Aharonov-bohm phase for an electric dipole on a noncommutative space

1. Introduction

Noncommutative space-time is introduced to the standard model of particle physics through noncommutative quantum field theory based on star products and Seiberg-Witten maps [ 1 ]. Since the standard model of particle physics has many unsolved problems and quantum field theory on a noncommutative space-time may provide some answers to these puzzles, certain amount of work have been done concerning noncommutative quantum field theory [ 2 – 6 ]. In addition, because finding the measurable effects of space noncommutativity is the central topic in noncommutative quantum mechanics (NCQM), various aspects of quantum mechanics (QM) have been studied to a very great extent on noncommutative space (NCS) and noncommutative phase space (NCPS). For example, NCQM is applied to the hydrogen-like atom and the corrections to the Lamb shift are calculated accordingly and the constraint on θ is obtained to be $1/\sqrt{\theta }\ge 10^{2}GeV$[ 7 – 9 ]. Transitions in the helium atom is studied and the constraint:$1/\sqrt{\theta }\ge 3GeV$ is provided in Haghighat and Loran [ 10 ]. The authors of Zhang [ 11 ] have suggested the possibility of testing spatial noncommutativity via cold Rydberg atoms. In Chaichian et al. [ 12 ], Chaichian et al. [ 13 ], Falomir et al. [ 14 ], Li and Dulat [ 15 ], Mirza and Zarei [ 16 ], Li and Wang [ 17 ], Mirza et al. [ 18 ], Singlton and Vagenas [ 19 ], Dulat and Ma [ 20 ] and Singleton and Ulbricht [ 21 ], the Aharonov-Bohm type phase is studied on a NCS and a NCPS. A lower bound is shown to be $1/\sqrt{\theta }\ge 10^{-6}GeV$ for the space noncommutativity parameter in Chaichian et al. [ 12 ]. In Mirza and Zarei [ 16 ], Li and Wang [ 17 ] and Mirza et al. [ 18 , 22 ], the Aharonov-Casher phase on a NCS and a NCPS has been studied for a spin-1/2 and a spin-1 particle. And in Mirza and Zarei [ 16 ] the limit for the space noncommutativity parameter is $1/\sqrt{\theta }\ge 10^{-7}GeV$. The authors of Harms and Micu [ 23 ], Dayi and Jellal [ 24 ] and Chakraborty et al. [ 25 – 30 ] have studied the noncommutative quantum Hall effect, and in Harms and Micu [ 23 ] a limit of $1/\sqrt{\theta }\ge 10GeV$ is given. The noncommutative spin Hall effect (SHE) is discussed through a semiclassical constrained Hamiltonian and interesting results are obtained in Dayi and Elbistan [ 31 ]. By studying the SHE in the framework of NCQM a lower limit for the noncommutative parameter is shown to be $1/\sqrt{\theta }\ge 10^{-12}GeV$ in Ma and Dulat [ 32 ]. Also Yakup and Dulat [ 33 ] studied the Harmonic oscillator influenced by gravitational wave in noncommutative quantum phase space. Ma et al. [ 34 ] investigated the time-dependent Aharonov-Bohm effect on the NCS. Studies about the NCQM in three dimensions and rotational symmetry can be found in Sinha et al. [ 35 ].

In this paper, we focus on the quantum effect of an electric dipole on a NCS. First, the definition of noncommutative space-time is given in the following. NCS is a deformation of ordinary space in which the space coordinate operators ${\hat{x}}_i$ satisfy the following relation:

$\begin{array}{ r}\left[{\hat{x}}_i,{\hat{x}}_j\right]=i{\theta }_{ij},& \left(1\right)\end{array}$$\begin{array}{ r}\left[{\hat{x}}_i,{\hat{p}}_j\right]=i\hslash {\delta }_{ij},& \left(2\right)\end{array}$$\begin{array}{ r}\left[{\hat{p}}_i,{\hat{p}}_j\right]=0,i,j=1,2,\cdots ,n.& \left(3\right)\end{array}$

θ _ij is the totally antisymmetric real tensor, which represent the noncommutativity of the space. In addition, the product between the external fields on a NCS is

$\begin{array}{ l}\hat{f}\left(\hat{\overrightarrow{x}}\right)\hat{g}\left(\hat{\overrightarrow{x}}\right)\equiv f\left(\overrightarrow{x}\right)\ast g\left(\overrightarrow{x}\right)=\exp \left[\frac{i{\theta }_{ij}}{2}{\partial }_{x_i}{\partial }_{y_j}\right]f\left(\overrightarrow{x}\right)g\left(\overrightarrow{y}\right){|}_{\overrightarrow{x}=\overrightarrow{y}},& \left(4\right)\end{array}$

where $f\left(\overrightarrow{x}\right)$ and $g\left(\overrightarrow{x}\right)$ are two arbitrary, infinitely differentiable functions on a commutative space, and $\hat{f}\left(\hat{\overrightarrow{x}}\right)$ and $ĝ\left(\hat{\overrightarrow{x}}\right)$ are the corresponding functions on a NCS. In NCQM, one replaces the position and momentum operators with the θ-deformed one,

$\begin{array}{ }{\hat{x}}_i=x_i+\frac{1}{\hslash }{\theta }_{ij}{p}_j,\\ {\hat{p}}_i=p_i,i,j=1,2\cdots ,n.& \left(5\right)\end{array}$

