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Time-dependent Aharonov–Casher effect on noncommutative space

  • Tao Wang ,
  • Kai Ma
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  • Department of Physics, Shaanxi University of Technology, Hanzhong 723000, Shaanxi, China

Received date: 2022-08-29

  Revised date: 2022-10-23

  Accepted date: 2022-11-10

  Online published: 2022-12-22

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© 2022 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

In this paper, we study the time-dependent Aharonov–Casher effect and its corrections due to spatial noncommutativity. Given that the charge of the infinite line in the Aharonov–Casher effect can adiabatically vary with time, we show that the original Aharonov–Casher phase receives an adiabatic correction, which is characterized by the time-dependent charge density. Based on Seiberg–Witten map, we show that noncommutative corrections to the time-dependent Aharonov–Casher phase contains not only an adiabatic term but also a constant contribution depending on the frequency of the varying electric field.

Cite this article

Tao Wang , Kai Ma . Time-dependent Aharonov–Casher effect on noncommutative space[J]. Communications in Theoretical Physics, 2023 , 75(1) : 015203 . DOI: 10.1088/1572-9494/aca1ac

1. Introduction

Successful observations of gravitational waves indicate that General Relativity is an excellent theory of gravitational force. However, it is very likely that spacetime is not only physical but also has quantum properties. Snyder pointed out that spacetime can be quantized, and Lorentz invariance can be preserved simultaneously [13]. A fundamental length is required to characterize the importance of the quantized spacetime and is naturally related to the gravitational effect [4]. Spacetime with a discrete spectrum was also motivated by string theory [5, 6] and quantum gravity [7]. The corresponding operators are noncommutative and described by the algebra [xμ, xν] = iθμν. Here θμν is a totally anti-symmetric constant with a dimension of length-squared and characterizes strength and relative directions of the spacetime noncommutativity [8, 9]. The above noncommutative algebra can also be realized by introducing a Moyal–Weyl product (also called ⋆-product) defined as
$\begin{eqnarray}f(x)\,\star \,g(x)=\exp \left[\displaystyle \frac{{\rm{i}}}{2}{\theta }_{\mu \nu }{\partial }_{{x}_{\mu }}{\partial }_{{y}_{\nu }}\right]f(x)g(y){| }_{x=y},\end{eqnarray}$
where f(x) and g(x) are two arbitrary infinitely differentiable functions on the commutative R3+1 space-time. Quantum field theory on the noncommutative spacetime can be obtained by replacing ordinary product by the ⋆-product [8].
Phenomenology of the noncommutative spacetime has been extensively studied, particularly the noncommutativily extended electromagnetic interactions [919]. It was shown that Lorentz symmetry was violated if the tenser θμν is non-dynamical [20]. Electromagnetic interactions of dipole moments can receive additional contributions [2128], and degenerate levels of hydrogen energy spectrum can be removed [2931]. Apart from the dynamical effects, topological properties of the ordinary electromagnetic theory can also be distorted, for instance, the Aharonov–Bohm (AB) effect [32] and the Aharonov–Casher (AC) effect [33] on noncommutative spacetime [3448] as well as in the space-time of topological defects [4951], can receive non-trivial corrections. Both AB and AC phases are purely quantum effects, and are closely related to gauge symmetry of the electromagnetic interactions. The topological property of the ordinary AB phase was clearly confirmed by the excellent electron holography experiment by Tonomura et al [52, 53], in which leakage of the magnetic fields is avoided by using a superconducting ring [53]. The ordinary AC phase has also been observed by using neutron interferometry [54] and atomic system [55]. However, it was pointed out that noncommutative corrections based on simple ⋆-product methods are not gauged invariant [36, 56]. This gauge symmetry problem can be solved by employing the Seiberg–Witten (SW) map [6], which is defined as a set of transformations of fields from the noncommmutative space to the ordinary one [57, 58]. Based on the SW map, noncommutative corrections on the AC phase were studied in [46], and corrections on the time-dependent AB effect were studied in [59].
In this paper, we study noncommutative corrections on the time-dependent AC effect. It was shown that to first order approximation, the time-dependent AC phase in a linearly polarized electromagnetic plane wave is identical to the static one [60]. Here we consider a different configuration, in which the time-dependence of the electromagnetic fields comes from a time-dependent charge density of the infinite charged line in the ordinary AC configuration. And we assume that variation of the charge density is adiabatic, such that the electric field does not change in the time of making a round for the moving particle. Based on this adiabatic approximation and employing the SW map, we will study the time-dependent AC effect on the noncommutative space.
The contents of this paper are organized as follows: in section 2, we introduce the effective U(1) gauge symmetry of the electromagnetic interaction of a magnetic dipole moment on 2 + 1 dimensional spacetime and interpretation of the AC effect. Then we study its adiabatic extension with a time-dependent electric field. In section 3, Employing the SW map, we study noncommutative corrections on the time-independent AC phase. our conclusions are given in the final section, section 4.

