Path Integral Confined Dirac Fermions in a Constant Magnetic Field
aa r X i v : . [ h e p - t h ] M a y Path Integral for Confined Dirac Fermions in a ConstantMagnetic Field
Abdeldjalil Merdaci ∗ a,b , Ahmed Jellal † c,d and Lyazid Chetouani ba Physics Department, College of Science, King Faisal University,PO Box 380, Alahsa 31982, Saudi Arabia b D´epartement de Physique, Facult´e des Sciences Exactes, Universit´e Mentouri,25000 Constantine, Alg´erie. c Saudi Center for Theoretical Physics, Dhahran, Saudi Arabia d Theoretical Physics Group, Faculty of Sciences, Choua¨ıb Doukkali University , PO Box 20, 24000 El Jadida, Morocco
Abstract
We consider Dirac fermion confined in harmonic potential and submitted to a constant magneticfield. The corresponding solutions of the energy spectrum are obtained by using the path integraltechniques. For this, we begin by establishing a symmetric global projection, which provides asymmetric form for the Green function. Based on this, we show that it is possible to end up withthe propagator of the harmonic oscillator for one charged particle. After some transformations, wederive the normalized wave functions and the eigenvalues in terms of different physical parametersand quantum numbers. By interchanging quantum numbers, we show that our solutions haveinteresting properties. The density of current and the non-relativistic limit are analyzed wheredifferent conclusions are obtained. Finally, the completeness of the Dirac oscillator eigenfunctionsis proved by using the standard properties of the generalized Laguerre polynomials.PACS numbers: 03.65.Pm, 03.65.GeKeywords: Dirac equation, confinement, magnetic field, path integral. ∗ [email protected] † [email protected] – [email protected] Introduction
The Dirac equation in (2+1)-dimensions is an important tool not only from mathematical point of viewbut also from the large applications in physics. In particular, several condensed matter phenomenapoint out to the existence of a (2 +1)-dimensional energy spectrum determined by the relativisticDirac equation [1]. For very recent works, one may consult references [2–7] and for early worksrelevant to our subject we cite [8, 9]. As example graphene [10], which is a single layer of carbonatoms arranged into a planar honeycomb lattice. This system has attracted a considerable attentionfrom both experimental and theoretical researchers since its experimental realization in 2004 [11].This is because of its unique and outstanding mechanical, electronic, optical, thermal and chemicalproperties [12]. Most of these marvelous properties are due to the apparently relativistic-like natureof its carriers, electrons behave as massless Dirac fermions in graphene systems. In fact starting fromthe original tight-binding Hamiltonian describing graphene it has been shown theoretically that thelow-energy excitations of graphene appear to be massless chiral Dirac fermions.On the other hand, due to recent technological advances in nano-fabrication there were a lot ofinterest in the study of low dimensional quantum systems such as quantum wells, quantum wires andquantum dots [13]. In particular, there has been considerable amount of work in recent years onsemiconductors confined structures, which finds applications in electronic and optoelectronic devices.These show the