Here x _i and p _i are coordinate and momentum operators in usual quantum mechanics. In the presence of the electro-magnetic field, we also need to replace the electromagnetic potentials with the shifts given as follows

$\begin{array}{ }\hat{\text{A}}\left({\hat{x}}_i\right)=\text{A}\left(x_i-\frac{1}{2\hslash }{\theta }_{ij}{p}_j\right)\\ =\text{A}\left(x_i\right)-\frac{1}{2\hslash }{\partial }_i\text{A}\left(x_i\right){\theta }_{ij}{p}_j+{O}\left({\theta }^2\right)& \left(6\right)\end{array}$$\begin{array}{ }\hat{\Phi }\left({\hat{x}}_i\right)=\Phi \left(x_i-\frac{1}{2\hslash }{\theta }_{ij}{p}_j\right)\\ =\Phi \left(x_i\right)-\frac{1}{2\hslash }{\partial }_i\Phi \left(x_i\right){\theta }_{ij}{p}_j+{O}\left({\theta }^2\right)& \left(7\right)\end{array}$

and then applies the usual rules of QM.

This paper is organized as follows: in Section 2, on a NCS the phase of an electric dipole is obtained as an application of the AB effect to a system composed of two charges. In Section 3, the behavior of an electric dipole is analyzed on a NCS by using a Lagrangian approach. In Section 4, we derive the phase for electric dipole once more by using the Lagrangian approach, furthermore we prove the gauge invariance of the phase on a NCS. Conclusions are given in the last section.

2. Aharonov-Bohm Phase for an Electric Dipole on a Noncommutative Space

A charged particle moving around a magnetic flux in force-free regions acquires the Aharonov and Bohm [ 36 ] topological phase,

$\begin{array}{ l}{\varphi }_{AB}=\frac{q}{\hslash c}\oint \text{A}·d\text{x}.& \left(8\right)\end{array}$

For an electric dipole with total mass m = m ₁ + m ₂ may be considered as being composed of two charges ± q of mass m ₁ , m ₂ separated by small distancer= x ₁ −x ₂ . The phase for an electric dipole obtains through summing the AB phase of two charges ± q [ 37 ],

$\begin{array}{ }\varphi =\frac{q}{\hslash c}\oint \text{A}\left({\text{x}}_1\right)·d{\text{x}}_1-\frac{q}{\hslash c}\oint \text{A}\left({\text{x}}_2\right)·d{\text{x}}_2\\ =\frac{1}{\hslash c}\oint \left(\text{d}·\nabla \right)\text{A}·d\text{x},& \left(9\right)\end{array}$

whered= qris electric dipole moment, x= ( m ₁ x ₁ + m ₂ x ₂ )/ m is the position of the center of mass.

We can obtain the topological AB phase for an electric dipole on a NCS by replacingA( x _i ) in Equation (9) with the $\hat{{\text{A}}}\left({\hat{x}}_i\right)$,

$\begin{array}{ l}\hat{\varphi }=\frac{1}{\hslash c}\oint \left(\hat{\text{d}}·\nabla \right)\text{A}\left(x\right)·d\text{x},& \left(10\right)\end{array}$

where

$\begin{array}{ l}{\hat{d}}_j=d_j-\frac{1}{2\hslash }{d}_j{\theta }_{ik}{p}_k{\partial }_i& \left(11\right)\end{array}$

is the electric dipole moment on a NCS; here the second term $-\frac{1}{2\hslash }{d}_j{\theta }_{ik}{p}_k{\partial }_i$ is the correction for the electric dipole moment on a NCS. Equation (12) can also be written as

$\begin{array}{ }\hat{\varphi }=\frac{1}{\hslash c}\oint \left(\hat{\text{d}}·\nabla \right)\text{A}\left(x\right)·d\text{x}\\ =\frac{1}{\hslash c}\oint d_j{\partial }_j{A}_{m}d{x}_m+\frac{1}{2{\hslash }^{2}c}{d}_j{\theta }_{ik}{p}_k\oint {\partial }_i{\partial }_j{A}_{m}d{x}_m\\ =\varphi +\delta {\varphi }_{NCS},& \left(12\right)\end{array}$

where the additional phase δϕ _NCS , related to the non-commutativity of space, is given by

$\begin{array}{ l}\delta {\varphi }_{NCS}=\frac{1}{2{\hslash }^{2}c}{d}_j{\theta }_{ik}{p}_k\oint {\partial }_i{\partial }_j{A}_{m}d{x}_m.& \left(13\right)\end{array}$

The result (Equation 12) represents the noncommutative extension of the electric dipole phase (Equation 9). In the next section we will derive it by using a different approach.