2. Effective U(1) symmetry and time-dependent AC effect

In this section, we discuss the effective U(1) gauge symmetry interpretation of the AC effect [59, 61], and its adiabatic extension. Following [59, 61], we restrict ourselves to the 2 + 1 dimensional spacetime. Hereafter index of the spatial vector in the 2D plane can take only the values of 1 and 2. A vector normal to the plane is defined as the third direction of the ordinary 3D space, and matches to a tensor of order 2 with index ‘12', for instance, T12, in the 2D plane. For a neutral particle of spin 1/2 having an anomalous magnetic dipole moment μm, its electromagnetic interaction on the ordinary spacetime is described by following Lagrangian
$\begin{aligned} \mathcal{L}= & \bar{\psi}(x)(\not p-m) \psi(x) \\ & -\frac{1}{2} \mu_{m} \bar{\psi}(x) \sigma^{\mu \nu} \psi(x) F_{\mu \nu}, \end{aligned}$
where m is the mass of the fermion ψ. On 2 + 1 dimensional spacetime, the metric and anti-symmetric tensors are defined as gμν = diag(1, −1, −1) and ε012 = 1, and the electromagnetic field strength tensor is given as
$\begin{eqnarray}{F}^{\mu \nu }=\left(\begin{array}{ccc}0 & -{E}^{1} & -{E}^{2}\\ {E}^{1} & 0 & -{B}^{3}\\ {E}^{2} & {B}^{3} & 0\end{array}\right),\end{eqnarray}$
where Ei and Bi are the electric and magnetic fields, respectively (the indices ‘i = 1, 2, 3' stands for the x, y and z directions on ordinary 3 + 1 dimensional spacetime, respectively). The Dirac matrixes can be chosen as follows [61]
$\begin{eqnarray}\begin{array}{rcl}{\gamma }^{0} & = & \left(\begin{array}{cc}{\sigma }^{3} & 0\\ 0 & {\sigma }^{3}\end{array}\right),\,{\gamma }^{1}=\left(\begin{array}{cc}{\rm{i}}{\sigma }^{2} & 0\\ 0 & -{\rm{i}}{\sigma }^{2}\end{array}\right),\\ {\gamma }^{2} & = & \left(\begin{array}{cc}{\rm{i}}{\sigma }^{1} & 0\\ 0 & {\rm{i}}{\sigma }^{1}\end{array}\right).\end{array}\end{eqnarray}$
We can see that all the γ-matrix are block diagonal in this representation, and satisfy an equation ${\gamma }_{\mu }{\gamma }_{\nu }={g}_{\mu \nu }+{\rm{i}}{ \mathcal S }{\epsilon }_{\mu \nu \alpha }{\gamma }^{\alpha }$ with ${ \mathcal S }=-{\rm{i}}{\gamma }^{0}{\gamma }^{1}{\gamma }^{2}=-{\gamma }^{0}{\sigma }^{12}$. On can easily find that ${ \mathcal S }=\mathrm{diag}(I,-I)$, i.e. the operator ${ \mathcal S }$ is also diagonal in this representation, and has eigenvalues s = ±1. Furthermore, with a straightforward calculation, one can obtain that ${\sigma }_{\mu \nu }=-{ \mathcal S }{\epsilon }_{\mu \nu \alpha }{\gamma }^{\alpha }$, which is again diagonal. Therefore, within this representation, dynamical evolutions of the ‘up' and ‘down' components of the 4-spinor, which can be defined as eigenstates of the operator ${ \mathcal S }$ with eigenvalues s = ±1, are completely independent. Without loss of generality, hereafter, we assume that the magnetic dipole moment is polarized along the z direction, i.e. ${\vec{\mu }}_{m}\parallel {\vec{n}}_{z}$ (${\vec{n}}_{z}$ is the unite vector along the z direction). In virtue of this property, the Lagrangian (2) can be written as
$\begin{aligned} \mathcal{L}= & \bar{\psi}_{s}(x)(\not p-m) \psi_{s}(x) \\ & -s \mu_{m} \bar{\psi}_{s}(x) \not S (x) \psi_{s}(x) \end{aligned}$
where the spinor ψs are eigenstates of the operator ${ \mathcal S }$, and the effective vector potential is defined as
$\begin{eqnarray}{S}^{\alpha }(x)=-\displaystyle \frac{1}{2}{\epsilon }^{\alpha \mu \nu }{F}_{\mu \nu }(x).\end{eqnarray}$
Within this formulation, the Lagrangian (2) can be rewritten into a minimal-coupling form
$\begin{eqnarray}\begin{array}{rcl}{ \mathcal L } & = & \bar{\psi }(x)\left({\rm{i}}{\gamma }_{\alpha }{{ \mathcal D }}^{\alpha }-m\right)\psi (x),\\ {{ \mathcal D }}^{\alpha } & = & {\partial }^{\alpha }+{\rm{i}}s{\mu }_{m}{S}^{\alpha },\end{array}\end{eqnarray}$
where ${{ \mathcal D }}^{\alpha }$ is the effective covariant derivative. Similar to the well-known U(1) gauge symmetry of electrodynamics for charged particles, this minimal coupling form means a new U(1) symmetry with the gauge potential Sμ, and s is the sign of the ‘charge' of the neutral particle ψ, μm is the magnitude of the ‘charge'. It is easy to check that the Lagrangian (5) is invariant under the following gauge transformations
$\begin{eqnarray}\begin{array}{rcl}{\psi }_{s}(x) & \to & {{\rm{e}}}^{-{\rm{i}}{\mu }_{m}s\lambda (x)}{\psi }_{s}(x),\\ {S}_{\mu }(x) & \to & {S}_{\mu }(x)-{\partial }_{\mu }\lambda (x)\,.\end{array}\end{eqnarray}$
