Tao Wang and Kai Ma

Department of Physics,Shaanxi University of Technology,Hanzhong 723000,Shaanxi,China

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.

Keywords: noncommutative space,Aharonov–Casher effect,minimal length

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 [1–3].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 θμνis 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

wheref(x) andg(x) are two arbitrary infinitely differentiable functions on the commutativeR3+1space-time.Quantum field theory on the noncommutative spacetime can be obtained by replacing ordinary product by the ?-product [8].

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 effectiveU(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 timedependent 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 effectiveU(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

wheremis the mass of the fermion ψ.On 2+1 dimensional spacetime,the metric and anti-symmetric tensors are defined asgμν=diag(1,-1,-1) and ?012=1,and the electromagnetic field strength tensor is given as

whereEiandBiare the electric and magnetic fields,respectively (the indices ‘i=1,2,3’ stands for thex,yandzdirections on ordinary 3+1 dimensional spacetime,respectively).The Dirac matrixes can be chosen as follows [61]

We can see that all the γ-matrix are block diagonal in this representation,and satisfy an equationγμ γv=gμv+i S?μvα γαwith S=-iγ0γ1γ2=-γ0σ12.On can easily find that S=diag (I,-I),i.e.the operatorS is also diagonal in this representation,and has eigenvaluess=±1.Furthermore,with a straightforward calculation,one can obtain thatσμv=-S?μvα γα,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 operatorS with eigenvaluess=±1,are completely independent.Without loss of generality,hereafter,we assume that the magnetic dipole moment is polarized along thezdirection,i.e.is the unite vector along thezdirection).In virtue of this property,the Lagrangian(2)can be written as

As soon as the sun rose the King came, and when he perceived the gold he was astonished and delighted, but his heart only lusted more than ever after the precious metal

where the spinor ψsare eigenstates of the operatorS,and the effective vector potential is defined as

Within this formulation,the Lagrangian (2) can be rewritten into a minimal-coupling form

whereαD is the effective covariant derivative.Similar to the well-knownU(1) gauge symmetry of electrodynamics for charged particles,this minimal coupling form means a newU(1) symmetry with the gauge potentialSμ,andsis the sign of the‘charge’of the neutral particle ψ,μmis the magnitude of the‘charge’.It is easy to check that the Lagrangian(5)is invariant under the following gauge transformations

By equations (3) and (6),one can easily find,.As long asSβ(t,) does not vary much at the typical spatial size of the wave functionψ s(t,) ,i.e.the magnetic dipole moment μmcan 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 fieldis non-zero and static.Then one can easily obtain that the phase shift after a cycler evolution is given as

which is similar to the Bernstein/scalar AB phase.Since only thez-component of the magnetic field can contribute,i.e.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 fieldlies 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 λehas a linear dependence on time

whereTis 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 parameterThas to be much larger than the time τ of making a round for the moving particle,i.e.T?τ.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 timedependent AC phase is simply given as As we have explained,the above result is valid whenT?τ.Deviation of the AC phase shift is characterized by the ratio k=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 effectiveU(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 [35,36,36–38].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 nonvanishing components can be expressed as a projection on thez?-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)

The corresponding effective gauge transformations are defined similarly

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]

whereWμν=?μSν-?νSμis the effective field strength tensor.Inserting equations(17)and(18)into(14),and neglecting total derivative terms we have

One can clearly see that the above Lagrangian is gauged invariant under both the ordinary electromagneticU(1) symmetry for a charged particle and the effectiveU(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 tocan 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

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

One can see that the noncommutative corrections depend on the tensorIn the case of the timedependent AC configuration studied here,and considering only the spatial noncommutativity,it is given as

is also a solution of the equation of motion.The noncommutative corrections on the time-dependent AC phase can be written as

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

where ω=1/Tis the frequency of the time-dependent electric field;Aeis the cross section of the charged line with a relation ρeAe=λe;Lis the length of the closed path;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

In the above expression,we have used the identitys2=1.Furthermore,one can see that the second term in the above equation vanishes because ofHence we have

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

where χe=θz/Aeis a dimensionless ratio andsμ mλe(t) is a dimensionless scale factor.Compared to noncommutative corrections to the static AC phase,there is an adiabatic correction characterized byand φAC(t).In addition,there is a constant correction depending on the dimensionless factor θz(ε-m)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 effectiveU(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.T?τ.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 effectiveU(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(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,ε-m≈p2/2m,the constant correction is proportional to the dimensionless factor,where ΛNCis the energy scale of the noncommutative spacetime

Acknowledgments

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