3. Lagrangian on a Noncommutative Space

In this part we construct the Lagrangian for an electric dipole on NCS. The dipole Lagrangian on a commutative space is [ 37 ]

$\begin{array}{ }L=\frac{1}{2}{m}_1{\dot{\text{x}}}_1^2+\frac{1}{2}{m}_2{\dot{\text{x}}}_2^2-q\left[\Phi \left({\text{x}}_1\right)-\Phi \left({\text{x}}_2\right)\right]\\ +\left(q/c\right)\left[{\dot{\text{x}}}_1·\text{A}\left({\text{x}}_1\right)-{\dot{\text{x}}}_2·\text{A}\left({\text{x}}_2\right)\right]-U\left(\text{r}\right),& \left(14\right)\end{array}$

Using the expansion of Φ(x ₁ ) = Φ(x ₂ ) +r∇Φ(x ₂ ) andA(x ₁ ) = A(x ₂ ) +r∇·A(x ₂ ), with dipole approximation $\text{v}=\left(m_1{\dot{{\text{x}}}}_1+m_2{\dot{{\text{x}}}}_2\right)/m$, Thus Equation (16) becomes

$\begin{array}{ }L=\frac{1}{2}m{\text{v}}^2+\frac{1}{2}{m}_r{\dot{\text{r}}}^2+c^{-1}{\text{v}}_1·\left[\left(q\text{r}·\nabla \right)\text{A}\right]\\ +c^{-1}q\dot{\text{r}}·\text{A}-\left(q\text{r}·\nabla \right)\Phi -U\left(\text{r}\right)& \left(15\right)\end{array}$

where m _r = m ₁ m ₂ /( m ₁ + m ₂ ) is the reduced mass and Φ andAare evaluated atx ₂ . This result is different from Gianfranco Spavieri’s work [ 37 ].

The Lagrangian for electric dipole in NCS can be obtained by using the Bopp’s to scalar potential Φ(x ₂ ) and vector potentialA(x ₂ ).

$\begin{array}{ }\hat{\text{A}}\left({\hat{x}}_i\right)=\text{A}\left(x_i-\frac{1}{2\hslash }{\theta }_{ij}{p}_j\right)\\ =\text{A}\left(x_i\right)-\frac{1}{2\hslash }{\partial }_i\text{A}\left(x_i\right){\theta }_{ij}{p}_j+{O}\left({\theta }^2\right)& \left(16\right)\end{array}$$\begin{array}{ }\hat{\Phi }\left({\hat{x}}_i\right)=\Phi \left(x_i-\frac{1}{2\hslash }{\theta }_{ij}{p}_j\right)\\ =\Phi \left(x_i\right)-\frac{1}{2\hslash }{\partial }_i\Phi \left(x_i\right){\theta }_{ij}{p}_j+{O}\left({\theta }^2\right)& \left(17\right)\end{array}$

By inserting Equations (18) and Equation (19) into Equation (17), the Equation (17) becomes

$\begin{array}{ }\hat{L}=L-\frac{m_2}{2\hslash c}\left\{\right(q\text{r}·\nabla \left)\right[({\text{v}}_2·\theta ·\nabla )({\text{v}}_1·\text{A})]\\ +q\dot{r}·\left[\right({\text{v}}_2·\theta ·\nabla \left)\text{A}\right]-c({\text{v}}_2·\theta ·\nabla )(q\text{r}·\nabla \Phi )\}\\ =L+\delta L_{NCS}& \left(18\right)\end{array}$

with,

$\begin{array}{ }\delta L_{NCS}=-\frac{m_2}{2\hslash c}\left\{\right(q\text{r}·\nabla \left)\right[({\text{v}}_2·\theta ·\nabla )({\text{v}}_1·\text{A})]\\ +q\dot{r}·\left[\right({\text{v}}_2·\theta ·\nabla \left)\text{A}\right]-c({\text{v}}_2·\theta ·\nabla )(q\text{r}·\nabla \Phi )\}& \left(19\right)\end{array}$

comparing with Gianfranco Spavieri’s work [ 37 ], this is the correction term for Lagrangian in NCS. The difference depend on the non-commutativity θ of the space.

The canonical momentum for the center of mass in NCS is

$\begin{array}{ l}\hat{\text{P}}=\frac{\partial \hat{L}}{\partial \text{v}}=\frac{\partial L}{\partial \text{v}}+\frac{\partial \delta L_{NCS}}{\partial \text{v}}=\text{P}+\delta {\text{P}}_{NCS}& \left(20\right)\end{array}$

where

$\begin{array}{ l}\text{P}=m\text{v}+\frac{1}{c}\left(q\text{r}·\nabla \right)\text{A}=m\text{v}+\text{Q}\prime ,\text{Q}\prime =\frac{1}{c}\left(q\text{r}·\nabla \right)\text{A}& \left(21\right)\end{array}$$\begin{array}{ l}\delta {\text{P}}_{NCS}=-\frac{m}{2\hslash c}\left\{\frac{m}{m_1}\left(\text{d}·\nabla \right)\left(\text{v}·\theta ·\nabla \right)\text{A}+\left(\dot{\text{d}}·\theta ·\nabla \right)\text{A}\right\}& \left(22\right)\end{array}$

$\hat{{\text{P}}}$ is the deformed momentum of electric dipole in NCS. The second term δP _NCS of Equation (22) is the correction momentum for electric dipole in NCS, and this also depend non-commutativity of space.