The above effective U(1) gauge symmetry implies a nontrivial topological phase. One can easily see that if ψs(x) is a solution of the free equation of motion, then ${\psi }_{s}^{{\prime} }(x)=\exp \{-{\rm{i}}s{\mu }_{m}\int {S}^{\alpha }{\rm{d}}{x}_{\alpha }\}{\psi }_{s}(x)$ is a solution of the equation $({\rm{i}}{\gamma }_{\mu }{{ \mathcal D }}^{\mu }-m){\psi }_{s}(x)=0$. Hence, the total phase accumulated after moving along a path is given as
$\begin{eqnarray}\phi =s{\mu }_{m}{\int }_{{\ell }}{\rm{d}}{x}_{\alpha }{S}^{\alpha }(t,\vec{x}).\end{eqnarray}$
By equations (3) and (6), one can easily find, ${S}^{\alpha }(t,\vec{x})=({B}^{3},{E}^{2},-{E}^{1})$. As long as ${S}^{\beta }(t,\vec{x})$ does not vary much at the typical spatial size of the wave function ${\psi }_{s}(t,\vec{x})$, i.e. the magnetic dipole moment μm can be excellently approximated as a constant with respect to spatial coordinates, the equation (9) is always valid. Hereafter, we assume that this condition is always satisfied. In the case of the ordinary AC effect, only the electric field $\vec{E}(\vec{x})$ is non-zero and static. Then one can easily obtain that the phase shift after a cycler evolution is given as
$\begin{eqnarray}\begin{array}{rcl}{\phi }^{\mathrm{AC}} & = & -s{\mu }_{m}{\oint }_{\ell }{\rm{d}}\vec{x}\cdot \vec{S}(\vec{x})\\ & = & -s{\mu }_{m}{\oint }_{\ell }{\rm{d}}\vec{{\rm{\Omega }}}\cdot (\vec{{\rm{\nabla }}}\times \vec{S})=s{\mu }_{m}{\lambda }_{e},\end{array}\end{eqnarray}$
where ${\rm{d}}\vec{{\rm{\Omega }}}$ is the surface element enclosed by the orbital of the moving particle, and λe is the charge density of the infinite charged line in the setup of the ordinary AC effect. This effect is topological because the torque acting on spin due to spin-orbital interaction ($\propto {\vec{\mu }}_{m}\cdot (\vec{p}\times \vec{E})$) is zero as long as ${\vec{\mu }}_{m}\parallel \vec{p}\times \vec{E}$. In the case that the magnetic field is also non-zero, the temporal component of the (9) has to be included [60]. For a constant magnetic field $\vec{B}$, the phase is simply given as
$\begin{eqnarray}{\phi }^{\mathrm{SAB}}=s{\mu }_{m}{\int }_{\ell }{\rm{d}}t\,{B}^{3}=s{\mu }_{m}{B}^{3}{\rm{\Delta }}t,\end{eqnarray}$
which is similar to the Bernstein/scalar AB phase. Since only the z-component of the magnetic field can contribute, i.e. $\vec{B}\parallel {\vec{\mu }}_{m}$, the torque $\vec{\mu }\times \vec{B}$ acts on spin is again zero, hence this contribution is also topological in the sense of torque-free.
For the time-dependent AC effect, the Bernstein/scalar AB phase can be particularly important, because both electric and magnetic fields are non-zero in this case. In [60], several different configurations of the electric and magnetic fields were studied, particularly when the magnetic field $\vec{B}$ lies in the moving plane of the neutral particle. However, the polarization direction of the magnetic moment can change with time in this case, and hence our approximation can be violated. In this paper, we assume that the time-dependence of the electromagnetic fields is much weaker such that adiabatic approximation can be used safely. We study a simple extension of the ordinary AC configuration, in which the charge density λe has a linear dependence on time
$\begin{eqnarray}{\lambda }_{e}(t)={\lambda }_{e}\left(1+\displaystyle \frac{t}{T}\right),\end{eqnarray}$
where T is the time scale that characterizes the strength of the charge variation. For concreteness, the above definition has assumed that the charge density is increasing with time. The case of decreasing with time can be defined similarly. Importantly, the parameter T has to be much larger than the time τ of making a round for the moving particle, i.e. Tτ. This is a necessary condition of our adiabatic approximation. Otherwise, the magnetic moment will have non-trivial precession due to the magnetic field generated by the charged line. Furthermore, it is clear that the magnetic field lies in the moving plane, i.e. B3 = 0. Therefore, as long as our approximation is valid, temporal contribution to the time-dependent AC phase is trivial, and the spatial component of the time-dependent AC phase is simply given as
$\begin{eqnarray}{\phi }^{\mathrm{AC}}(t)=s{\mu }_{m}{\lambda }_{e}\left(1+\frac{t}{T}\right).\end{eqnarray}$
As we have explained, the above result is valid when Tτ. Deviation of the AC phase shift is characterized by the ratio κ = τ/T. In practice, the time-dependent AC effect in our configuration can be observed by a time series measurement on the phase shift. In the next section, we will show how this linear dependence was distorted on noncommutative space.

