a definition which is valid for n 1 n [73]. − i Et If the function e h¯ is an eigenfunction of the fractional derivation operators Dα and D2α, so that 7.2 The fractional Schrödinger equation

α − i − i α − i − i h¯ Et = h¯ Et, 2 h¯ Et = h¯ Et, In the present section we will consider the evolution of a Dt e aαe Dt e a2αe (150) system in the energy operator representation of the energy- dependent noncommutative quantum mechanics. We will where aα and a2α are constants, the separation of the time restrict again our analysis to the two-dimensional case only. variable can be performed in the noncommutative fractional The generalized fractional two-dimensional Schrödinger Schrö dinger equation with energy-dependent noncommuta- equation can then be expressed as tive strengths. 123 300 Page 18 of 22 Eur. Phys. J. C (2019) 79 :300

α = σξ2 >> ξ 4 ξ<<( )1/4 7.3 The free particle: the case 1 When or, equivalently, 4m 1/2 1/4 1 + mφ BI / (K I ) , that is, when the noncommutative For simplicity, in the following we investigate the quantum effects can be neglected, in the standard quantum mechanical dynamics in the energy operator representation only in the limit we obtain the solution of Eq. (157): case of the free particle, by assuming V (x,y) = 0. Therefore = k 0, and the effective mass of the particle coincides with R(ξ) = c1 Jn(ξ) + c2Yn(ξ), (158) ∗ = the ordinary mass, ml m. Moreover, for simplicity we will restrict our analysis to the choice α = 1. Then D (t) =−i E 1 h¯ where c1 and c2 are arbitrary integration constants, and Jn(ξ) and D (t) =−1 E2, respectively. (ξ) 2 h¯ 2 and Yn are the Bessel function of the first kind and the Explicitly, the Schrödinger equation describing the motion Bessel function of the second kind [68], respectively. We of the free particle in the energy-dependent noncommutative have already discussed in detail the behavior of the wave geometry takes the form function in this case in the previous section. ∂ η ∂ ξ 4 σξ2 i 0 ˆ If the term cannot be neglected as compared to , ih¯ (t, x, y) − Lz (t, x, y) ∂t 2mE0 ∂t then the general solution of Eq. (157) is given by η2  ∂2 ¯ 2  0 2 2 h (mφ ) −ξ 2/2 mφ (mφ ) 2 + x + y (t, x, y) =− 2(t, x, y). R (ξ) = ce ξ L ξ , (159) 2 ∂t2 2m n n 8mE0 (151) that is, the same form of the solution as the one already con- sidered when discussing the evolution of the free particle and In the polar coordinate system (r,φ) with r = r cos φ, of the harmonic oscillator in the energy coupling model of y = r sin φ,wehave the energy-dependent noncommutative quantum mechanics. The wave function must be finite at the origin. The normal- ˆ h¯ ∂ Lz = , (152) ization and other properties of the wave function are similar i ∂φ to the ones already investigated in the previous section. and

1 ∂ ∂ 1 ∂2 8 Discussions and final remarks  = r + , (153) 2 r ∂r dr r 2 ∂φ2 In the present paper we have considered the quantum respectively. mechanical implications of a noncommutative geometric By introducing for the wave function the representation model in which the strengths of the noncommutative parame- −(i/h¯ )Et (t, x, y) = e ψ (r,φ), and, similarly to the previous ters are energy dependent. From a physical point of view such section, representing the reduced wave function as ψ (r,φ) = an approach may be justified since the effects of the noncom- imφ φ R(r)e , mφ ∈ N, it follows that Eq. (148) takes the form mutativity of the spacetime are expected to become apparent  at extremely high energies, of the order of the Planck energy, 2 2m 2 r R (r) + rR(r) + E 1 + mφ BI r and at distance scales of the order of the Planck length. By h¯ 2  assuming an energy-dependent noncommutativity we obtain mKI 2 4 2 a smooth transition between the maximally noncommuta- − E r − mφ R(r) = 0. (154) h¯ 4 tive geometry at the Planck scale, and its commutative ordi- By introducing a new independent variable ξ, defined as nary quantum mechanical version, which can be interpreted as the low-energy limit of the noncommutative high-energy ¯ quantum mechanics. Hence this approach unifies in a sin- = h ξ, r / (155) 2 1 4 gle formalism two apparently distinct approaches, the non- mKI E commutative and commutative versions of quantum mechan- and by denoting ics, respectively, and generally leads to an energy-dependent Schrödinger equation, as