Expression (Equation 20) leads to the equation of motion

$\begin{array}{ }\frac{d}{dt}\left(m\text{v}\right)=\nabla [{\text{v}}_1·\text{Q}\prime +\frac{1}{c}\dot{d}·\text{A}-(\text{d}·\nabla \left)\Phi \right]\\ -\frac{m_2}{2\hslash c}\nabla [{\text{v}}_1·(\text{d}·\nabla \left)\right({\text{v}}_2·\theta ·\nabla )\text{A}\\ +\dot{d}·({\text{v}}_2·\theta ·\nabla )\text{A}-c(\text{d}·\nabla )({\text{v}}_2·\theta ·\nabla )\Phi ]-\frac{d}{dt}\text{Q}\prime \\ +\frac{m}{2\hslash c}\left[\frac{m}{m_1}\right(\dot{d}·\nabla \left)\right(\text{v}·\theta ·\nabla )\text{A}-c(\dot{d}·\nabla \left)\right(\theta ·\nabla \left)\Phi \right]& \left(23\right)\end{array}$

with d / dt = ∂ _t +v·∇, and ∇ × ∇Φ = 0, and in terms of fields, Equation (25) reads

$\begin{array}{ }\frac{d}{dt}\left(m\text{v}\right)=[1-\frac{m}{2\hslash }({\text{v}}_2·\theta ·\nabla \left)\right]\left\{\right(\text{d}·\nabla )\text{E}\\ +\frac{1}{c}{\text{v}}_1\times [\nabla \times (\text{B}\times \text{d}\left)\right]-\frac{1}{c}\text{B}\times \dot{d}\}\\ +\frac{m}{2\hslash c}\left\{\frac{m}{m_1}\right(\text{v}·\theta ·\nabla \left)\right[\nabla (\dot{d}·\text{A})+(\text{B}\times \dot{d})]\\ +(\theta ·\nabla )(\dot{d}·\text{E})\}& \left(24\right)\end{array}$

The second, third term is also the correction term for equation of motion in NCS.

Dipole moves in force-free region where the fields are uniform, so the force free condition $\frac{d}{dt}\left(m{\text{v}}\right)=0$, and this gives $\left({\text{d}}·\nabla \right){\text{E}}+c^{-1}{{\text{v}}}_1\times \left[\nabla \times \left({\text{B}}\times {\text{d}}\right)\right]=0$; v· θ · ∇B= 0. This result is very important, we will make use of this in Section 4 to prove the gauge invariance of the phase.

The canonical momentum for relative coordinates is

$\begin{array}{ }{\hat{\text{P}}}^{\prime }=\frac{\partial \hat{L}}{\partial \dot{\text{r}}}=m_r\dot{\text{r}}+\frac{q}{c}\text{A}-\frac{m_2}{2\hslash c}q\left({\text{v}}_2·\theta ·\nabla \right)\text{A}\\ =m_r\dot{\text{r}}+\frac{q}{c}\text{A}+\delta {\text{P}}_{NCS}^{\prime }& \left(25\right)\end{array}$

where $\delta {{\text{P}}}_{NCS}^{\prime }=-\frac{m_2}{2\hslash c}q\left({{\text{v}}}_2·\theta ·\nabla \right){\text{A}}$.

The equation of motion for relative coordinates is

$\begin{array}{ l}\frac{d}{dt}\left(m_r\dot{\text{r}}\right)=q\left(\text{E}+\frac{{\text{v}}_1}{c}\times \text{B}\right)-\frac{q{m}_2}{2\hslash }\left({\text{v}}_2·\theta ·\nabla \right)\left(\text{E}+\frac{{\text{v}}_1}{c}\times \text{B}\right)& \left(26\right)\end{array}$

for the angular momentum ${\text{L}}={\text{r}}\times \left(m_r\dot{{\text{r}}}\right)$

$\begin{array}{ l}\frac{d}{dt}\text{L}=\text{d}\times \left[\left(\text{E}+\frac{{\text{v}}_1}{c}\times \text{B}\right)-\frac{m_2}{2\hslash }\left({\text{v}}_2·\theta ·\nabla \right)\left(\text{E}+\frac{{\text{v}}_1}{c}\times \text{B}\right)\right]& \left(27\right)\end{array}$

If the dipole moves in a region of space where fields are uniform, (d· ∇)E+ c ⁻¹ v× [∇ × (B×d)] = 0 in Equation (26) and ( d / dt )( m v+ c ⁻¹ B×d). In this case, v= −(1/ mc )B×d) + const may be inserted in Equation (28), which becomes an equation in the variabler′ only (here we assumev= v ₁ = v ₂ ). In most cases the solution of this equation represents a bound oscillatory motion and will contain terms of the type sin(ω t + φ) or cos(ω t + φ), where ω is the frequency and φ is a constant phase. The average of a dynamical variable is obtained by performing the average over the phase constants and one may expect that the variabled= q roscillated about the constant average equilibrium position

$\begin{array}{ }{\langle \text{d}\rangle }_{\phi }\simeq {\text{d}}_0\left(\theta \right)\simeq \alpha \left[\right(\text{E}+\frac{{\text{v}}_1}{c}\times \text{B})\\ -\frac{m_2}{2\hslash }({\text{v}}_2·\theta ·\nabla )(\text{E}+\frac{{\text{v}}_1}{c}\times \text{B})]+q\overline{x}\\ ={\text{d}}_0+{\text{d}}_0^{\prime }& \left(28\right)\end{array}$

where ${{\text{d}}}_0=\alpha \left({\text{E}}+c^{-1}{{\text{v}}}_1\times {\text{B}}\right)+q\overline{{\text{x}}};{{\text{d}}}_0^{\prime }=-\alpha \frac{m_2}{2\hslash }\left({{\text{v}}}_2·\theta ·\nabla \right)\left({\text{E}}+c^{-1}{{\text{v}}}_1\times {\text{B}}\right)$ is the correction term in NCS; α is the polarizability and $\overline{{\text{x}}}$ is the equilibrium position in the absence of external fields.

