The concept of spatial noncommutativity has a long history in physics.^[1-2] It has attracted considerable attention in recent years due to superstring theories^[3-5] since it arises naturally in the D-branes at the presence of background NS-NS B-field.^[6-9] Fluctuations of D-branes are described by noncommutative gauge field theories. As a result, there are tremendous papers about quantum field theories in noncommutative space.^[10-13] Noncommutative quantum mechanics has also been studied extensively.^[14-20] The general study method is to map the noncommutative variables to the commutative ones, which satisfy the standard Heisenberg algebra by Bopp-shift (or generalized Bopp-shift), and then to solve dynamical equations in commutative space.^[21] Exactly solvable models, such as noncommutative harmonic oscillator, Landau problem and some relativistic quantum mechanics models^[22-23] are studied by using this method. The path integral formulation in noncommutative quantum mechanics has also been investigated.^[24-25] Recently, Chaichian et al. studied the relativistic hydrogen atom with noncommutative corrections perturbatively and found that the degeneracy of several energy levels was lifted due to spatial noncommutativity.^[26] Corrections to various quantum phases due to the spatial noncommutativity had also attracted many interests.^[27-31] Interestingly, based on the path integral formulation in noncommutative space, Refs. [32$-$33] proposed a semi-classical effective Lagrangian to study the Aharonov-Bohm effect^[34] in noncommutative space and presented explicit corrections due to the spatial noncommutativity.

The fractional angular momentum (FAM) has become a popular research topic since the early of 1980s^[35-36] because of its applications both in quantum Hall effect and high $T_c$ superconductivity.^[37-40] It has received renewed interests in recent years.^[41-43] As we know, eigenvalues of the canonical angular momentum should be quantized in three-dimensional space because of the non-Abelian rotation group. However, this conclusion does not hold any more in the $(2+1)$-dimensional space-time since the rotation group in two-dimensional space is an Abelian one which cannot impose any constraints on eigenvalues of the canonical angular momentum. Due to the dynamical nature of the Chern-Simons gauge field and in the absence of the Maxwell term, one can realize the FAM in $(2+1)$-dimensional space-time by coupling a charged particle to the Chern-Simons gauge field.^[44-47] Reference ^[48] found that it is possible to realize the FAM by coupling a cold ion to magnetic fields. This work was generalized to a noncommutative space in Ref. ^[49] It is argued that the FAM can also be generated by the spatial noncommutativity.^[50] The purpose of this work is to realize the FAM on the noncommutative plane. Different from Ref. ^[49] in which the FAM is realized by a trapped charged particle on the noncommutative plane, we realize it by a trapped neutral particle, i.e., a trapped atom which possesses a permanent magnetic dipole moment in the background of electric fields.

This paper is organized as follows. For the purpose of fixing our conventions and further studies, we start from the commutative plane in Sec. 2. Although this model has been investigated in Ref. ^[51], we shall analyze it by applying a different method. In Sec. 3, we generalize the model studied in Sec. 2 to the noncommutative plane. We show that there are two different mechanisms to realize the FAM on the noncommutative plane. Some concluding remarks will be given in last section.

In order to fix our convention, we re-exam the model which was proposed in Ref. ^[51] in this section by applying a different method. The Hamiltonian that describes dynamics of an atom with a permanent magnetic dipole moment in the background of an electric field is given by

where $m$, $\pmb p = -{\rm i} \hbar \mathbf \nabla$, $\mu$, $c$, $\pmb n$, and $\pmb E$ are the mass of the atom, canonical momentum, magnitude of the permanent magnetic dipole moment, speed of light in vacuum, unit vector along magnetic dipole moment, and the electric field respectively.

Hamiltonian (1) is the non-relativistic limit of a spin-half relativistic neutral particle, which possesses a permanent magnetic dipole moment interacting with an electric field. We assume that the atom is trapped by a harmonic potential and confined in a plane perpendicular to the magnetic dipole moment. Apart from these, two electric fields $ {\pmb E = {\pmb E}^{(1)} + {\pmb E}^{(2)}}$ are applied simultaneously, i.e.,

where ${\pmb E}^{(1)}$ is generated by a long filament with a uniform charges $\lambda_e$ per unit length, $\epsilon_0$ is the dielectric constant, $\rho_e$ is the density of electric charges which is the source of electric field ${\pmb E}^{(2)}$, ${\pmb e}_r$ is the unit vector along radial direction on the plane. In fact, only the electric field ${\pmb E}^{(2)}$ contributes to the term $\mathbf \nabla \cdot {\pmb E}$ since $ {\nabla \cdot {\pmb E}^{(1)}} =0$ in the area $r \neq 0$ where the atom moves. Therefore, one has ${\nabla \cdot {\pmb E}} ={\nabla \cdot {\pmb E}^{(2)} } = \rho_e/\epsilon_0$.

