Energydependent noncommutative quantum mechanics
Abstract
We propose a model of dynamical noncommutative quantum mechanics in which the noncommutative strengths, describing the properties of the commutation relations of the coordinate and momenta, respectively, are arbitrary energydependent functions. The Schrödinger equation in the energydependent noncommutative algebra is derived for a twodimensional system for an arbitrary potential. The resulting equation reduces in the small energy limit to the standard quantum mechanical one, while for large energies the effects of the noncommutativity become important. We investigate in detail three cases, in which the noncommutative strengths are determined by an independent energy scale, related to the vacuum quantum fluctuations, by the particle energy, and by a quantum operator representation, respectively. Specifically, in our study we assume an arbitrary powerlaw energy dependence of the noncommutative strength parameters, and of their algebra. In this case, in the quantum operator representation, the Schrö dinger equation can be formulated mathematically as a fractional differential equation. For all our three models we analyze the quantum evolution of the free particle, and of the harmonic oscillator, respectively. The general solutions of the noncommutative Schrödinger equation as well as the expressions of the energy levels are explicitly obtained.
1 Introduction
It is generally believed today that the description of the spacetime as a manifold M, locally modeled as a flat Minkowski space \(M_0 =\mathbb {R} \times \mathbb {R}^3\), may break down at very short distances of the order of the Planck length \(l_P=\sqrt{G\hbar /c^3}\approx 1.6\times 10^{33}\) cm, where G, \(\hbar \) and c are the gravitational constant, Planck’s constant, and the speed of light, respectively [1]. This assumption is substantiated by a number of arguments, following from quantum mechanical and general relativistic considerations, which point towards the impossibility of an arbitrarily precise location of a physical particle in terms of points in spacetime.
One of the basic principles of quantum mechanics, Heisenberg’s uncertainty principle, requires that a localization \(\Delta x\) in spacetime can be reached by a momentum transfer of the order of \(\Delta p=\hbar /\Delta x\), and an energy of the order of \(\Delta E=\hbar c/\Delta x\) [2, 3, 4]. On the other hand, the energy \(\Delta E\) must contain a mass \(\Delta m\), which, according to Einstein’s general theory of relativity, generates a gravitational field. If this gravitational field is so strong that it can completely screen out from observations some regions of spacetime, then its size must be of the order of its Schwarzschild radius \(\Delta R\approx G\Delta m/c^2\). Hence we easily find \(\Delta R\approx G\Delta E/c^4=G\hbar /c^3\Delta x\), giving \(\Delta R\Delta x\approx G\hbar /c^3\). Thus the Planck length appears to give the lower quantum mechanically limit of the accuracy of position measurements [5]. Therefore the combination of the Heisenberg uncertainty principle with Einstein’s theory of general relativity leads to the conclusion that at short distances the standard concept of space and time may lose any operational meaning.
From a general physical point of view we can interpret the noncommutativity parameters \(\theta ^{\mu \nu }\) and \(\eta _{\mu \nu }\) as describing the strength of the noncommutative effects exerted in an interaction. In this sense they are the analogues of the coupling constants in standard quantum field theory.
It is a fundamental assumption in quantum field theory that the properties of a physical system (including the underlying force laws) change when viewed at different distance scales, and these changes are energy dependent. This is the fundamental idea of the renormalization group method, which has found fundamental applications in quantum field theory, elementary particle physics, condensed matter etc. [60].
It is the main goal of the present paper to introduce and analyze a dynamic noncommutative model of quantum mechanics, in which the noncommutative strengths \(\theta ^{\mu \nu }\) and \(\eta _{\mu \nu }\) are energydependent quantities. This would imply the existence of several noncommutative scales that range from the energy level of the standard model, where the lowenergy scales of the physical systems reduce the general noncommutative algebra to the standard Heisenberg algebra, and ordinary quantum mechanics, to the Planck energy scale. On energy scales of the order of the Planck energy, \(E_P=\sqrt{\hbar c^5/G}\approx 1.22\times 10^{19}\) GeV, the noncommutative effects become maximal. Under the assumption of the energy dependence of the noncommutative parameters, with the help of the generalized Seiberg–Witten map, we obtain the general form of the Schrödinger equation describing the quantum evolution in an energydependent geometry. The noncommutative effects can be included in the equation via a generalized quantum potential, which contains an effective (analogue) magnetic field, and an effective elastic constant, whose functional forms are determined by the energydependent noncommutative strengths.