Because of Equation (30),${\hat{{\text{d}}}}_0$ is constant and independent ofxonly when the fields are uniform. In this case $\hat{{\text{d}}}$ is no longer a dynamical variable and, withE ₀ = −∇Φ, the Lagrangian assumes the simple form

$\begin{array}{ }{\hat{L}}_0=\frac{1}{2}m{\text{v}}^2+\frac{1}{c}\text{v}·\left[\right({\text{d}}_0·\nabla \left)\text{A}\right]+{\text{d}}_0·{\text{E}}_0\\ -\frac{m_2}{2\hslash c}\{{\text{v}}_1·({\text{d}}_0·\nabla \left)\right({\text{v}}_2·\theta ·\nabla )\text{A}-c({\text{d}}_0·\nabla \left)\right({\text{v}}_2·\theta ·\nabla \left)\Phi \right\}& \left(29\right)\end{array}$

4. The Hamiltonian and Gauge Invariance of the Phase on Noncommutative Space

Gauge invariance of the AB phase in the time dependent AB efect is discussed in MacDougall and Singleton [ 38 ]. In this paper, we discuss the quantum phase for an electric dipole on a NCS by solving the Schroedinger equation. Let H ( x, p ) be the Hamiltonian operator of the usual quantum system, then the static Schrödinger equation on NC space is usually written as

$\begin{array}{ l}H\left(x,p\right)*\psi =E\psi ,& \left(30\right)\end{array}$

where the Moyal-Weyl (or star) product between two functions is defined in Equation (4). On a NCS the star product can be changed into the ordinary product by replacing H ( x, p ) with $H\left(\hat{x},\hat{p}\right)$. Thus the Schrödinger equation can be written as,

$\begin{array}{ l}H\left({\hat{x}}_i,{\hat{p}}_i\right)\psi =H\left(x_i-\frac{1}{2\hslash }{\Theta }_{ij}{p}_j,p_i\right)\psi =E\psi .& \left(31\right)\end{array}$

Thus the Equation (31) is actually defined on a commutative space, and the noncommutative effects can be evaluated through the Θ related terms. Note that the Θ term always can be treated as a perturbation in QM, since Θ _ij < < 1. When magnetic field is involved, the Schrödinger equation (Equation 30) becomes

$\begin{array}{ l}H\left(x_i,p_i,A_i\right)*\psi =E\psi .& \left(32\right)\end{array}$

To replace the star product in Equation (32) with a usual product, we need to replace x _i , p _i , and A _i with the shifts given in Equations (5) and (7). Thus the Schrödinger Equation (32) in the presence of magnetic field becomes

$\begin{array}{ l}H(x_i-\frac{1}{2\hslash }{\Theta }_{ij}{p}_j,p_i,A_i+\frac{1}{2}{\Theta }_{lj}{p}_l{\partial }_j{A}_i)\psi =E\psi .& \left(33\right)\end{array}$

Now let us consider a particle of mass m and charge q moving in a magnetic field with magnetic potential A _i , then the Schrödinger equation is (we choose unit of ℏ = c = 1),

$\begin{array}{ l}\frac{1}{2m}{(p}_i-q{A}_i-\frac{1}{2}q{\Theta }_{lj}{p}_l{\partial }_j{A}_i){}^2\psi =E\psi .& \left(34\right)\end{array}$

Using the expressions of the canonical momenta Equations (22) and (27), the Hamiltonian Ĥ may be derived from the Lagrangian Equation (20). the corresponding Schrödinger equation $Ĥ\left(\hat{{\text{p}}},\hat{{\text{x}}}\right)\psi \left({\text{x}}\right)=E\psi \left({\text{x}}\right)$ in NCS reads

$\begin{array}{ }\{\frac{{(\hat{P}-\text{Q}\prime -\delta {\text{P}}_{NCS})}^2}{2m}+\frac{{({\hat{P}}^{\prime }-\frac{q}{c}\text{A}+\delta {\text{P}}_{NCS}^{\prime })}^2}{2m_r}\\ +q\text{r}·\nabla \Phi +U\left(\text{r}\right)\}\Psi \\ =E\Psi & \left(35\right)\end{array}$

The main difference between classical and quantum behavior is due to the existence of the quantum phase ϕ of the wave function which, through the process of interference, may lead to an observable phase shift △ϕ. We obtain the solution of the Schrödinger equation (Equation 32) by direct substitution

$\begin{array}{ }\text{Ψ}=exp\left(\frac{i}{\hslash }\int \text{Q}\prime ·d\text{x}\right)exp\left(\frac{i}{\hslash }\int \delta {\text{P}}_{NCS}·d\text{x}\right){\text{Ψ}}_0\\ =exp\left(i\varphi \right)exp\left(i\delta {\varphi }_{NCS}\right){\text{Ψ}}_0\\ =exp\left[i\left(\varphi +\delta {\varphi }_{NCS}\right)\right]{\text{Ψ}}_0& \left(36\right)\end{array}$

where $\delta {\varphi }_{NCS}=\frac{i}{\hslash }\int \delta {{\text{P}}}_{NCS}·d{\text{x}}$ is the correction term in NCS,${{\text{Q}}}^{\prime }=\frac{1}{c}\left({\text{d}}·\nabla \right){\text{A}}$, and Ψ ₀ solves the equation withA= 0, Φ = 0. Thus, the quantum phase ϕ coincides with Equation (14) in the Previous section.