relevance of the confinements in physics and therefore deserve to be analyzed inother systems like relativistic ones. Furthermore, an applied magnetic field perpendicular to theheterostructure systems quantizes the energy levels in the plane, drastically affecting the density ofstates giving rise to the famous quantum Hall effect [14].A confined relativistic system subjected to a constant magnetic field was studied normal to theplane in [15]. In fact, the exact solution of the Dirac equation in (2 + 1)-dimensions was obtaineddepending on various parameters. The solution space consists of positive and negative energy solutions,each of which splits into two disconnected subspaces depending on the sign of an azimuthal quantumnumber l = 0 , ± , ± l , the relativistic energy spectrum was found to be in finitely degenerate due to the factthat it is dependent of l .Motivated by different investigations on the Dirac fermions in (2+1)-dimensions, we give an exactsolution of a problem that has been studied at various levels by researchers dealing with differentphysical phenomena. More precisely, we return to the problem studied in [15]and use another tech-niques to determine the solutions of the energy spectrum as well dealing with related issues. Then,we consider a relativistic particle subjected to an external magnetic field as well as to a confiningpotential. By using the path integral formalism and making different transformations, we show thatthe causal Green function can be written in terms of the propagator for the harmonic oscillator for acharged particle. This allowed us to obtain various solutions, in terms of different physical parametersand quantum numbers, and emphasis similarities to, and differences from, already published workelsewhere [15, 16].Furthermore, we give discussions of our results based on different physical settings. The full richspace of solutions suggested enabled us to carry out a deeper analysis in relation to various physical1uantities. For instance, we show that our energy remains invariant under the changes of the quantumnumbers characterizing our system behavior, reveals a non trivial symmetry of the problem. Also, weobtain, as expected in the absence of applied voltage, a null current density for both directions in theCartesian representation. However, this is not the case in polar coordinate. In fact, we show that theradial current vanishes whereas the angular component does not. It is dependent on various physicalparameters in the problem. Additionally, we discuss the non-relativistic limit of the problem.The paper is organized as follows. In section 2, we show that how one can use the global projectionrather than local one for (2 + 1)-dimensional Dirac equation in electromagnetic field. In section 3,we give the theoretical formulation of the problem where different changes are introduced to simplifythe process for obtaining the solutions of the energy spectrum. More precisely, we will use the causalGreen function as well as different techniques to solve our problem. We determine the eigenvaluesand eigenspinors in terms of different physical parameters and quantum numbers in section 4. Insection 5, we analysis our results by showing that our system has hidden symmetries, those can beused to deal with different issues. In section 6, we calculate the