The possibility of an energydependent Schrödinger equation was first suggested by Pauli [61], and it was further considered and investigated extensively (see [62, 63, 64, 65, 66, 67] and the references therein). Generally, the nonlinearity induced by the energy dependence requires modifications of the standard rules of quantum mechanics [62]. In the case of a linear energy dependence of the potential for confining potentials the saturation of the spectrum is observed, which implies that with the increase of the quantum numbers the eigenvalues reach an upper limit [65]. The energydependent Schrödinger equation was applied to the description of heavy quark systems in [63], where for a linear energy dependence the harmonic oscillator was studied as an example of a system admitting analytical solutions. A new quark interaction was derived in [64], by means of a Tamm–Dancoff reduction, from an effective field theory constituent quark model. The obtained interaction is nonlocal and energy dependent. Moreover, it becomes positive and rises up to a maximum value when the interquark distance increases. The quantum mechanical formalism for systems featuring energydependent potentials was extended to systems described by generalized Schrödinger equations that include a positiondependent mass in [67]. Modifications of the probability density and of the probability current need the adjustments in the scalar product and the norm. The obtained results have been applied to the energydependent modifications of the Mathews–Lakshmanan oscillator, and to the generalized Swanson system.
From a physical point of view we can assume that the energy dependent noncommutative effects can be described by two distinct energy scales. One is the energy scale of the spacetime quantum fluctuations, generated by the vacuum background and the zero point energy of the quantum fields. The noncommutativity is then essentially determined by this energy scale, which is independent of the particle energy. This is the first explicit model we are considering, a two energy scales model, in which the energy of the quantum fluctuations and the particle energy evolve in different and independent ways. The alternative possibility, in which the noncommutative strengths are dependent on the particle energy only, is also investigated. We consider the quantum evolution of the free particle and of the harmonic oscillator in these cases, and the resulting energy spectrum and wave functions are determined. The particle oscillation frequencies are either dependent on the vacuum fluctuation energy scale, or they have an explicit dependence on the particle energy. In the limiting case of small energies we recover the standard results of quantum mechanics.
As a simple application of the developed general formalism we consider the case in which the noncommutative strengths \(\eta \) and \(\theta \) are powerlaw functions of energy, with arbitrary exponents. However, the quantization of such systems, in which we associate an operator to the energy, requires the mathematical/physical interpretation of operators of the form \(\partial ^{\alpha }/\partial t^{\alpha }\), where \( \alpha \) can have arbitrary real) values, like, for example, \(\alpha =1/2\), \( \alpha =5/4\) etc. These types of problems belong to the field of fractional calculus [71, 72, 73], whose physical applications have been intensively investigated. In particular, the mathematical and physical properties of the fractional Schrö dinger equation, whose introduction was based on a purely phenomenological or abstract mathematical approach, have been considered in detail in [74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93]. It is interesting to note that the present approach gives a physical foundation for the mathematical use of fractional derivatives in quantum mechanics, as resulting from the noncommutative and energydependent structure of the spacetime. We present in full detail the fractional Schrödinger equations obtained by using two distinct quantization of the energy (the time operator and the Hamiltonian operator approach, respectively), and we investigate the quantum evolution of the free particle and of the harmonic oscillator in the time operator formalism for a particular simple choice of the energy dependent noncommutativity strength parameters.
The present paper is organized as follows. We introduce the energydependent noncommutative quantum geometry, its corresponding algebra, and the Seiberg–Witten map that allows one to construct the noncommutative set of variables from the commutative ones in Sect. 2. The Schrö dinger equation describing the quantum evolution in the energydependent noncommutative geometry is obtained in Sect. 3, where the form of the effective potential induced by the noncommutative effects is also obtained. Three relevant physical and mathematical mechanisms that could induce energydependent quantum behaviors in noncommutative geometry are discussed in Sect. 4, and their properties are explored in the framework of a particular model in which the noncommutativity parameters have a powerlaw dependence on energy. The quantum dynamics of a free particle and of the harmonic oscillator in the spacetime quantum fluctuations model is analyzed in Sect. 5, while the same physical systems are analyzed in the energy coupling model in Sect. 6. The fractional Schrödinger equations for the quantum evolution of general quantum systems in the energy operator approach are presented in Sect. 7, where the dynamics of the free particles and of the harmonic oscillator are analyzed in detail. A brief review of the fractional calculus is also presented. We discuss and conclude our results in Sect. 8.
2 Energydependent noncommutative geometry and algebra
In the following we will not consider timelike noncommutative relations, that is, we take \(\theta ^{0i}=0\) and \(\eta ^{0i}=0\), since otherwise the corresponding quantum field theory is not unitary.