In the interference experiments with particles possessing an electric dipole moment, the observable quantity is the phase shift $\left(1/\hslash c\right)\oint {\text{Q}}·d{\text{x}}=\left(1/\hslash c\right)\oint \left(\hat{{\text{d}}}·\nabla \right){\text{A}}·d{\text{x}},$ Since

$\begin{array}{ }\langle \text{Ψ}|\oint \left(\hat{\text{d}}·\nabla \right)\text{A}·d\text{x}|\text{Ψ}\rangle =\oint \left(\langle {\text{Ψ}}^{\prime }\left|\hat{\text{d}}\right|{\text{Ψ}}^{\prime }\rangle ·\nabla \right)\text{A}·d\text{x}\\ =\oint \left({\text{d}}_0·\nabla \right)\text{A}·d\text{x}+\oint \left({\text{d}}_0^{\prime }·\nabla \right)\text{A}·d\text{x}& \left(37\right)\end{array}$

where $\oint \left({{\text{d}}}_0^{\prime }·\nabla \right){\text{A}}·d{\text{x}}=\oint \left(\langle {\text{Ψ}}^{\prime }\left|{{\text{d}}}^{\prime }\right|{\text{Ψ}}^{\prime }\rangle ·\nabla \right){\text{A}}·d{\text{x}}$. its expectation value reads

$\begin{array}{ }△\hat{\varphi }=\frac{1}{\hslash c}\oint \left({\text{d}}_0·\nabla \right)\text{A}·d\text{x}+\frac{1}{\hslash c}\oint \left({\text{d}}_0^{\prime }·\nabla \right)\text{A}·d\text{x}\\ =\frac{1}{\hslash c}\oint \left[\text{B}\times {\text{d}}_0+\nabla \left({\text{d}}_0·\text{A}\right)\right]·d\text{x}\\ +\frac{1}{\hslash c}\oint \left({\text{d}}_0^{\prime }·\nabla \right)\text{A}·d\text{x}& \left(38\right)\end{array}$

the relevant termB×d ₀ known as Röntgen interaction. the second term $\frac{1}{\hslash c}\oint \left({{\text{d}}}_0^{\prime }·\nabla \right){\text{A}}·d{\text{x}}$ is our correction term for Gianfranco Spavieri’s work [ 37 ] in NCS.

Our result (Equation 35) for the phase shift of an electric dipole in NCS differs from that proposed by other authors (Gianfranco Spavieri) for the presence of the extra term $\frac{1}{\hslash c}\oint \left({{\text{d}}}_0^{\prime }·\nabla \right){\text{A}}·d{\text{x}}$.

It can be shown that the phase shift $△\hat{\varphi }$ is also gauge independent in NCS. Gauge theory in NCS is different from the case in usual commutative space. Gauge transformation in NCS is $A_{\mu }^{\prime }=A_{\mu }+{\delta }_{\lambda }{A}_{\mu },{\delta }_{\lambda }{A}_{\mu }={\partial }_{\mu }\lambda \left(x\right)+i{\left[\lambda \left(x\right),A_{\mu }\right]}_{*}$. Here we consider the three dimensional case: δ _λ A= ∇λ( x ) + i [λ( x ), A]_*.

Using the expression (Equation 35) in order to point out some properties of the AB phase shift, we write the contribution to the phase shift due to the gauge transformation as

$\begin{array}{ }\delta \hat{\varphi }=\frac{1}{\hslash c}\oint \left[\left({\text{d}}_0+{\text{d}}_0^{\prime }\right)·\nabla \right]{\delta }_{\lambda }\text{A}·d\text{x}\\ =\frac{q}{\hslash c}\oint \left(\nabla \lambda \right)·d\text{x}+\frac{q}{\hslash c}\oint \left(\text{v}·\theta ·\nabla \right)\left(\nabla \lambda \right)·d\text{x}\\ -\frac{1}{\hslash c}\oint \left({\text{d}}_0·\nabla \right)\nabla \lambda ·\theta ·\left(\nabla \text{B}\right)·d\text{s}\\ -\frac{1}{\hslash c}\oint \left[\left(\text{v}·\theta ·\nabla \right){\text{d}}_0·\nabla \right]\left[\nabla \lambda ·\theta ·\left(\nabla \text{B}\right)\right]·d\text{s}\\ =0& \left(39\right)\end{array}$

This gauge independence based on the fact that the scalar function λ is a monovalued function for which ∮(∇λ)· d x= 0, and θ·(∇B) = 0 which we obtained under the force free condition.