3. SW map and noncommutative corrections

As we have shown in the last section, in the case that the spin of the neutral particle is polarized, there is an effective U(1) gauge symmetry in its electromagnetic interaction. This symmetry has to be preserved in a noncommutative extension of both the time-dependent and static AC effect. Following [59], we will employ the SW map to obtain noncommutative corrections to the time-dependent AC effect. Unitarity can be violated if the time and position operators are noncommutative [35, 36, 3638]. With this in mind, we assume that all the temporal components are zero, i.e. θ0i = 0. Furthermore, it was pointed out that spatial noncommutativity can induce a time-modulated variation of physical measurements in the laboratory frame located on the Earth (due to the rotation of the Earth) [62]. Instead of studying physics in the rotating reference frame, we assume that the spatial direction of the noncommutative parameter is fixed in the laboratory frame, and then one can measure the physical observable by rotating the experimental equipment. Without loss of generality, we will consider only the corrections on 2 + 1 dimensional noncommutative space, i.e. θ12 ≠ 0 and all other components are zero. Noncommutativity with the other non-vanishing components can be expressed as a projection on the $\hat{z}$-direction, and then induce a time-modulated variation of the observables. The noncommutative extension of the Lagrangian (7) can be obtained by replacing the ordinary product with the ⋆ product defined in equation (1)
$\begin{aligned} \mathcal{L}_{\mathrm{NC}}= & \bar{\psi}_{s}(x)(\not p-m) \star \psi_{s}(x) \\ & -s \mu_{m} \bar{\psi}_{s}(x) \star \not S (x) \star \psi_{s}(x) . \end{aligned}$
The corresponding effective gauge transformations are defined similarly
$\begin{eqnarray}{\psi }_{s}^{{\prime} }(x)=U(x)\,\star \,{\psi }_{s}(x),\end{eqnarray}$
$\begin{eqnarray}S{{\prime} }_{\mu }(x)=U(x)\,\star \,\left({S}_{\mu }(x)-\displaystyle \frac{1}{s{\mu }_{m}}{p}_{\mu }\right)\,\star \,{U}^{-1}(x).\end{eqnarray}$
The SW map for the neutral particle is defined in analogy to the SW map for charged particle, and the corresponding transformations are given as [59]
$\begin{eqnarray}{\psi }_{s}\to {\psi }_{s}-\displaystyle \frac{1}{2}s{\mu }_{m}{\theta }^{\alpha \beta }{S}_{\alpha }{\partial }_{\beta }{\psi }_{s},\end{eqnarray}$
$\begin{eqnarray}{S}_{\mu }\to {S}_{\mu }-\displaystyle \frac{1}{2}s{\mu }_{m}{\theta }^{\alpha \beta }{S}_{\alpha }({\partial }_{\beta }{S}_{\mu }+{W}_{\beta \mu }),\end{eqnarray}$
where Wμν = ∂μSν − ∂νSμ is the effective field strength tensor. Inserting equations (17) and (18) into (14), and neglecting total derivative terms we have
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal L }}_{\mathrm{NC}} & = & \left(1-\frac{1}{4}s{\mu }_{m}{\theta }^{\alpha \beta }{W}_{\alpha \beta }\right){\bar{\psi }}_{s}(x)({\rm{i}}{\gamma }_{\mu }{{ \mathcal D }}^{\mu }-m){\psi }_{s}(x)\\ & & +\frac{{\rm{i}}}{2}s{\mu }_{m}{\theta }^{\alpha \beta }{\bar{\psi }}_{s}(x){\gamma }^{\mu }{W}_{\mu \alpha }{{ \mathcal D }}_{\beta }{\psi }_{s}(x).\end{array}\end{eqnarray}$
One can clearly see that the above Lagrangian is gauged invariant under both the ordinary electromagnetic U(1) symmetry for a charged particle and the effective U(1) symmetry for a neutral spinor. In this sense, noncommutative corrections on the AC phase have an unambiguous physical meaning.
There are two kinds of corrections in (19). In the case of a charged particle [20], the term proportional to $1-\tfrac{1}{4}s{\mu }_{m}{\theta }^{\alpha \beta }{W}_{\alpha \beta }$ can induce a correction on strength of the dipole moment. However, it turns out that the contribution is of second order of the noncommutative parameter, and hence can be ignored safely in our study. This property can be observed by investigating the equation of motion of the neutral spinor, which can be written as
$\begin{eqnarray}\begin{array}{l}({\rm{i}}{\gamma }_{\mu }{{ \mathcal D }}^{\mu }-m){\psi }_{s}(x)+\displaystyle \frac{{\rm{i}}}{2}s{\mu }_{m}\\ \quad \times {\left(1-\displaystyle \frac{1}{4}s{\mu }_{m}{\theta }^{\alpha \beta }{W}_{\alpha \beta }\right)}^{-1}{\theta }^{\alpha \beta }{\gamma }^{\mu }{W}_{\mu \alpha }{{ \mathcal D }}_{\beta }{\psi }_{s}(x)=0.\end{array}\end{eqnarray}$
One can easily see that, apart from a total renormalization effect, the first kind of correction can be neglected in first order expansion of the noncommutative parameter. In view of this, here and after, we will consider only corrections due to the second term which depends on the effective covariant derivative. Then, the Lagrangian (19) can be approximated as follows
$\begin{eqnarray}{{ \mathcal L }}_{\mathrm{NC}}={\bar{\psi }}_{s}(x)({\rm{i}}{\gamma }_{\mu }{{ \mathcal D }}_{\mathrm{NC}}^{\mu }-m){\psi }_{s}(x),\end{eqnarray}$
$\begin{eqnarray}{{ \mathcal D }}_{\mathrm{NC}}^{\mu }=\left({g}_{\,\beta }^{\mu }+\frac{1}{2}s{\mu }_{m}{W}^{\mu \alpha }{\theta }_{\alpha \beta }\right){{ \mathcal D }}^{\beta }.\end{eqnarray}$