3 The Schrödinger equation in the energydependent noncommutative geometry
3.1 Probability current and density in the energydependent potential
3.2 The free particle
3.3 The harmonic oscillator
4 Physical mechanisms generating energydependent noncommutative algebras, and their implications

We assume first that there exists an intrinsic and universal energy scale \(\varepsilon \), different of the particle energy scale E, which induces the noncommutative effects, and the corresponding algebra. This intrinsic universal energy scale could be related to the spacetime quantum fluctuations (SQF), and to the Planck energy scale, respectively. Therefore the energydependence in the commutation relations is determined by the \(\epsilon \) energy scale, or by the magnitude of the quantum fluctuations. Hence in this approach the dynamics of the quantum particle is determined by two independent energy scales.

For the second mechanism we assume that there is an energy coupling (EC) between the noncommutative evolution, and the dynamical energy E of the quantum systems. Hence in this approach the interaction between the particle dynamics and the spacetime fluctuations is fully determined by the particle energy, and all physical processes related to the energydependent noncommutativity are described in terms of the particle energy scale E.

Finally, the third mechanism we are going to consider follows from the possibility that the energy of a quantum system can be mapped to an energy operator, which modifies the Hamiltonian of the system, and the corresponding Schrödinger equation. This approach we call the EO (energy operator) approach; it assumes again that the dominant energy scale describing noncommutative effects is the particle energy scale, E.
4.1 The noncommutative algebra of the spacetime quantum fluctuations (SQF) model
4.2 The noncommutative algebra of the energy coupling (EC) model
4.3 The noncommutative algebra of the energy operator (EO) model
Finally, we consider the model in which the energydependent noncommutative geometry can be mapped to a quantum mechanical representation . This can be realized by associating a quantum operator to the considered energy scales \(\epsilon \) or E. There are two possibilities to construct such a mapping between energy and operators.
Since \(\alpha ,\beta \) are real variables, the energydependent noncommutative geometry in the EO model now involves fractional derivative differential equations.
Hence the energy operator (EO) representation of the energydependent noncommutative quantum mechanics leads to the emergence of fractional calculus for the physical description of the highenergy scale quantum processes. In general there are several definitions of the fractional derivatives, which we will discuss briefly in Sect. 7.
5 Quantum evolution in the spacetime quantum fluctuation (SQF) energydependent noncommutative model
In the present section we explore the physical implications of the SQF noncommutative algebra and the underlying quantum evolution. To gain some insights into the effects of the energydependent noncommutativity on the dynamics of quantum particles we analyze two basic models of quantum mechanics – the free particle and the harmonic oscillator, respectively.
5.1 Quantum mechanics of the free particle in the SQF model
5.2 The harmonic oscillator
6 Quantum dynamics in the energy coupling model
In the present section we investigate the two basic quantum mechanical models, the free particle, and the harmonic oscillator, respectively, in the energy coupling (EC) noncommutative algebra, by assuming that the noncommutativity parameters \(\theta \) and \(\eta \) are functions of the particle energy E only, and independent of energy scale of the quantum spacetime fluctuations.
6.1 Quantum evolution: the Schrödinger equation
6.2 Quantum evolution: wave function and energy levels
In the interval \(\left[ 0,\xi _{s}\right] \), the Bessel functions satisfy the condition \(\int _{0}^{\xi _{s}}J_{m_{\phi }}\left( j_{m_{\phi }l}\frac{x}{ \xi _{s}}\right) J_{m_{\phi }}\left( j_{m_{\phi }n}\frac{x}{\xi _{s}}\right) x{\text {d}}x=\left( 1/2\right) \xi _{s}^{2}\left[ J_{m_{\phi }+1}\left( j_{m_{\phi }l}\right) \right] ^{2}\delta _{ln }\), where \(j_{m_{\phi }l}\) is the lth zero of \(J_{m_{\phi }}\left( \xi \right) \) [68]. There is a large literature on the zeros of the Bessel functions; for a review and some recent results see [69].
6.3 The case of the free particle
6.4 The harmonic oscillator
7 Quantum evolution in the energy operator (EO) energydependent noncommutative geometry
Finally, we will consider in detail the third possibility of constructing quantum mechanics in the framework of energydependent noncommutative geometry. This approach consists in mapping the energy in the noncommutative algebra to an operator. As we have already discussed, we have two possibilities to develop such an approach, by mapping the energy to the time operator, or to the particle Hamiltonian. Under the assumption of a powerlaw dependence on energy of the noncommutative strengths, in the general case these maps lead to a fractional Schrödinger equation. In the following we will first write down the basic fractional Schrö dinger equations for the free particle and the harmonic oscillator case, and after that we will proceed to a detailed study of their properties. We will concentrate on the models obtained by the time operator representation of the energy. But before proceeding to discuss the physical implications of the generalized Schrödinger equation with fractional operators, we will present o very brief summary of the basic properties of the fractional calculus.