already considered in the literature √ 2 m 1 + mφ BI [61–67]. σ = √ , (156) One of the important question related to the formalism K I developed in the present paper is related to the physical impli- Eq. (154) takes the form cations of the obtained results. In the standard approach to noncommutative geometry, by using the linearity of the D 2  map, one could find a representation of the noncommutative 2 d R dR 2 4 2 ξ + ξ + σξ − ξ − mφ R = 0, (157) dξ 2 dξ observables as operators acting on the conventional Hilbert 123 Eur. Phys. J. C (2019) 79 :300 Page 19 of 22 300 space of ordinary quantum mechanics. More exactly, the D extension of the results of [26] to the energy-dependent case map converts the noncommutative system into a modified would help to clarify the mathematical structure and phys- commutative quantum mechanical system that contains an ical properties of energy-dependent noncommutative quan- explicit dependence of the Hamiltonian on the noncommu- tum mechanics. tative parameters, and on the particular D map used to obtain In order to implement the idea of energy-dependent non- the representation. The states of the considered quantum sys- commutativity we need to specify the relevant energy scales. tem are then wave functions in the ordinary Hilbert space; the In the present work we have assumed a two scale and a single dynamics is determined by the standard Schrödinger equa- energy scale model. Moreover, we have limited our investi- tion with a modified Hamiltonian that depends on the non- gations to the case in which the noncommutative strengths commutative strengths θ and η [26]. Even though the mathe- have a simple power-law dependence on the energy. In the matical formalism is dependent on the functional form of the first approach the energy dependence of the noncommutative adopted D map that is used to realize the noncommutative- strengths is determined by a specific energy scale, which is commutative conversion, this is not the case for physical pre- related to the energy of the quantum fluctuations that modify dictions of the theory such as expectation values and proba- the geometry. This approach may be valid to describe physics bility distributions [26]. On the other hand it is important to very nearby the Planck scale, where the vacuum energy may point the fact that the standard formalism in which the energy be the dominant physical effect influencing the quantum evo- dependence is ignored is not manifestly invariant under a lution of particles in the noncommutative geometric setting. modification of the D map. In this context we have considered the dynamics of two sim- In the energy-dependent approach to noncommutative ple but important quantum systems, the free particle, and the geometry after performing the D mapwearriveatan harmonic oscillator, respectively. The physical characteris- energy-dependent Schrödinger equation, which contains tics of the evolution are strongly dependent on the energy of some energy-dependent potentials V (x, E,θ,η). In order to the quantum fluctuations, with the oscillations frequencies obtain a consistent physical interpretation we need to redefine effectively determined by the spacetime fluctuation scale. the probability density, the normalization condition and the In our second model we have assumed that all the non- expectation values of the physical observables. For example, commutative effects can be described by means of the particle in order to be sure that a solution of the Schrödinger equa- energy scale, which is the unique scale determining the phys- tion associated with a stationary energy E is normalizable ical implications of noncommutative geometry. The choice the following two conditions must hold simultaneously [62]: of a single energy scale allows the smooth transition from the noncommutative algebra of the Planck length to the commu- ∂V (x, E,θ,η) tative Heisenberg algebra of the ordinary quantum mechanics 1 − ≥ 0, x ∈ D, N(ψ) < ∞. (160) ∂ E that gives an excellent description of the physical processes on the length and energy scales of the atoms and molecules, Moreover, in opposition to the standard case of an energy- and for the standard model of elementary particles. In this independent potential, the nonnegativity and the existence of case the basic physical