One can see that the noncommutative corrections depend on the tensor ${C}_{\,\beta }^{\mu }\equiv \tfrac{1}{2}s{\mu }_{m}{W}^{\mu \alpha }{\theta }_{\alpha \beta }$. In the case of the time-dependent AC configuration studied here, and considering only the spatial noncommutativity, it is given as
$\begin{eqnarray}{C}^{\mu \beta }=\left(\begin{array}{ccc}0 & -s{\mu }_{m}{\theta }_{z}{\partial }_{t}{E}^{1}/2 & -s{\mu }_{m}{\theta }_{z}{\partial }_{t}{E}^{2}/2\\ s{\mu }_{m}{\theta }_{z}{\partial }_{t}{E}^{1}/2 & s{\mu }_{m}{\rho }_{e}{\theta }_{z}/2 & 0\\ s{\mu }_{m}{\theta }_{z}{\partial }_{t}{E}^{2}/2 & 0 & s{\mu }_{m}{\rho }_{e}{\theta }_{z}/2\end{array}\right),\end{eqnarray}$
where ${\rho }_{e}=\vec{{\rm{\nabla }}}\cdot \vec{E}$ is charge density of the external electric field, and by definition θz = θ12. The off-diagonal components describe the time-dependent corrections. Assuming that ψ(x) is a solution of the equation of motion, $({\rm{i}}{\gamma }_{\mu }{{ \mathcal D }}_{{NC}}^{\mu }-m)\psi (x)\,=0$, then
$\begin{eqnarray}{\psi }_{s}^{{\prime} }(x)=\exp \left\{{\rm{i}}\int \left(s{\mu }_{m}{S}^{\mu }+{\rm{i}}{C}^{\mu \nu }{{ \mathcal D }}_{\nu }\right){\rm{d}}{x}_{\mu }\right\}{\psi }_{s}(x)\end{eqnarray}$
is also a solution of the equation of motion. The noncommutative corrections on the time-dependent AC phase can be written as
$\begin{eqnarray}{\phi }_{\mathrm{NC}}^{\mathrm{AC}}(t)={\phi }^{\mathrm{AC}}(t)+{\phi }_{\theta -v}^{\mathrm{AC}}(t)+{\phi }_{\theta -g}^{\mathrm{AC}}(t).\end{eqnarray}$
The first term is the time-dependent AC phase on ordinary spacetime. The second term is a correction depending on the momentum of the neutral spinor
$\begin{eqnarray}\begin{array}{rcl}{\phi }_{\theta -v}^{\mathrm{AC}} & = & {\oint }_{\ell }{C}^{\mu \nu }{p}_{\nu }{\rm{d}}{x}_{\mu }=\frac{s}{2}{\theta }_{z}{\mu }_{m}\\ & & \times {\oint }_{\ell }\left({\rho }_{e}\vec{p}-\varepsilon {\partial }_{t}\vec{E}\right)\cdot {\rm{d}}\vec{x}\\ & & +\frac{s}{2}{\theta }_{z}{\mu }_{m}{\oint }_{\ell }\left[{\partial }_{t}\vec{E}\cdot \vec{p}\right]{\rm{d}}t,\end{array}\end{eqnarray}$
where ϵ and $\vec{p}$ are the energy and momentum of the moving particle, respectively. By the relation $m{\rm{d}}\vec{x}=\vec{p}{\rm{d}}t$ for the moving particle, after straightforward calculations we have
$\begin{eqnarray}\begin{array}{rcl}{\phi }_{\theta -v}^{\mathrm{AC}}(t) & = & s{\lambda }_{e}(1+\omega t){\mu }_{m}\frac{{pL}{\theta }_{z}}{2{A}_{e}}\\ & & -\frac{s}{2}{\theta }_{z}{\mu }_{m}(\varepsilon -m)\omega {\lambda }_{e},\end{array}\end{eqnarray}$
where ω = 1/T is the frequency of the time-dependent electric field; Ae is the cross section of the charged line with a relation ρeAe = λe; L is the length of the closed path; $p=| \vec{p}| $ is the magnitude of the momentum. One can see that apart from a time-dependent correction which is directly related to the adiabatic approximation, there is a constant contribution that is proportional to the frequency of the time-dependent electric field and energy of the moving particle. The third term is a correction depending on the effective gauge potential
$\begin{eqnarray}\begin{array}{rcl}{\phi }_{\theta -g}^{\mathrm{AC}}(t) & = & -s{\mu }_{m}{\oint }_{\ell }{C}^{\mu \nu }{S}_{\nu }{\rm{d}}{x}_{\mu }=-\frac{1}{2}{\mu }_{m}^{2}{\theta }_{z}\\ & & \times {\oint }_{\ell }{\rho }_{e}^{2}\vec{S}\cdot {\rm{d}}\vec{x}-\frac{1}{2}{\mu }_{m}^{2}{\theta }_{z}{\oint }_{\ell }\left[{\partial }_{t}\vec{E}\cdot ({\vec{n}}_{z}\times \vec{E})\right]{\rm{d}}t.\end{array}\end{eqnarray}$
In the above expression, we have used the identity s2 = 1. Furthermore, one can see that the second term in the above equation vanishes because of ${\partial }_{t}\vec{E}\cdot ({\vec{n}}_{z}\times \vec{E})=0$. Hence we have
$\begin{eqnarray}{\phi }_{\theta -g}^{\mathrm{AC}}(t)=-\frac{{\mu }_{m}^{2}{\lambda }_{e}^{2}{\theta }_{z}}{2{A}_{e}}{\left(1+\omega t\right)}^{2}.\end{eqnarray}$
This means that, for the effective gauge potential dependent correction, there is only contribution due to adiabatic approximation. Combine all the contributions, the noncommutative correction to the time-dependent AC phase can be written as
$\begin{eqnarray}\begin{array}{rcl}{\phi }_{\theta }^{\mathrm{AC}}(t) & = & {\phi }_{\theta -v}^{{AC}}(t)+{\phi }_{\theta -g}^{{AC}}(t)={\chi }_{e}{K}_{m}^{\hat{s}}(t){\phi }^{{AC}}(t)\\ & & -\frac{1}{2}{\theta }_{z}(\varepsilon -m)\omega {\phi }^{{AC}},\end{array}\end{eqnarray}$
where χe = θz/Ae is a dimensionless ratio and ${K}_{m}^{s}(t)={pL}\,-s{\mu }_{m}{\lambda }_{e}(t)$ is a dimensionless scale factor. Compared to noncommutative corrections to the static AC phase, there is an adiabatic correction characterized by ${K}_{m}^{s}(t)$ and φAC(t). In addition, there is a constant correction depending on the dimensionless factor θz(ϵm)ω. While the adiabatic correction can only be observed by a time-series measurement of phase shift, the additional correction can be seen in a single measurement.