The Lagrangian which corresponds to Hamiltonian (1) is

Starting from this Lagrangian, Ref. ^[51] shows that eigenvalues of the canonical angular momentum of the reduced model which is obtained by cooling down the atom to the negligible kinetic energy, i.e., neglecting the kinetic energy term in the Lagrangian (2), will take fractional values. Here, we apply an alternative method to analyze this problem.

Observing the term ${\pmb n} \times {\pmb E}$ in Lagrangian (3) plays the same role as the magnetic vector potential in describing a charged particle in the background of a magnetic field, it is convenient to introduce the effective vector potential ${\pmb A}^{\rm eff} = {\pmb n} \times {\pmb E}$. The direction of the effective magnetic potential is parallel to the plane since the magnetic dipole moment and electric fields (2) are perpendicular and parallel to the plane respectively. As a result, the effective magnetic field which is given by ${\pmb B}_{\rm eff} ={\nabla \times {\pmb A}^{\rm eff}}$ obviously is perpendicular to the plane.

In terms of the effective vector potential, we write Lagrangian (3) in the form

Since we only focus on the dynamics on the plane, we write the above Lagrangian as

where a harmonic potential $(1/2)K x_i ^2$, which is applied to trap the atom is included and the summation convention is applied. In two-dimensional space, the effective magnetic vector potential and magnetic field are expressed explicitly as

Instead of contributing to the effective magnetic field, the electric field ${\pmb E}^{(1)}$ only has contributions to the effective vector potential due to its topological nature.

We shall show that the result in Ref. ^[51] can be reproduced by basing on the first-order Lagrangian (5). The first-order form of the Lagrangian (5) is

in which $H$ is given by

For the sake of further studies, we quantize model (7) canonically. To this end, we introduce canonical momenta $(\pi_i ^x, \ \pi_i ^p)$ with respect to variables ($x_i, \ p_i$) in the standard way. They are

in which we replace $p_i \dot x_i$ by a symmetric form $({1/2}) p_i \dot x_i - ({1/2}) x_i \dot p_i$ and drop a total time derivative term in Lagrangian (7). The non-vanishing Poisson brackets among canonical variables $(x_i, \ p_i, \ \pi_i ^x, \ \pi_i ^p)$ are $\{x_i, \ \pi_j ^x \} =\{p_i, \ \pi_j ^p \}= \delta_{ij}$. The canonical angular momentum is defined by

Since there are no "velocities" on the right-hand sides of Eq. (9), the introduction of canonical momenta leads to primary constraints.^[52] They are

where "$\approx $" is weak equivalence, which means equivalent on the constraint surface.

Since we are interested in the case of cooling down the atom to the limit of negligibly small kinetic energy, there are two additional constraints $\chi_i = p_i \approx 0$ appear in this limit. We treat all constraints $\chi_i \approx 0$, $\phi_i ^{(0)} \approx 0$, and $\ \psi_i ^{(0)} \approx 0$ on the same footing although they originate differently from model (7). We label them in a unified way as $\Phi_I = (\phi_i, \ \psi_i, \ \chi_i), \ I=1, 2, \ldots, 5, 6$. One must make sure whether there are secondary constraints before proceeding on. For this purpose, we apply the consistency condition to $\Phi_I \approx 0$,

where $H_T = H + \lambda_I \Phi_I$ is the total Hamiltonian with $\lambda_I$ being Lagrange multipliers.^[52]

The matrix of Poisson brackets among constraints $\Phi_I$, i.e., $\{\Phi_I, \ \Phi_J \}$ is $\{\Phi_I, \ \Phi_J \}$ which is the Poisson brackets among the constraints $\Phi_I$ is

from which we can calculate the determinant of the matrix $\{\Phi_I, \ \Phi_J \}$. It gives ${\rm det} \{\Phi_I, \ \Phi_J \}=(({\mu}/{c^2})B_{\rm eff})^2$. Thus, the consistency condition of the primary constraints (12) can only determine Lagrangian multipliers. According to Ref. ^[52], there are no secondary constraints and all constraints $\Phi_I \approx 0$ belong to the second class. Therefore, they can be regarded as "strong" equivalence and can be used to eliminate dependent degrees of freedom. After substituting constraints $\Phi_I \approx 0$ into the canonical angular momentum (10), we get

In order to get its eigenvalues, we must know commutators between variables $x_i$. The classical version of commutators, i.e., Dirac brackets between $x_i$ can be calculated according to the definition,$\{x_i, \ x_j\}_D = \{x_i, \ x_j\} - \{x_i, \Phi_I \} \{\Phi_I, \ \Phi_J \}^{-1} \{\Phi_J, \ x_j \}$, in which $\{\Phi_I, \ \Phi_J \}^{-1}$ is the inverse of the matrix $\{\Phi_I, \ \Phi_J \}$ and can be written explicitly as

After some direct algebraic calculations, we get $\{x_i, \ x_j \}_D = - {c^2 \epsilon_{ij}}/{\mu B_{\rm eff}}$.Thus, commutators between $x_i$ are $[x_i, \ x_j ]= - {{\rm i} c^2 \epsilon_{ij}}/{\mu B_{\rm eff}}$ (in the unit of $\hbar=1$).