parameters of the of the quantum the norm integral must hold at the same time. The modified dynamics of the free particles and of the harmonic oscilla- forms of the probability density and of the probability current tor are energy dependent, with the oscillation frequencies also lead to adjustments in the scalar product and the norm described by complicated functions of the particle energy. that do not appear in standard quantum mechanics. Similarly Such an energy dependence of the basic physical parameters to the case of standard noncommutative quantum mechan- of the quantum processes may have a significant impact on ics we also expect that, like in the energy-independent case, the high-energy evolution of the quantum particles. our present formalism is not invariant under a change in the Perhaps the most interesting physical implications are functional form of the D map. The phase-space formula- obtained in the framework of our third approach, which con- tion of a noncommutative extension of quantum mechanics sists in mapping the energy with a quantum operator. There in arbitrary dimensions, with both spatial and momentum are two such possibilities we have briefly discussed, namely, noncommutativities, was considered in [26]. By consider- mapping the energy with the time derivative operator, and ing a covariant generalization of the Weyl–Wigner transform with the Hamiltonian of the free particle (its kinetic energy). and of the Darboux D map, an isomorphism between the The corresponding Schrödinger equation changes its math- operator and the phase-space representations of the extended ematical form, content an interpretation, becoming a frac- Heisenberg algebra was constructed. This map allows one to tional differential equation, in which the ordinary derivatives develop a systematic approach to deriving the entire struc- of quantum mechanics are substituted by fractional ones. ture of noncommutative quantum mechanics in phase space. Fractional Schrödinger equations have been introduced More importantly, it turns out that the entire formalism is some time ago in the physical literature [74], and presently independent of the particular choice of the Darboux map. The they are becoming a very active field of research in both 123 300 Page 20 of 22 Eur. Phys. J. C (2019) 79 :300 physics and mathematics [74–93]. For an extensive review energy. Moreover, in the present approach to noncommuta- of fractional quantum mechanics see [94]. A typical example tive quantum mechanics, the noncommutative strength θ is of a fractional Schrödinger equation is given by the equation an explicit function of the energy, and therefore an explicit [74] dependence of the nonlocality on the energy always appears. As a particular case we consider f (σ) = 1, that is, the non-  ∂ψ ( , ) α/2 commutative strengths depend only on the energy. Then, by r t 2 ih¯ = Dα −h¯  ψ (r , t) + V (r , t) ψ (r , t) , ∂t taking into account the normalization of the wave function (161) we obtain (x) (x) ≥ (θ(E)/2) or, by considering the explicit choice of the energy dependence of θ adopted in () () ≥ (θ / )( / )β where Dα is a constant, and α is an arbitrary number. The the present study, x  x  0 2 E E0 . Hence << () () ≈ fractional Hamilton operator is Hermitic, and a parity con- when E E0, x  x  0, and we recover the servation law for fractional quantum mechanics can also be standard quantum mechanical result. In the case of the har- established. The energy spectra of a hydrogenlike atom can monic oscillator the uncertainty relations for the noncom- (σ) = σ also be obtained, while in this approach the fractional oscil- mutative coordinates can be obtained for f as () () ≥ θ 4 lator with the Hamiltonian x  x  O , that is, nonlocality does not appear 2 in higher orders of θ [50]. For the case f (σ) = σ , one finds 3 α 2 β (x) (x) ≥ (θ/2ωσ )(n + 1/2) + O θ , where ωη is Hα,β = Dα|p| + q |x| , (162) the oscillation frequency of the σ -dependent potential term in the total Hamiltonian, given by V (σ ) = ω2 σ 2/2[50]. where α, β, and q are constants, has the energy levels quan- σ A central question in the noncommutative extensions of tized according to the rule [74] quantum mechanics is the likelihood of its observational or ⎡ ⎤ αβ experimental testing. A possibility of detecting the