5. Conclusion

In this paper we studied the noncommutative non-relativistic behavior of a neutral particle, which possessing electric dipole moments, in the presence of external electric and magnetic fields. For a special configuration of the field, we derived the phase of an electric dipole as an application of the AB effect to a system composed of two charges in NCS. We have shown that a topological phase of the AB type is a generic effect in dipole moment of neutral particles.

The result of this paper indicates that there is some difference between usual and NCQM studying same problem. For example, our Lagrangian, canonical momentum, phase, equation of motions are different from Gianfranco Spavieri’s work [ 37 ] who studied the same problem in usual commutative space. The difference depend on the noncommutativity θ and $\overline{\theta }$ of the space. We may say that, our extra terms for theses quantities are the correction term for Gianfranco Spavieri’s work [ 37 ] in NCS. Obviously, if noncommutative parameter θ = 0, the phase in NCS is to be that in commutative space. If the present experimental situation is attainable, this effect in NCS might be tested at very high energy level, and the experimental observation of the effect remains to be further studied.

Author Contributions

MA and RR designed the study; AA and MH carried out the theoretical calculations independently; MA performed the theoretical analysis; RR wrote the manuscript; MA edited the final manuscript.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of China Grant Nos. 11465018, 61501026, 11347031, and XJEDU2014I058. We thank Prof. Sayipjamal Dulat for many helpful discussions.

References

1. Seiberg N, Witten E. String theory and noncommutative geometry. JHEP (1999)32: 9909.

2. Godfrey S, Doncheski MA. Signals for noncommutative QED in eY and YY collisions. Phys Rev.(2001)D65: 015005. doi: 10. 1103/PhysRevD. 65. 015005

CrossRef Full Text | Google Scholar

3. Haghighat M, Ettefaghi MM. Parton model in Lorentz invariant noncommutative space. Phys Rev.(2004)D70: 034017. doi: 10. 1103/PhysRevD. 70. 034017

CrossRef Full Text | Google Scholar

4. Devoto A, DiChiara S, Repko WW. Noncommutative QED corrections to e+e- at linear collider energies. Phys Rev.(2005)D72: 056006. doi: 10. 1103/PhysRevD. 72. 056006

CrossRef Full Text | Google Scholar

5. Nicolini P. Noncommutative black holes, the final appeal to quantum gravity: a review. Int J Modern Phys.(2009)4: 1229. doi: 10. 1142/S0217751X09043353

CrossRef Full Text | Google Scholar

6. Calmet X. Quantum electrodynamics on noncommutative spacetime. Eur Phys J.(2007)C50: 113. doi: 10. 1140/epjc/s10052-006-0192-4

CrossRef Full Text | Google Scholar

7. Chaichian M, Sheikh-Jabbari MM, Tureanu A. Hydrogen atom spectrum and the lamb shift in noncommutative QED. Phys Rev Lett.(2001)86: 2716. doi: 10. 1103/PhysRevLett. 86. 2716

PubMed Abstract | CrossRef Full Text | Google Scholar

8. Dossa AF, Avossevou GYH. Noncommutative space and the two dimensional quantum dipole in background electric and magnetic fields. J Modern Phys.(2013)4: 1400. doi: 10. 4236/jmp. 2013. 410168

CrossRef Full Text | Google Scholar

9. Adorno TC, Baldiotti MC, Chaichian M, Gitman DM, Tureanu A. Dirac equation in noncommutative space for hydrogen atom. Phys Lett.(2009)B682: 235. doi: 10. 1016/j. physletb. 2009. 11. 003

CrossRef Full Text | Google Scholar

10. Haghighat M, Loran F. Three body bound state in noncommutative space. Phys Rev.(2003)D67: 096003. doi: 10. 1103/physrevd. 67. 096003

CrossRef Full Text | Google Scholar

11. Zhang J-Z. Testing spatial noncommutativity via Rydberg atoms. Phys Rev Lett.(2004)93: 043002. doi: 10. 1103/PhysRevLett. 93. 043002

PubMed Abstract | CrossRef Full Text | Google Scholar

12. Chaichian M, Demichev A, Presnajder P, Sheikh-Jabbari MM, Tureanu A. Aharonov-Bohm effect in noncommutative spaces. Phys Lett.(2002)B527: 149. doi: 10. 1016/S0370-2693(02)01176-0

CrossRef Full Text | Google Scholar

13. Chaichian M, Demichev A, Presnajder P, Sheikh-Jabbari MM, Tureanu A. Quantum theories on noncommutative spaces with nontrivial topology: Aharonov-Bohm and Casimir effects. Nucl Phys.(2001)B611: 383. doi: 10. 1016/S0550-3213(01)00348-0

CrossRef Full Text | Google Scholar

14. Falomir H, Gamboa J, Loewe M, Méndez F, Rojas JC. Testing spatial noncommutativity via the Aharonov-Bohm effect. Phys Rev.(2002)D66: 045018. doi: 10. 1103/PhysRevD. 66. 045018

CrossRef Full Text | Google Scholar

15. Li K, Dulat S. The Aharonov-Bohm effect in noncommutative quantum mechanics. Eur Phys J.(2006)C46: 825. doi: 10. 1140/epjc/s2006-02538-2

CrossRef Full Text | Google Scholar

16. Mirza B, Zarei M. Non-commutative quantum mechanics and the Aharonov-Casher effect. Eur Phys J.(2004)C32: 583. doi: 10. 1140/epjc/s2003-01522-8