density current in polar coordinatesystem. We show how to recover the nonrelativistic limit from our results in section 7. The proofof the completeness relation of the Dirac oscillator eigenfunctions will be presented in section 8. Weconclude our findings in the final section. Before embarking on our problem and investigate its basic features, let us establish some mathematicaltool that needed to deal with different issues. For this, we start by considering a relativistic chargedparticle in electromagnetic field to show that the global projection giving rises to an new symmetricform for the Green function. For such system the causal Green function S c ( x b , x a ) satisfies the twoDirac equations [17] ( γ µ ( i∂ bµ − eA µ ( x b )) − m ) S c ( x b , x a ) = δ ( x b − x a ) (1) S c ( x b , x a ) (cid:16) γ µ (cid:16) − i ←− ∂ aµ − eA µ ( x a ) (cid:17) − m (cid:17) = δ ( x b − x a ) (2)where the matrices γ µ are defined by the relations generating Clifford algebra[ γ µ , γ ν ] = − iσ µν , { γ µ , γ ν } = 2 η µν . (3)Actually, we have η µν = diag (1 , − , − µ, ν = 0 , ,
2) and γ µ = i ǫ µνλ γ ν γ λ or equivalently γ = iγ γ , γ = − iγ γ , γ = iγ γ . (4)Formally, S c ( x b , x a ) is the matrix element in the coordinate space S c ( x b , x a ) = h x b | S c | x a i (5)of the inverse Dirac operator O − − S c = ( γ µ ( p µ − eA µ ) − m ) − = O − − = O + ( O − O + ) − (6)2here the operators O ± are given by O ± = γ µ ( p µ − eA µ ) ± m. (7)We recall that Alexandrou et al. [18] have been described a massive Dirac particle in externalvector and scalar fields by making use of the asymmetric form. In the present study, we consider thesymmetric form to write the Dirac propagator in order to straightforwardly derive the correspondingspinors. Indeed, the local inverse Dirac operator (6) can be written in two equivalent forms S c = ( γ µ ( p µ − eA µ ) − m ) − = O − − = O + S cg = S cg O + (8)where the Green operator for global projection is S cg = ( O − O + ) − = ( O + O − ) − (9)and the label g stands for global. It is clear that the matrix element of S cg verifies the quadratic Diracequation O − O + S cg ( x b , x a ) = O + O − S cg ( x b , x a ) = δ ( x b − x a ) . (10)Now let us insert the completeness relation R | z ih z | d z = I in the matrix element (5) and use (8)to end up with S c ( x b , x a ) = Z h x b | O + | z ih z | S cg | x a i d z = Z h x b | S cg | z ih z | O + | x a i d z (11)which implies the equality( γ µ −→ π µ ( b ) + m ) S cg ( x b , x a ) = S cg ( x b , x a ) ( γ µ ←− π µ ( a ) + m ) (12)where the momentum operators read as −→ π µ = i −→ ∂∂x µ − eA µ ( x ) , ←− π µ = − i ←− ∂∂x µ − eA µ ( x ) . (13)The right and left derivatives are defined byˆ A −→ ∂∂x ˆ B = ˆ A (cid:16) ∂∂x ˆ B (cid:17) , ˆ A ←− ∂∂x ˆ B = (cid:16) ∂∂x ˆ A (cid:17) ˆ B. (14)Therefore, we can deduce the following new symmetric form S c ( x b , x a ) = ( γ µ −→ π µ ( b ) + m ) S cg ( x b , x a ) + S cg ( x b , x a ) ( γ µ ←− π µ ( a ) + m ) (15)which is an interesting result and can be generalized to any dimension. We will see how it can be usedto determine explicitly the solution of the energy spectrum. redTo be specific, let us consider a confined system of (2 + 1)-dimensional Dirac fermions in a constantmagnetic field −→ B . This confinement can be realized by modifying the operators O ± , given in (7), by3he non-minimal substitution such that ~p −→ ~p − imω~rγ , where ω is the oscillator frequency. Bychoosing the symmetric gauge the symmetric gauge ~A = B ( − y, x ), we can write (7) O ± = γ p − γ (cid:18) p x − | e | B y − imωxγ (cid:19) − γ (cid:18) p y + | e | B x − imωyγ (cid:19) ± m (16)and using γ = iγ γ to obtain in the compact form O ± = γ p − γ ( p x + Gy ) − γ ( p y − Gx ) ± m (17)where we have set the parameter G = m ( ω − ω c ) and the cyclotron frequency ω c = | e | B m = ω . It isclearly seen that both frequencies ω and ω c come from different sources but regrouped in he sameplace, which is an unique property of (2 + 1)-dimensional space.Now, by using Schwinger proper time representation, we can express S cg ( x b , x a ) in an integralexpressions form S cg ( x b , x a ) = h x b | ( O + O − ) − | x a i = − i Z ∞ dλ h x b | exp (cid:18) i λ H + iε ) (cid:19) | x a i (18)where the Hamiltonian H is given by H = O + O − = ˆ p − ˆ p x − ˆ p y − G (cid:0) ˆ x + ˆ y (cid:1) − m + 2 G (ˆ x ˆ p y − ˆ y ˆ p x ) + 2 iGγ γ (19)for G = 0. Similar to [19], we represent S cg ( x b , x a ) via the path integral S cg ( x b , x a ) = − i (cid:18) iγ · ∂ l ∂θ (cid:19) Z ∞ dλ Z D x D p Z D ψ (20) × exp (cid:26) i Z (cid:20) − ˙x . p + λ (cid:0) p − m − G x (cid:1) +2 G x × p − iGψ ψ − iψ µ ˙ ψ µ i dτ + ψ µ (1) ψ µ (0) o(cid:12)(cid:12)(cid:12) θ =0 where x = ( t, x, y ) and ( ψ, θ ) refer respectively to even and odd variables. They satisfy boundaryconditions x (0) = x a , x (1) = x b , ψ µ (0) + ψ µ (1) = θ µ . (21)The path integration measure D ψ = Dψ "Z ψ (1)+ ψ (0)=0 Dψ exp (cid:18)Z ψ µ ˙ ψ µ dτ (cid:19) − (22)has been considered above. Note that, by integrating over the path t , we can see that the momentumsbecome constants, i.e. p =const.To go further, let us simplify our formalism by changing the momentum variables to define twoothers as p x = ¯ p x − Gy, p y = ¯ p y + Gx (23)which allow to write S cg ( x b , x a ) = − i (cid:18) iγ · ∂ l ∂θ (cid:19) Z ∞ dλ Z + ∞−∞ dp π e − ip ( t b − t a )+ iλ ( p − m ) Z DxDyD ¯ p x D ¯ p y × Z D ψ exp (cid:26) i Z (cid:20) ¯ p x ˙ x + ¯ p y ˙ y − λ p x − λ p y + G ( x ˙ y − y ˙ x ) − iλGψ ψ − iψ µ ˙ ψ µ i dτ + ψ µ (1) ψ µ (0) o(cid:12)(cid:12)(cid:12) θ =0 . (24)4ow integrating over ¯ p x and ¯ p y to find S g ( x b , x a ) = − i (cid:18) iγ. ∂ l ∂θ (cid:19) Z ∞ dλ Z + ∞−∞ dp π e − ip ( t b − t a )+ iλ ( p − m ) Z DxDy × Z D ψe (cid:26) i R (cid:18) ˙ x λ + ˙ y λ + G ( x ˙ y − y ˙ x ) − iλGψ ψ − iψ µ ˙ ψ µ (cid:19) dτ + ψ µ (1) ψ µ (0) (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) θ =0 . (25)In order to integrate over ψ µ ( τ ), let us first eliminate the constants θ µ appearing in the boundaryconditions (21) by performing the following changes ψ µ ( τ ) = ξ µ ( τ ) + 12 θ µ , µ = 0 , , . (26)By replacing in (25), we obtain S cg ( x b , x a ) = − i (cid:18) iγ · ∂ l ∂θ (cid:19) Z ∞ dλ Z + ∞−∞ dp π e − ip ( t b − t a )+ iλ ( p − m ) Z DxDy × Z D ξ exp (cid:26) i Z (cid:20) ˙ x λ + ˙ y λ + G ( x ˙ y − y ˙ x ) (cid:27) (27) − iλG (cid:0) ξ ξ − θ ξ ( τ ) + θ ξ ( τ ) + θ θ (cid:1) − iξ µ ˙ ξ µ i dτ o(cid:12)(cid:12)(cid:12) θ =0 where the antiperiodic boundary conditions become ξ µ (1) + ξ µ (0) = 0 . (28)By considering ξ = ξ ξ ! and σ = − ii ! , one can easily see the relation ξ ξ = i ξσ ξ (29)Now using (29) and being able to undertake easily the integration over Grassmann variables, one canseparate (27) as S cg ( x b , x a ) = − i (cid:18) iγ · ∂ l ∂θ (cid:19) Z ∞ dλ Z + ∞−∞ dp π e − ip ( t b − t a )+ iλ ( p − m ) (30) × Z Z
