We review the oscillator with Aharonov-Casher system and study some
mathematical foundation about factorization method. The factorization method helps us to obtain the energy spectrum and general wave function for the corresponding system in some spin condition. The factorization method leads us to obtain the raising and lowering operators for the Aharonov-Casher system. The corresponding operators give us the generators of the algebra.

1. Introduction

As we know, the relativistic quantum dynamics of a natural particle which describes the relativistic Aharonov-Casher system [1] is given by introducing minimal coupling into the Dirac equation [2, 3] in Cartesian coordinates which is given by
(1)iγμ∂μ⟶iγμ∂μ+μ2∑μνFμν(x),
where we consider the natural units ℏ=c=1. Also, μ corresponds to the magnetic dipole moment of natural particle, and Fμν(x) corresponds to the electromagnetic tensor, which are defined by F0i=-Fi0=Ei and Fij=-Fji=-ϵijkBk and ∑ab=(i/2)[γa,γb]. The γa matrices correspond to the Dirac matrices in Minkowski space-time [4]:
(2)γ0=β^=(100-1),γi=β^α^i=(0σi-σi0),∑i=(σi00σi),
with γaγb+γbγa=-2ηab, with ∑→ being the spin vector and σi being Pauli matrices. The tensor ηab=diag(-+++) is the Minkowski tenor. Moreover, by introducing the coupling that describes the Dirac oscillator P→→P→-imωρβ^ρ^ into the nonminimal coupling (1), we can see that the whole system is cylindrically symmetric. So, we can work with curvilinear coordinates x=ρcosφ and y=ρsinφ. Therefore, we write the line element of the Minkowski space-time in the following form:
(3)ds2=-dt2+dρ2+ρ2dφ2+dz2.
Here, we note that, in curvilinear coordinate (both flat and curved space-time background), the relativistic quantum dynamics of a neutral particle with a permanent magnetic dipole moment interacting with the external field is not described by the Dirac equation with the introduction of nonminimal coupling (1) anymore. Based on the spinor theory in curved space-time, nonminimal coupling (1) plus the coupling describing the Dirac oscillator becomes
(4)iγμ∇μ⟶iγμ∂μ+iγμΓμ(x)+iγμmωργ0+δμρ+μ2∑μνFμν(x),
where ∇μ+∂μ+Γμ(x) corresponds to the components of the covariant derivative of a spinor, with Γμ=(i/4)ωμab(x)∑ab being the spinnorial connection [5, 6], and ∑ab=(1/2)[γa,γb]. In the spinor theory in curved space-time, the γa matrices are defined in the local reference frame of the observers and are identical to the Dirac matrices defined in Minkowski space-time (3). In this notation, the indices (a,b,c=0,1,2,3) indicate the local reference frame, while the indices (μ,ν) indicate the space-time indices. Thus, the γμ matrices given by (4) are related to the γa via γμ=eaμ(x)γa, where components eaμ(x) are called tetrades and give rise to the local reference frame of the observers. The tetrades satisfy the following equation [5, 6]:
(5)gμν(x)=eμa(x)eνb(x)ηab.
The tetrades also have an inverse defined as dxμ=eaμ(x)θa, where
(6)eμa(x)ebμ(x)=δba,eaμ(x)eνa(x)=δνμ.
Furthermore, the components of the spinnorial connection can be obtained by solving the Cartan structure [6] in the absence of torsion:
(7)dθ^a+ωbaθ^b=0,