7.1 Fractional calculus: a brief review
7.2 The fractional Schrödinger equation
7.3 The free particle: the case \(\alpha =1\)
For simplicity, in the following we investigate the quantum dynamics in the energy operator representation only in the case of the free particle, by assuming \(V(\widehat{x},\widehat{y})=0\). Therefore \(k=0\), and the effective mass of the particle coincides with the ordinary mass, \(m_l^*=m\). Moreover, for simplicity we will restrict our analysis to the choice \(\alpha =1\). Then \( \mathcal {D}_{1}(t)=\frac{i}{\hbar }E\) and \(\mathcal {D}_{2}(t)=\frac{1}{ \hbar ^{2}}E^{2}\), respectively.
8 Discussions and final remarks
In the present paper we have considered the quantum mechanical implications of a noncommutative geometric model in which the strengths of the noncommutative parameters are energy dependent. From a physical point of view such an approach may be justified since the effects of the noncommutativity of the spacetime are expected to become apparent at extremely high energies, of the order of the Planck energy, and at distance scales of the order of the Planck length. By assuming an energydependent noncommutativity we obtain a smooth transition between the maximally noncommutative geometry at the Planck scale, and its commutative ordinary quantum mechanical version, which can be interpreted as the lowenergy limit of the noncommutative highenergy quantum mechanics. Hence this approach unifies in a single formalism two apparently distinct approaches, the noncommutative and commutative versions of quantum mechanics, respectively, and generally leads to an energydependent Schrödinger equation, as already considered in the literature [61, 62, 63, 64, 65, 66, 67].
One of the important question related to the formalism developed in the present paper is related to the physical implications of the obtained results. In the standard approach to noncommutative geometry, by using the linearity of the D map, one could find a representation of the noncommutative observables as operators acting on the conventional Hilbert space of ordinary quantum mechanics. More exactly, the D map converts the noncommutative system into a modified commutative quantum mechanical system that contains an explicit dependence of the Hamiltonian on the noncommutative parameters, and on the particular D map used to obtain the representation. The states of the considered quantum system are then wave functions in the ordinary Hilbert space; the dynamics is determined by the standard Schrödinger equation with a modified Hamiltonian that depends on the noncommutative strengths \(\theta \) and \(\eta \) [26]. Even though the mathematical formalism is dependent on the functional form of the adopted D map that is used to realize the noncommutativecommutative conversion, this is not the case for physical predictions of the theory such as expectation values and probability distributions [26]. On the other hand it is important to point the fact that the standard formalism in which the energy dependence is ignored is not manifestly invariant under a modification of the D map.
In order to implement the idea of energydependent noncommutativity we need to specify the relevant energy scales. In the present work we have assumed a two scale and a single energy scale model. Moreover, we have limited our investigations to the case in which the noncommutative strengths have a simple powerlaw dependence on the energy. In the first approach the energy dependence of the noncommutative strengths is determined by a specific energy scale, which is related to the energy of the quantum fluctuations that modify the geometry. This approach may be valid to describe physics very nearby the Planck scale, where the vacuum energy may be the dominant physical effect influencing the quantum evolution of particles in the noncommutative geometric setting. In this context we have considered the dynamics of two simple but important quantum systems, the free particle, and the harmonic oscillator, respectively. The physical characteristics of the evolution are strongly dependent on the energy of the quantum fluctuations, with the oscillations frequencies effectively determined by the spacetime fluctuation scale.
In our second model we have assumed that all the noncommutative effects can be described by means of the particle energy scale, which is the unique scale determining the physical implications of noncommutative geometry. The choice of a single energy scale allows the smooth transition from the noncommutative algebra of the Planck length to the commutative Heisenberg algebra of the ordinary quantum mechanics that gives an excellent description of the physical processes on the length and energy scales of the atoms and molecules, and for the standard model of elementary particles. In this case the basic physical parameters of the of the quantum dynamics of the free particles and of the harmonic oscillator are energy dependent, with the oscillation frequencies described by complicated functions of the particle energy. Such an energy dependence of the basic physical parameters of the quantum processes may have a significant impact on the highenergy evolution of the quantum particles.
Perhaps the most interesting physical implications are obtained in the framework of our third approach, which consists in mapping the energy with a quantum operator. There are two such possibilities we have briefly discussed, namely, mapping the energy with the time derivative operator, and with the Hamiltonian of the free particle (its kinetic energy). The corresponding Schrödinger equation changes its mathematical form, content an interpretation, becoming a fractional differential equation, in which the ordinary derivatives of quantum mechanics are substituted by fractional ones.