4. Conclusions

The AC effect is one of the most profound phenomena in quantum mechanics. As we have shown that the AC phase is closely related to an effective U(1) gauge symmetry of the electromagnetic interactions of a magnetic dipole moment. Because of this, the AC effect is an ideal probe for any deviation of the standard Quantum Electrodynamics. In this paper, we studied corrections on the time-dependent AC effect on a noncommutative space. Noncommutative corrections on static AC effect have been studied extensively. Here, we define the time-dependent AC effect as an adiabatic extension of the static one by allowing the charge density of the straight line can change with time. This kind of extension is meaningful when the frequency of the varying electric field is sufficiently small such that polarization of the incident neutron does not change within a signal round evolution, i.e. Tτ. Based on this condition, noncommutative corrections on the time-dependent AC phase are studied by employing an SW map. The advantage of this approach is that the noncommutative corrections are gauge invariant with respect to the effective U(1) symmetry, and hence can give unambiguous physical results. Compared to noncommutative corrections to the static AC phase, there are two kinds of corrections to the time-dependent AC effect. The first one depends on λe(t) = λe(1 + ωt), and is characterized by functions ${K}_{m}^{s}(t)$ and φAC(t). Such a kind of correction is purely adiabatic. The second one is a constant (in time) and depends on the frequency of the varying electric field. While the adiabatic correction can only be observed by a time-series measurement of phase shift, the additional correction can be seen in a single measurement. In the non-relativistic limit, ϵmp2/2m, the constant correction is proportional to the dimensionless factor ${p}^{2}\omega /(m{{\rm{\Lambda }}}_{\mathrm{NC}}^{2})$, where ΛNC is the energy scale of the noncommutative spacetime $| {\theta }^{\mu \nu }| \propto 1/{{\rm{\Lambda }}}_{{NC}}^{2}$.

This work is supported by the Innovation Capability Support Program of Shaanxi Province (Program No. 2021KJXX-47).