where ωba=ωμba(x)dxμ and ωμba(x) are called connection 1-form. For instance, we can choose the tetrades for line element (3) being
(8)θ^θ=dt,θ^1=dρ,θ^2=ρdφ,θ^3=dz.
By solving the Cartan structure equations in the absence of torsion, we can obtain ωφ21(x)=-ωφ12(x)=-1 and γμΓμ(x)=γ1/2ρ. Hence, the Dirac equation describing the interaction between the Dirac oscillator and Aharonov-Casher system is
(9)mΨ=iγ0∂Ψ∂t+iγ1[∂∂ρ+12ρ+mωργ0]Ψ+iγ2ηρ∂Ψ∂φ+iγ3∂Ψ∂z+iμα→·E→Ψ-μ∑⟶B→Ψ.
Here, we consider radial electric field as E→=(λ/ρ)ρ^ and also consider the magnetic dipole moment parallel to the z-axis. In that case we can rewrite the Dirac equation as
(10)i∂Ψ∂t=mβ^Ψ-iα^1[∂∂ρ+12ρ+mωρβ^]Ψ-iα^2ρ∂Ψ∂φ-iα^3∂Ψ∂z-iΦAC2πρβ^·α^1Ψ,
where ΦAC=±2π(μλ/ℏc)=(1/ℏc)∮(M→×E→)νdxν is Aharonov-Casher geometric phase [1]. In order to solve (10) one can write Ψ in terms of two components of spinors as
(11)Ψ=e-iEt(ϕψ),
where ϕ and ψ are spinors of two components. We substitute (11) in (10) to obtain two coupled equations for ϕ and ψ. Now, we are going to write the first coupled equation which is given by
(12)(ɛ-m)ϕ=-iδ1[∂∂ρ+12ρ+ΦAC2πρ-mωρ]ψ-iδ2ρ∂ψ∂φ-iδ3∂ψ∂z,
and the second coupled equation is
(13)(ɛ+m)ψ=-iδ1[∂∂ρ+12ρ-ΦAC2πρ+mωρ]ϕ-iδ2ρ∂ϕ∂φ-iδ3∂ϕ∂z.
By using (12) and (13) one can obtain the following second order differential equation:
(14)(ɛ2-m2)ϕ=-∂2ϕ∂ρ2-1ρ∂ϕ∂ρ-1ρ2∂2ϕ∂φ2-∂2ϕ∂z2+iδ3ρ2∂ϕ∂φ+14ρ2ϕ+2iδ3mω∂ϕ∂φ-mωϕ-2iδ3ΦAC2πρ2∂ϕ∂φ-ΦAC2πρ2ϕ+m2ω2ρ2ϕ-2mωΦAC2πϕ+(ΦAC2π)2ϕρ2.
Here, ϕ is eigenfunction of the Pauli matrix δ3 and total angular momentum J^z=-i∂ϕ, and the z-component of the momentum p^z=-i∂z commutes with the Hamiltonian of (14). In that case, we can write the solution of (14) in terms of the eigenvalues of the operators p^z=-i∂z and J^z=-i∂ϕ,
(15)ϕs(ρ,φ,z)=Aei(ℓ+(1/2))φeikzRz(φ).s=±1 give δ3ϕ0=sϕ0, where ϕ0=(ϕ+,ϕ-)T, with ℓ=0,±1,±2,…,k being constant and A being normalization factor. Thus, substituting the solution (15) into the second order differential equation (14), we obtain the following radial equation:
(16)Rs′′(ρ)+1ρRs′(ρ)+(βs-ξs2ρ2-m2ω2ρ2)Rs(ρ)=0,
where
(17)ξs=ℓ+12(1-s)+sΦAC2π,βs=ɛ2-m2-k2+2mω(1+sξs).
In order to solve (16) we change variables given by x=mωρ2 and obtain
(18)xRs′′(x)+Rs′(x)+(βs4mω-ξs24x-x4)Rs(x)=0.
Again, we choose the change of variable as follows:
(19)Rs(x)=F(x)Ln,m′α,β(x),
so, (18) changes as follows:
(20)xLn,m′′(α,β)(x)+(2xF′(x)F(x)+1)Ln,m′(α,β)(x)+[xF′′(x)F(x)+F′(x)F(x)+βs4mω-ξs24x-x4]Ln,m(α,β)(x)=0.
In order to obtain Rs(x) in (19), we have to compare (20) with known polynomial. For this reason, first we introduce the Laguerre polynomial which is corresponding to (20). So, in this paper first we will try to review some mathematical foundation about factorization method [7]. In Section 3 we take advantage of factorization method and obtain the energy spectrum and general wave function for the corresponding system. Also, we show that the corresponding equation can be factorized first with respect to m′ and then with respect to n and m′. These lead us to obtain the raising and lowering operators. Note that the shape invariant equation (27) can be written as the raising and lowering relations for the Aharonov-Casher system. These operators will be generators algebra.