Using above commutators between $x_i$, one finds that apart from a constant ${\mu \lambda_e}/{2 \pi c^2 \epsilon_0}$, the canonical angular momentum (14) is analogous to the Hamiltonian of a one-dimensional harmonic oscillator with unit frequency. Thus, eigenvalues of the canonical angular momentum are

Obviously, besides the "normal" part $n+ 1/2$, the last term which is proportional to $\lambda_e$ can take fractional values because of the classic parameter $\lambda_e$. Therefore, the eigenvalues of canonical angular momentum can take fractional values when the atom is cooled down to the limit of negligible kinetic energy. The result obtained by using this method is in accordance with previous work.^[51] The advantage of this method is that it can be easily generalized to the noncommutative plane as to be shown below.

We now generalize above studies to the noncommutative plane. We shall show that there are two different mechanisms to get the FAM on the noncommutative plane.

The noncommutative plane is characterized by the algebraic relations

$$[x_i, \ x_j] = {\rm i} \theta \epsilon_{ij}, \ [x_i, \ p_j] = i \delta_{ij}, \ [p_i, \ p_j] =0,$$

in which $\theta$ is the noncommutative parameter. It can be checked that the classical version of the above commutators can be realized by the first-order Lagrangian^[25,53-54]

in which $H$ is a specific Hamiltonian.

There are two different ways to incorporate a magnetic field in noncommutative quantum mechanics. One is to modify the commutators between momenta directly,^[14] the other is by the minimal coupling.^[53] It is shown that due to the Jacobi identity, the former way is only valid for the uniform magnetic field.^[54] In order to have a wider application, we choose the latter to introduce the magnetic field. The minimal coupling is achieved by substituting $p_i$ by $p_i + ({\mu}/{c^2}) A_i ^{\rm eff}$, i.e., $p_i \to p_i + ({\mu}/{c^2}) A_i ^{\rm eff}$ in the first term of Lagrangian (17). Thus, the Lagrangian in our model takes the following form

where we have symmetrized the term $p_i \dot x_i$ as $(1/2) p_i \dot x_i - (1/2) x_i \dot p_i$ and dropped a total time derivative term, $H$ is given in Eq. (8). Similar to the commutative case, in order to quantize it canonically, we introduce canonical momenta with respect to variables $x_i, \ p_i$ as

By definition, the canonical angular momentum takes the same form as Eq. (10).

The introduction of canonical momenta (19) will lead to primary constraints which are labeled as

where

$$H_T = H + \zeta_i \phi_i ^{(0)} + \xi_i \psi_i ^{(0)} $$

is the total Hamiltonian with $\zeta_i$, $\xi_i$ being Lagrange multipliers. After some algebraic calculations, we arrive at

in which

Obviously, Lagrange multipliers $\zeta_i, \ \xi_i$ can be determined provided $\kappa \neq 0$. In this case, there are no secondary constraints. On the contrary, %if $\kappa=0$,constraint chains are not ended and the consistency condition of primary constraints will produce further constraints. We will analyze these two cases separately and show that the FAM can arise from both cases by different mechanisms.

Based on the first-order Lagrangian (18), we study rotation properties of an atom which possesses a permanent magnetic dipole moment in the background of electric fields on a noncommutative plane. The interaction between a magnetic dipole moment and an electric field is similar with the minimal coupling between a charged particle and a magnetic field. Thus, it is convenient to introduce the effective magnetic vector potential and the corresponding effective magnetic field. Compared with previous work which realizes the FAM by charged particles, we realize it by an atom which possesses a permanent magnetic dipole moment. The present work can be regarded as a noncommutative generalization of the study in Ref. ^[51].

We show that there are two different ways to realize the FAM on the noncommutative plane. One is to cool down the atom to the negligible kinetic energy,the other is to modulate the intensity of the effective magnetic field to the critical value. The effective magnetic field relates the density of the electric charge by $B_{\rm eff} = {\rho_e}/{\epsilon_0}$, it means that the later way of getting the FAM is in fact, by modulating the density of the electric charges to the critical value $\rho_e = {c^2 \epsilon_0} /{\theta \mu}$.

These two ways belong to different mechanisms. In the former mechanism, two additional constraints $\chi _i = p_i \approx 0$ arise during the process of cooling down the atom. These two additional constraints originate differently from the ones $\phi_i \approx 0 \ {\rm and} \ \psi_i \approx 0$ which originate from the singularities of the first-order Lagrangian (18). This procedure of getting the FAM is analogous to the commutative counterpart of the model (18). The latter mechanism is related to the spatial noncommutativity of the plane where the atom moves. When the intensity of the effective magnetic field approaches the critical value, the consistency condition of primary constraints will lead to two secondary constraints. These two secondary constraints play crucial roles in producing the FAM.