CrossRef Full Text | Google Scholar

17. Li K, Wang J-H. The topological AC effect on noncommutative phase space. Eur Phys J.(2007)C50: 1007. doi: 10. 1140/epjc/s10052-007-0256-0

CrossRef Full Text | Google Scholar

18. Mirza B, Narimani R, Zarei M. Aharonov-Casher effect for spin-1 particles in a non-commutative space. Eur Phys J.(2006)C48: 641. doi: 10. 1140/epjc/s10052-006-0047-z

CrossRef Full Text | Google Scholar

19. Singlton D, Vagenas E. The covariant, time-dependent Aharonov-Bohm Effect. Phys Lett.(2013)B723: 241. doi: 10. 1016/j. physletb. 2013. 05. 014

CrossRef Full Text | Google Scholar

20. Dulat S, Ma K. Aharonov-Casher and scalar Aharonov-Bohm topological effects. Phys Rev Lett.(2012)108: 070405. doi: 10. 1103/PhysRevLett. 108. 070405

PubMed Abstract | CrossRef Full Text | Google Scholar

21. Singleton D, Ulbricht J. The time-dependent Aharonov-Casher effect. Phy Lett.(2016)B753: 91. doi: 10. 1016/j. physletb. 2015. 12. 002

CrossRef Full Text | Google Scholar

22. Dulat S, Li K. The Aharonov-Casher effect for spin-1 particles in non-commutative quantum mechanics. Eur Phys J.(2008)C54: 333. doi: 10. 1140/epjc/s10052-008-0522-9

CrossRef Full Text | Google Scholar

23. Harms B, Micu O. Noncommutative quantum Hall effect and Aharonov-Bohm effect. J. Phys.(2007)A40: 10337. doi: 10. 1088/1751-8113/40/33/024

CrossRef Full Text | Google Scholar

24. Dayi OF, Jellal A. Hall effect in noncommutative coordinates. J Math Phys.(2002)43: 4592. doi: 10. 1063/1. 1504484

CrossRef Full Text | Google Scholar

25. Chakraborty B, Gangopadhyay S, Saha A. Seiberg-Witten map and Galilean symmetry violation in a noncommutative planar system. Phys Rev.(2004)D70: 107707. doi: 10. 1103/PhysRevD. 70. 107707

CrossRef Full Text | Google Scholar

26. Kokado A, Okamura T, Saito T. Noncommutative phase space and the Hall effect. Prog Theor Phys.(2003)110: 975. doi: 10. 1143/PTP. 110. 975

CrossRef Full Text | Google Scholar

27. Dulat S, Li K. Quantum Hall effect in noncommutative quantum mechanics. Eur Phys J.(2009)C60: 163. doi: 10. 1140/epjc/s10052-009-0886-5

CrossRef Full Text | Google Scholar

28. Scholtz FG, Chakraborty B, Gangopadhyay S, Hazra AG. Dual families of non-commutative quantum systems. Phys Rev.(2005)D71: 085005. doi: 10. 1103/PhysRevD. 71. 085005

CrossRef Full Text | Google Scholar

29. Scholtz FG, Chakraborty B, Gangopadhyay S, Govaerts J. Interactions and non-commutativity in quantum Hall systems. J Phys.(2005)A38: 9849. doi: 10. 1088/0305-4470/38/45/008

CrossRef Full Text | Google Scholar

30. Dayi OF, Jellal A. Erratum: Hall effect in noncommutative coordinates. J. Math. Phys.(2004)45: 827.

Google Scholar

31. Dayi OF, Elbistan M. Spin Hall effect in noncommutative coordinates. Phys Lett.(2009)A373: 131. doi: 10. 1016/j. physleta. 2009. 02. 029

CrossRef Full Text | Google Scholar

32. Ma K., Dulat S. Spin Hall effect on a noncommutative space. Phys Rev.(2011)A84: 012104. doi: 10. 1103/PhysRevA. 84. 012104

CrossRef Full Text

33. Yakup R, Dulat S. The harmonic oscillator influenced by gravitational wave in noncommutative quantum phase space. Int J Theor Phys.(2014)53: 1404. doi: 10. 1007/s10773-013-1937-0

CrossRef Full Text | Google Scholar

34. Ma K, Wang J-H, Yang H-X. Time-dependent Aharonov-Bohm effect on the noncommutative space. (2016). ARXIV: 1604. 02110

Google Scholar

35. Sinha D, Chakraborty B, Scholtz FG. Non-commutative Quantum Mechanics in Three Dimensions and Rotational Symmetry. J Phys.(2012)A45: 105308. doi: 10. 1088/1751-8113/45/10/105308

CrossRef Full Text | Google Scholar

36. Aharonov Y, Bohm D. Significance of electromagnetic potentials in the quantum theory. Phys Rev.(1959)115: 48. doi: 10. 1103/PhysRev. 115. 485

CrossRef Full Text | Google Scholar

37. Spavieri G. Quantum effect for an electric dipole. Phys Rev.(1998)A59: 3194.

Google Scholar

38. MacDougall J, Singleton D. Baryogenesis via Hawking-like radiation in the FRW space-time. J Math Phys.(2014)55: 042101. doi: 10. 1063/1. 4869324

CrossRef Full Text