An interesting theoretical question in the field of noncommutative quantum mechanics is the problem of the nonlocality generated by the dynamical noncommutativity. This problem was investigated in [50] for a noncommutative quantum system with the coordinates satisfying the commutation relations \(\left[ \widehat{q}^i,\widehat{q}^j\right] =i\theta f(\sigma )\epsilon ^{ij}\), where \(f(\sigma )\) is a function of the physical parameter \(\sigma \), which could represent, for example, position, spin, or energy. Then for the uncertainty relation between the coordinate operators \(\widehat{x}\) and \(\widehat{y}\) we obtain the uncertainty relation \(\left( \Delta \widehat{x}\right) _{\Psi }\left( \Delta \widehat{x}\right) _{\Psi }\ge \left( \theta /2\right) \left \left<\Psi \left f(\sigma )\right \Psi \right>\right \) [50]. As was pointed out in [50], if \(\Psi>\) represents a stationary state of the quantum system, that is, an eigenstate of the Hamiltonian, it follows that the nonlocality induced by the noncommutativity of the coordinates will be a function of the energy. Moreover, in the present approach to noncommutative quantum mechanics, the noncommutative strength \(\theta \) is an explicit function of the energy, and therefore an explicit dependence of the nonlocality on the energy always appears. As a particular case we consider \(f(\sigma )=1\), that is, the noncommutative strengths depend only on the energy. Then, by taking into account the normalization of the wave function we obtain \(\left( \Delta \widehat{x}\right) _{\Psi }\left( \Delta \widehat{x}\right) _{\Psi }\ge \left( \theta (E)/2\right) \) or, by considering the explicit choice of the energy dependence of \(\theta \) adopted in the present study, \(\left( \Delta \widehat{x}\right) _{\Psi }\left( \Delta \widehat{x}\right) _{\Psi }\ge \left( \theta _0 /2\right) \left( E/E_0\right) ^{\beta }\). Hence when \(E<<E_0\), \(\left( \Delta \widehat{x}\right) _{\Psi }\left( \Delta \widehat{x}\right) _{\Psi }\approx 0\), and we recover the standard quantum mechanical result. In the case of the harmonic oscillator the uncertainty relations for the noncommutative coordinates can be obtained for \(f(\sigma )=\sigma \) as \(\left( \Delta \widehat{x}\right) _{\Psi }\left( \Delta \widehat{x}\right) _{\Psi }\ge O\left( \theta ^4\right) \), that is, nonlocality does not appear in higher orders of \(\theta \) [50]. For the case \(f(\sigma )=\sigma ^2\), one finds \(\left( \Delta \widehat{x}\right) _{\Psi }\left( \Delta \widehat{x}\right) _{\Psi }\ge \left( \theta /2\omega _{\sigma }\right) \left( n+1/2\right) +O\left( \theta ^3\right) \), where \(\omega _{\eta }\) is the oscillation frequency of the \(\sigma \)dependent potential term in the total Hamiltonian, given by \(V\left( \sigma \right) =\omega _{\sigma }^2 \sigma ^2/2\) [50].
A central question in the noncommutative extensions of quantum mechanics is the likelihood of its observational or experimental testing. A possibility of detecting the existence of the noncommutative phase space by using the Aharonov–Bohm effect was suggested in [33]. As we have already seen the noncommutativity of the momenta leads to the generation of an effective magnetic field and of an effective flux. In a mesoscopic ring this flux induces a persistent current. By using this effect it may be possible to detect the effective magnetic flux generated by the presence of the noncommutative phase space, even if it is very weak. Persistent currents and magnetic fluxes in mesoscopic rings can be studied by using experimental methods developed in nanotechnology [33]. The dynamics of a free electron in the twodimensional noncommutative phase space is equivalent to the evolution of the electron in an effective magnetic field, induced by the effects of the noncommutativity of the coordinates and momenta.
The investigation of the spacetime structure and physical processes at very high energies and small microscopic length scales may open the possibility of a deeper understanding of the nature of the fundamental interactions and of their mathematical description. In the present work we have developed some basic tools that could help to give some new insights into the complex problem of the nature of the quantum dynamical evolution processes at different energy scales, and of their physical implications.
Notes
Acknowledgements
We would like to thank the two anonymous reviewers for their comments and suggestions, which helped us to significantly improve our manuscript. T. H. would like to thank the Yat Sen School of the Sun Yat Sen University in Guangzhou, P. R. China, for the kind hospitality offered during the preparation of this work. S.D. L. thanks the Natural Science Foundation of Guangdong Province for financial support (grant No. 2016A030313313).
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