existence α+β αβ 1/α 2/β of the noncommutative phase space by using the Aharonov– ⎣ πh¯ β Dα q ⎦ 1 α+β En =  n + , (163) Bohm effect was suggested in [33]. As we have already seen 1 , 1 + 2 2B β α 1 the noncommutativity of the momenta leads to the genera- tion of an effective magnetic field and of an effective flux. where the B(a, b) function is defined by the integral repre- In a mesoscopic ring this flux induces a persistent current. ( , ) = 1 a−1( − )b−1 α sentation B a b 0 duu 1 u [68], and and By using this effect it may be possible to detect the effective β are arbitrary numerical parameters. As for the physical magnetic flux generated by the presence of the noncommu- origin of the fractional derivatives, in [95,96]itwasshown tative phase space, even if it is very weak. Persistent currents that it originates from the path integral approach to quan- and magnetic fluxes in mesoscopic rings can be studied by tum mechanics. More exactly, the path integral over Brown- using experimental methods developed in nanotechnology ian trajectories gives the standard Schr ödinger equation of [33]. The dynamics of a free electron in the two-dimensional quantum mechanics, while the path integral over Levy´ tra- noncommutative phase space is equivalent to the evolution jectories generates the fractional Schrödinger equation. In of the electron in an effective magnetic field, induced by the present paper we have outlined the possibility of another the effects of the noncommutativity of the coordinates and physical path towards the fractional Schrödinger equation, momenta. namely, the framework of the quantum operator approach to For the motion of a free electron in the noncommutative energy-dependent noncommutative geometry. phase space, the Hamiltonian can be obtained: An interesting theoretical question in the field of non-    commutative quantum mechanics is the problem of the non- 1 2 2 1 2 2 Hnc = p + p = ∗ (px + eAx ) + py + eAy , locality generated by the dynamical noncommutativity. This 2m x y 2m problem was investigated in [50] for a noncommutative quan- (164) tum system with the coordinates satisfying the commutation   ∗ relations qi ,q j = iθ f (σ) ij, where f (σ) is a function of where we have denoted by m = m/α the effective mass in the physical parameter σ, which could represent, for exam- the noncommutative phase space. The parameter α is defined ple, position, spin, or energy. Then for the uncertainty rela- through the relation θη = 2h¯ 2α2 1 − α2 . The components tion between the coordinate operators x and y we obtain the Ax and Ay of the effective vector potential A are given by 2 2 uncertainty relation (x) (x) ≥ (θ/2) | | f (σ)| | Ax = η/2eα h¯ y and Ay =− η/2eα h¯ x, respectively [50]. As was pointed out in [50], if |>represents a station- [33], while the effective magnetic field is obtained in the form 2 ary state of the quantum system, that is, an eigenstate of the Bz = η/eα h¯ . Hamiltonian, it follows that the nonlocality induced by the A possibility of experimentally implementing a method noncommutativity of the coordinates will be a function of the that could detect noncommutative quantum mechanical 123 Eur. Phys. J. C (2019) 79 :300 Page 21 of 22 300 effects consists in considering a one-dimensional ring in an Data Availability Statement This manuscript has no associated data external magnetic field B , oriented along the axis of the ring. or the data will not be deposited. [Authors’ comment: Since this is a theoretical study no data are available.] B is constant inside rc < R (ring radius), which implies that the electrons are located only in the field-free region of Open Access This article is distributed under the terms of the Creative the small ring. Moreover, the quantum electronic states are Commons Attribution 4.0 International License (http://creativecomm functions of the total magnetic flux crossing the ring only. ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, By introducing a polar coordinate system by means of the and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative definitions x = R cos ϕ, y = R sin ϕ, we obtain for the Commons license, and indicate if changes were made. Hamiltonian of the electrons the expression [33] Funded by SCOAP3.

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