2. Mathematical Foundation

Using the factorization approach, we compute the energy spectrum ɛ and also bound states Ψ through the comparison of the differential equation given in (18) with associated Laguerre differential equation in an appropriate manner. We also factorize the second order differential equations into new sets of operators A-,A+ and shape invariant form, which are the first order differential equations. This process is called factorization method. Before anything else we will try to explain the associated Laguerre differential equation Ln,m′(α,β)(x) in factorization method point of view. To start, we need to recall that, for the real parameters α>-1 and β>0, the associated Laguerre differential equation corresponding to Ln,m′(α,β)(x) in the interval x∈(0,∞) is introduced as [8, 9]
(21)xLn,m′′′(α,β)(x)+(1+α-βx)Ln,m′′(α,β)(x)+[(n-m′2)β-m′2x(α+m′2)]Ln,m′(α,β)(x)=0.
Here, the indices n and m′ are nonnegative integers for 0≤m′<n. The associated Laguerre function, Ln,m′(α,β)(x) as a solution of the differential (21), has the following Rodrigues representation:
(22)Ln,m(α,β)(x)=αn,m′(α,β)xα+(m′/2)e-βx(ddx)n-m′(xn+αe-βx),
where αn,m′ is the normalization coefficient and will be obtained later. As mentioned in [10, 11], we can write the associated Laguerre differential equation (21) as the following shape invariant equations with respect to the parameter m′:
(23)Am′+(x)Am′-(x)Ln,m′(α,β)(x)=(n-m′+1)βLn,m′(α,β)(x),Am′-(x)Am′+(x)Ln,m′-1(α,β)(x)=(n-m′+1)βLn,m′-1(α,β)(x),
where operators Am′+(x) and Am′-(x) are given by the following equation:
(24)Am′+(x)=xddx-m′-12x,Am′-(x)=-xddx-2α+m′-2βx2x.
One may write down the shape invariance equation (23) as the raising and lowering relations:
(25)Am′+(x)Ln,m′(α,β)(x)=(n-m′+1)βLn,m′(α,β)(x),Am′-(x)Ln,m′(α,β)(x)=(n-m′+1)βLn,m′-1(α,β)(x).
On the other hand, the associated Laguerre differential equation (21) can be factorized with respect to the parameter n for a given m′ as
(26)An,m′+(x)An,m′-(x)Ln,m′(α,β)(x)=(n-m′)(n+α)βLn,m′(α,β)(x),An,m′-(x)An,m′+(x)Ln-1,m′(α,β)(x)=(n-m′)(n+α)βLn-1,m′(α,β)(x),
where the differential operators are functions of parameters n and m′ which are obtained as
(27)An,m′+(x)=xddx-βx+12(2n+2α-m′),An,m′-(x)=-xddx+12(2n-m′).
Note that the shape invariant equation (28) can be written as the raising and lowering relations
(28)An,m′+(x)Ln-1,m′(α,β)(x)=(n-m′)(n+α)Ln,m′(α,β)(x),An,m′-(x)Ln,m′(α,β)(x)=(n-m′)(n+α)βLn-1,m′(α,β)(x).
The above mentioned method with some calculations leads to the following normalization coefficient:
(29)αn,m′(α,β)=(-1)nβα+m′+1Γ(n-m′+1)Γ(n+α+1).
Also, the normalization coefficient equation (29) has been so chosen that the associated Laguerre functions Ln,m′(α,β)(x), with the same m′ but with different n with respect to the inner product with the weight functions xαe-βx, form an orthonormal set in the interval 0≤x<∞:
(30)∫0∞Ln,m′(α,β)(x)Ln′,m′(α,β)(x)xαe-βxdx=δn,n′.
Now, we go back to the Dirac oscillator with Aharonov-Casher system and compare it with associated Laguerre equation.