1
Snyder H S 1947 Quantized space-time Phys. Rev. 71 38

DOI

2
Snyder H S 1947 The electromagnetic field in quantized space-time Phys. Rev. 72 68

DOI

3
Yang C N 1947 On quantized space-time Phys. Rev. 72 874

DOI

4
Mead C A 1964 Possible connection between gravitation and fundamental length Phys. Rev. 135 B849

DOI

5
Witten E 1986 Noncommutative geometry and string field theory Nucl. Phys. B 268 253

DOI

6
Seiberg N Witten E 1999 String theory and noncommutative geometry J. High Energy Phys. JHEP09(1999)032

DOI

7
Freidel L Livine E R 2006 3D quantum gravity and effective noncommutative quantum field theory Phys. Rev. Lett. 96 221301

DOI

8
Douglas M R Nekrasov N A 2001 Noncommutative field theory Rev. Mod. Phys. 73 977

DOI

9
Szabo R J 2003 Quantum field theory on noncommutative spaces Phys. Rep. 378 207

DOI

10
Zhang J-Z 2004 Testing spatial noncommutativity via Rydberg atoms Phys. Rev. Lett. 93 043002

DOI

11
Rafiei A Rezaei Z Mirjalili A 2022 Electromagnetic form factors in noncommutative space time Eur. Phys. J. C 82 62

DOI

12
Mehdaoui N Khodja L Haouat S 2021 Noncommutative scalar field in de Sitter space–time and pair creation process Int. J. Mod. Phys. A 36 2150011

DOI

13
Wang B-Q Long Z-W Long C-Y Wu S-R 2018 Solution of the spin-one DKP oscillator under an external magnetic field in noncommutative space with minimal length Chin. Phys. B 27 010301

DOI

14
Wang K Zhang Y-F Wang Q Long Z-W Jing J 2017 Quantum speed limit for a relativistic electron in the noncommutative phase space Int. J. Mod. Phys. A 32 1750143

DOI

15
Tizchang S Batebi S Haghighat M Mohammadi R 2016 Cosmic microwave background polarization in non-commutative space-time Eur. Phys. J. C 76 478

DOI

16
Deriglazov A A Tereza D M 2019 Covariant version of the Pauli Hamiltonian, spin-induced noncommutativity, Thomas precession, and the precession of spin Phys. Rev. D 100 105009

DOI

17
Abyaneh M Z Farhoudi M 2021 Electron dynamics in noncommutative geometry with magnetic field and Zitterbewegung phenomenon Eur. Phys. J. Plus 136 863

DOI

18
Ma K 2017 Constrains of charge-to-mass ratios on noncommutative phase space Adv. High Energy Phys. 2017 1945156

DOI

19
Wang K Zhang Y F Wang Q Long Z W Jing J 2017 Quantum speed limit for relativistic spin-0 and spin-1 bosons on commutative and noncommutative planes Adv. High Energy Phys. 2017 4739596

DOI

20
Carroll S M Harvey J A Kostelecky V A Lane C D Okamoto T 2001 Noncommutative field theory and Lorentz violation Phys. Rev. Lett. 87 141601

DOI

21
Franchino-Viñas S Vega F 2021 Magnetic properties of a Fermi gas in a noncommutative phase space Eur. Phys. J. Plus 136 877

DOI

22
Heddar M Falek M Moumni M Lütfüoğlu B C 2021 Pauli oscillator in noncommutative space Mod. Phys. Lett. A 36 2150280

DOI

23
Muhuri A Sinha D Ghosh S 2021 Entanglement Induced by Noncommutativity: anisotropic harmonic oscillator in noncommutative space Eur. Phys. J. Plus 136 35

DOI

24
Dutta M Ganguly S Gangopadhyay S 2020 Exact solutions of a damped harmonic oscillator in a time dependent noncommutative space Int. J. Theor. Phys. 59 3852

DOI

25
Wu S-R Long Z-W Long C-Y Wang B-Q Liu Y 2017 (2+1)-dimensional Klein–Gordon oscillator under a magnetic field in the presence of a minimal length in the noncommutative space Int. J. Mod. Phys. A 32 1750148

DOI

26
Zhong L Chen H Ran Q-K Long C-Y Long Z-W Analysis of the Dirac equation with the Killingbeck potential in non-commutative space arXiv:2101.08406

27
Ababekri M Anwar A Hekim M Rashidin R 2016 Aharonov–Bohm phase for an electric dipole on a noncommutative space Front. Phys. 4 22

DOI

28
Ma K Dulat S 2011 Spin Hall effect on a noncommutative space Phys. Rev. A 84 012104

DOI

29
Chaichian M Sheikh-Jabbari M M Tureanu A 2001 Hydrogen atom spectrum and the Lamb shift in noncommutative QED Phys. Rev. Lett. 86 2716

DOI

30
Gnatenko K P Tkachuk V M 2017 Noncommutative phase space with rotational symmetry and hydrogen atom Int. J. Mod. Phys. A 32 1750161

DOI

31
Jing J Zhang Q-Y Wang Q Long Z-W Dong S-H 2019 Fractional angular momentum of an atom on a noncommutative plane Commun. Theor. Phys. 71 1353

DOI

32
Aharonov Y Bohm D 1959 Significance of electromagnetic potentials in the quantum theory Phys. Rev. 115 485

DOI

33
Aharonov Y Casher A 1984 Topological quantum effects for neutral particles Phys. Rev. Lett. 53 319

DOI

34
Chaichian M Langvik M Sasaki S Tureanu A 2008 Gauge covariance of the aharonov-bohm phase in noncommutative quantum mechanics Phys. Lett. B 666 199

DOI

35
Chaichian M Demichev A Presnajder P Sheikh-Jabbari M M Tureanu A 2001 Quantum theories on noncommutative spaces with nontrivial topology: Aharonov–Bohm and Casimir effects Nucl. Phys. B 611 383

DOI

36
Chaichian M Demichev A Presnajder P Sheikh-Jabbari M M Tureanu A 2002 Aharonov–Bohm effect in noncommutative spaces Phys. Lett. B 527 149

DOI

37
Li K Dulat S 2006 The Aharonov–Bohm effect in noncommutative quantum mechanics Eur. Phys. J. C 46 825

DOI

38
Liang S-D Li H Huang G-Y 2014 Detecting noncommutative phase space by the Aharonov–Bohm effect Phys. Rev. A 90 010102

DOI

39
Rodriguez R M E 2018 Quantum effects of Aharonov–Bohm type and noncommutative quantum mechanics Phys. Rev. A 97 012109

DOI

40
Jing J Kong L-B Wang Q Dong S-H 2020 On the noncommutative Aharonov–Bohm effects Phys. Lett. B 808 135660

DOI

41
Mirza B Zarei M 2004 Noncommutative quantum mechanics and the Aharonov–Casher effect Eur. Phys. J. C 32 583

DOI

42
Li K Wang J 2007 The Topological AC effect on noncommutative phase space Eur. Phys. J. C 50 1007

DOI

43
Wang J-H Li K 2007 The HMW effect in noncomutative quantum mechanics J. Phys. A 40 2197

DOI

44
Mirza B Narimani R Zarei M 2006 Aharonov–Casher effect for spin one particles in a noncommutative space Eur. Phys. J. C 48 641

DOI

45
Li K Sayipjamal D Wang J-H 2008 The He-McKellar–Wilkens effect for spin-1 particles on non-commutative space Chin. Phys. B 17 1716

DOI

46
Ma K Wang J-H Yang H-X 2016 Seiberg–Witten map and quantum phase effects for neutral Dirac particle on noncommutative plane Phys. Lett. B 756 221

DOI

47
Konstantinou G Moulopoulos K 2017 The ‘forgotten' pseudomomenta and gauge changes in generalized Landau Level problems: spatially nonuniform magnetic and temporally varying electric fields Int. J. Theor. Phys. 56 1484

DOI

48
Konstantinou G Moulopoulos K 2016 Generators of dynamical symmetries and the correct gauge transformation in the Landau level problem: use of pseudomomentum and pseudo-angular momentum Eur. J. Phys. 37 065401

DOI

49
Yang Y Long Z-W Chen H Zhao Z-L Long C-Y 2021 Aharonov-Bohm effect on the generalized Duffin–Kemmer–Petiau oscillator in the Som–Raychaudhuri spacetime Mod. Phys. Lett. A 36 2150059

DOI

50
Chen H Long Z W Ran Q K Yang Y Long C Y 2020 Aharonov–Bohm effect on the generalized Dirac oscillator in a cosmic dislocation space-time EPL 132 50006

DOI

51
Chen H Long Z-W Long C-Y Zare S Hassanabadi H 2022 The influence of Aharonov–Casher effect on the generalized Dirac oscillator in the cosmic string space-time Int. J. Geom. Meth. Mod. Phys. 19 2250133

DOI

52
Tonomura A 1986 Evidence for aharonov-bohm effect with magnetic field completely shielded from electron wave Phys. Rev. Lett. 56 792

DOI

53
Osakabe N 1986 Experimental confirmation of Aharonov–Bohm effect using a toroidal magnetic field confined by a superconductor Phys. Rev. A 34 815

DOI

54
Cimmino A 1989 Observation of the topological Aharonov–Casher phase shift by neutron interferometry Phys. Rev. Lett. 63 380

DOI

55
Sangster K Hinds E A Barnett S M Riis E 1993 Measurement of the Aharonov–Casher phase in an atomic system Phys. Rev. Lett. 71 3641

DOI

56
Bertolami O Leal P 2015 Aspects of phase-space noncommutative quantum mechanics Phys. Lett. B 750 6

DOI

57
Brace D Cerchiai B L Pasqua A F Varadarajan U Zumino B 2001 A cohomological approach to the nonabelian Seiberg–Witten map J. High Energy Phys. JHEP06(2001)047

DOI

58
Barnich G Grigoriev M A Henneaux M 2001 Seiberg–Witten maps from the point of view of consistent deformations of gauge theories J. High Energy Phys. JHEP10(2001)004

DOI

59
Ma K Wang J-H Yang H-X 2016 Time-dependent Aharonov–Bohm effect on the noncommutative space Phys. Lett. B 759 306

DOI

60
Singleton D Ulbricht J 2016 The time-dependent Aharonov–Casher effect Phys. Lett. B 753 91

DOI

61
He X-G McKellar B H J 2001 The Topological AC and HMW effects, and the dual current in (2+1)-dimensions Phys. Rev. A 64 022102

DOI

62
Kamoshita J-I 2007 Probing noncommutative space-time in the laboratory frame Eur. Phys. J. C 52 451

DOI

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