3. Factorization Method and Dirac Oscillator with Aharonov-Casher System

We wish to get a solution for (18) which will be regular at the origin; then we have to compare (20) to (21) and obtain F(x) as
(31)F(x)=xα/2e-(β/2)x,
where x=mωρ2. Comparing equation (20) and (21), yields to the following conditions:
(32)β=±12,ξs=m′+α,Bs=4mω(n-m′2)+2mω(α+1).
The associated Laguerre polynomial and regular solution at the origin lead us to take β=1/2. Equation (32) helps us to obtain the energy spectrum for the Dirac oscillator with Aharonov-Casher system. In order to specify the energy spectrum, we consider two cases. First, for obtaining the positive-energy solution of Dirac equation (Ψ), we consider the component parallel to z-axis of the space-time; we must take s=+1 and consider φ-=0. The energy spectrum is
(33)ɛ2=4mω(n-m′)+m2+k2.
We see the energy is positive because always we have n≥m′>0. In second case, we consider the positive-energy solutions corresponding to the s=-1 and φ+=0. So, we can obtain,
(34)ɛ2=4mω(n+α)+m2+k2.
We can see in two cases the energy compound to Aharonov-Casher effect in the Minkowski space-time. So, the general wave function with x=mωρ2 and condition (32) will be as
(35)ϕs(x)=Aei(ℓ+(1/2))φeikzF(x)Ln,m′(α,(1/2))(x)=Aei(ℓ+(1/2))φeikz(mω)α/2xαe-(1/4)mωρ2Ln,m′(α,1/2)(x),ϕs(φ)=A(mω)α/2ei(ℓ+(1/2))φeikzραe-(1/4)mωρ2Ln,ξs-α(α,(1/2))(ρ).
So, the associated Laguerre function obtained as a solution of the differential (20) has the following Rodrigues representation:
(36)Ln,ξs-α(ρ)=αn,ξs-α(α,1/2)ρα+ξse-(mωρ2/2)(1ρddρ)n-ξs+α(ρ2n+2αe-mωρ2/2),
where αn,ξs-α(α,1/2) will be as
(37)αn,ξs-α(α,12)=(-1)n23ξs-2n-2αΓ(n-ξs+α)Γ(n+α+1)(mω)α-ξs.
As mentioned in [10, 11], we can write the associated Laguerre differential equation (20) as the following shape invariant equations with respect to the parameter ξs-α=m′:
(38)Aξs-α+(ρ)Aξs-α-(ρ)Ln,ξs-α(α,1/2)(ρ)=(n-ξs+α+1)12Ln,ξs-α(α,1/2)(ρ),Aξs-α-(ρ)Aξs-α+(ρ)Ln,ξs-α-1(α,1/2)(ρ)=(n-ξs+α+1)12Ln,ξs-α-1(α,1/2)(ρ),
where operators Aξs-α+(ρ) and Aξs-α-(ρ) are given by
(39)Aξs-α+(ρ)=12mω(ddρ-ξs-α-1ρ),Aξs-α-(x)=-12mω(ddρ+α+ξs-mωρ2ρ).
One may write down the shape invariance equation (1) as the raising and lowering relations
(40)Aξs-α+(ρ)Ln,ξs-α(α,1/2)(ρ)=(n-ξs+α+1)12Ln,ξs-α(α,1/2)(ρ),Aξs-α-(ρ)Ln,ξs-α(α,1/2)(ρ)=(n-ξs+α+1)12Ln,ξs-α-1(α,1/2)(ρ).
On the other hand, the associated Laguerre differential equation (20) can be factorized with respect to the parameter n for a given n as
(41)An,ξs-α+(ρ)An,ξs-α-(ρ)Ln,ξs-α(α,1/2)(ρ)=12(n-ξs+α)(n+α)Ln,ξs-α(α,1/2)(ρ),An,ξs-α-(ρ)An,ξs-α+(ρ)Ln-1,ξs-α(α,1/2)(ρ)=12(n-ξs+α)(n+α)Ln-1,ξs-α(α,1/2)(ρ),
where the differential operators are functions of parameters n and ξs-α which are obtained by
(42)An,ξs-α+(ρ)=12ρddρ-12mωρ2+12(2n+3α-ξs),An,ξs-α-(ρ)=-12ρddρ+12(2n-ξs+α).
Note that the shape invariant equation (27) can be written as the raising and lowering relations
(43)An,ξs-α+(ρ)Ln-1,ξs-α(α,1/2)(ρ)=(n-ξs+α)(n+α)Ln,ξs-α(α,1/2)(ρ),An,ξs-α-(ρ)Ln,ξs-α(α,1/2)(ρ)=(n-ξs+α)(n+α)Ln-1,ξs-α(α,1/2)(ρ).
Here, we note that the factorization method and shape invariance condition help us to factorize the second order oscillator with the Aharonov-Casher equation. It means that such equation factorized in terms of two first order operators which are known by raising and lowering operators.

4. Conclusion

In this paper, first we introduced the nonlinear equation corresponding with oscillator and Aharonov-Casher system. We could easily solve such system by using factorization method and shape invariance condition. Also, the factorization method helps us to obtain the general form of wave function and energy spectrum which is hard to obtain with ordinary methods. Finally, we achieved the first order differential equation as raising and lowering operators. It may be interesting to show that such operators can be a form of generators of N=2 supersymmetry algebras. Another problem for such system is partner Hamiltonian with new and modified potential. We can do such complicated problem in future.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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