DxDy exp (cid:26) i Z (cid:18) ˙ x λ + ˙ y λ + G ( x ˙ y − y ˙ x ) (cid:19)(cid:27) I ( λ, θ ) | θ =0 where the fermionic part is given by I ( λ, θ ) = e λGθ θ Z D ξ Z Z D ξ (31) × e R ξ ( τ ) R ( λ | τ,τ ′ ) ξ ( τ ′ ) dτdτ ′ + R J ( τ ) ξ ( τ ) dτ + R ξ ( τ ) δ ′ ( τ − τ ′ ) ξ ( τ ′ ) dτdτ ′ and we have R (cid:0) λ | τ, τ ′ (cid:1) = − δ ′ (cid:0) τ − τ ′ (cid:1) + 2 iλGσ δ (cid:0) τ − τ ′ (cid:1) , J T ( τ ) = 2 λG (cid:0) − θ , θ (cid:1) . (32)Since the integration over ξ yields trivially the unity, then integrating over ξ and ξ to obtain I ( x, y, λ ; θ ) = q det R ( λ )det R (0) e λGθ θ e R R J ( τ ) . R − ( λ | τ,τ ′ ) . J ( τ ′ ) dτdτ ′ (33)5here R − ( λ | τ, τ ′ ) is the inverse matrix of R ( λ | τ, τ ′ ), which can be considered as an operatoracting on the space of antiperiodic functions R − ( λ | , τ ) + R − ( λ | , τ ) = 0 , ∀ τ ∈ [0 , . (34)It obeys the first order differential equation (cid:18) − ∂∂τ + 2 iGλσ (cid:19) R − (cid:0) λ | τ, τ ′ (cid:1) = δ (cid:0) τ − τ ′ (cid:1) (35)giving the solution R − (cid:0) λ | τ, τ ′ (cid:1) = 12 e iλGσ ( τ − τ ′ ) (cid:2) iσ tan ( λG ) − ε (cid:0) τ − τ ′ (cid:1)(cid:3) , ε (cid:0) τ − τ ′ (cid:1) = sgn (cid:0) τ − τ ′ (cid:1) . (36)Using similar calculations as in [20] to get q det R ( λ )det R (0) = e T r R λ R − ( λ ′ ) ddλ ′ R ( λ ′ ) dλ ′ = e T r R λ [ iσ tan( λ ′ G ) ] [2 iGσ ] dλ ′ = e − R λ tan( λ ′ G ) Gdλ ′ = cos ( λG ) . (37)Combining all to show that (30) takes the form S cg ( x b , x a ) = − i Z ∞ dλ Z + ∞−∞ dp π e − ip ( t b − t a )+ iλ ( p − m )Φ ( λ ) K ( x a , x b , y a , y b ; λ ) (38)where K is the propagator of the harmonic oscillator for one charged particle K ( x a , x b , y a , y b ; λ ) = Z Z
DxDye i R (cid:18) ˙ x λ + ˙ y λ + G ( x ˙ y − y ˙ x ) (cid:19) dτ (39)and the function Φ ( λ ) reads asΦ ( λ ) = cos ( λG ) exp (cid:18) iγ · ∂ l ∂θ (cid:19) exp (cid:0) tan ( λG ) θ θ (cid:1)(cid:12)(cid:12) θ =0 . (40)In order to evaluate the propagator K , we first decouple x and y by introducing a rotation of coordi-nates to define new variables q and q as x = q cos ( λGτ ) + κq sin ( λGτ ) (41) y = − κq sin ( λGτ ) + q cos ( λGτ ) (42)where κ = sgn ( G ) = 0. Under the above rotation, K can be written in separable form as K ( x a , x b , y a , y b ; λ ) = Z Z Dq Dq e i R (cid:18) ˙ q λ + ˙ q λ − λ | G | q − λ | G | q (cid:19) dτ (43)which represents the kernel of two propagators corresponding to two harmonic oscillators and haswell-known form K ( x a , x b , y a , y b ; λ ) = | G | iπ sin( | G | λ ) e i | G | | G | λ [( ( x b − x a ) +( y b − y a ) ) cos( | G | λ )+2 κ sin( | G | λ )( x a y b − y a x b ) ] (44)6here the following coordinates have been used q a = x a , q b = cos ( Gλ ) x b − κ sin ( Gλ ) y b (45) q a = y a , q b = κ sin ( Gλ ) x b + cos ( Gλ ) y b . (46)To explicitly determine the matrix Φ ( λ ), let us proceed by using derivation over the variables θ . Thecalculations performed by adopting the prevailing approach [19] such that acting the operator ∂ l ∂θ andthen replacing θ by the matrices γ . We can write the expression of Φ ( λ ) by using the identitiesexp (cid:16) iγ µ ∂ l ∂θ µ (cid:17) f ( θ ) | θ =0 = f (cid:16) ∂ l ∂ζ (cid:17) exp( iζ µ γ µ ) | ζ =0 , (47)exp ( iζ µ γ µ ) = 1 + iζ µ γ µ + 12 ζ µ ζ ν γ µ γ ν + iζ ζ ζ γ γ γ (48)which are valid for (2+1)-dimensional, with γ = iσ , γ = iσ , γ = σ (49)where ζ µ are odd variables. In this case, we can simplify Φ( λ ) asΦ ( λ ) = e iκ | G | λσ (50)and therefore (38) becomes S cg ( x b , x a ) = − i Z ∞ dλ Z + ∞−∞ dp π e − ip ( t b − t a )+ iλ ( p − m ) e iκ | G | λσ K ( x a , x b , y a , y b ; λ ) (51)which can be implemented in (15) to obtain the causal Green function. This task will be done toextract the solutions of the energy spectrum. We would like to determine the eigenvalues and corresponding eigenspinors of our system.To this end,we introduce the polar coordinates x = r cos ϕ, y = r sin ϕ. Thus, the causal Green function (15)becomes S c ( r a , r b , ϕ a , ϕ b , t b , t a ) = − i Z ∞ dλ Z + ∞−∞ dp π e − ip ( t b − t a )+ iλ ( p − m ) (52) × (cid:26)(cid:18) p σ + σ e iσ ϕ b ∂∂r b + 1 r b σ e iσ ϕ b ∂∂ϕ b − iκ | G | σ r b e iσ ϕ b + m (cid:19) e iκ | G | λσ K + e iκ | G | λσ (cid:18) p σ − σ e iσ ϕ a ∂∂r a − r a σ e iσ ϕ a ∂∂ϕ a − iκ | G | σ r a e iσ ϕ a + m (cid:19) K (cid:27) where K = K ( r a , r b , ϕ a , ϕ b ; λ ) = | G | iπ sin( | G | λ ) e i | G | | G | λ [( r b + r a ) cos( | G | λ ) − r b r a (cos( ϕ b − ϕ a + κ | G | λ )) ] . (53)In order to deduce the wave functions, we use the expansion and Hille formulas [22] e (cid:16) − i | G | sin | G | λ r b r a cos( ϕ b − ϕ a + κ | G | λ ) (cid:17) = + ∞ X l = −∞ I | l | (cid:18) − i | G | sin | G | λ r b r a (cid:19) e il ( ϕ b − ϕ a + κ | G | λ ) (54) ( xyt ) − α − t exp (cid:16) − t x + y − t (cid:17) I α (cid:16) √ xyt − t (cid:17) = ∞ X n =0 n ! L αn ( x ) L αn ( y ) t n Γ( n + α +1) , | t | < . (55)7y taking t = e − iλ | G | , x = | G | r b , y = | G | r a and α = | l | , the propagator K can be expressed as K ( r a , r b , ϕ a , ϕ b ; λ ) = | G | π + ∞ X n =0 + ∞ X l = −∞ e − iλ | G | n e − i | l || G | λ e il ( ϕ b − ϕ a + κ | G | λ ) n !Γ( n + | l | +1) × e − i | G | λ z | l | b z | l | a e − z b e − z a L | l | n (cid:0) z b (cid:1) L | l | n (cid:0) z a (cid:1) (56)where z = p | G | r , L | l | n (cid:0) z (cid:1) is a generalized Laguerre polynomial and | l | is an integer number. Afterinserting (56) into (52) we obtain S c ( x b , x a ) = − i | G | √ | G | π + ∞ X n =0 + ∞ X l = −∞ n !( n + | l | )! Z + ∞−∞ dp π e − ip ( t b − t a ) e il ( ϕ b − ϕ a ) × Z ∞ dλe i λ ( p − m − | G | [2 n +1+ | l |− κl ] ) × (cid:26)(cid:18)(cid:16) − iκσ z b + σ ∂∂z b + iσ lz b (cid:17) e iσ ϕ b + m + p σ √ | G | (cid:19) e iκ | G | λσ + e iκ | G | λσ (cid:18) e − iσ ϕ a (cid:16) − iκσ z a − σ ∂∂z a + iσ lz a (cid:17) + m + p σ √ | G | (cid:19)(cid:27) × e − z b z | l | b L | l | n (cid:0) z b (cid:1) e − z a z | l | a L | l | n (cid:0) z a (cid:1) . (57)Let us introduce the spin operator σ and insert the identity P s = ± χ s χ + s = I , where σ χ s = sχ s and χ + s σ = sχ + s . We can easily check the relations σ χ s = χ − s , σ χ s = isχ − s (58) χ + s σ = χ + − s , χ + s σ = − isχ + − s (59)for the vectors χ +1 = ! , χ − = ! (60)and the identities σ e iσ ϕ = e − iσ ϕ σ , σ e iσ ϕ = e − iσ ϕ σ . (61)These can be used to write the Green function (57) relative to our particle as S c ( x b , x a ) = − i | G | √ | G | π + ∞ X n =0 + ∞ X l = −∞ X s = ± n !( n + | l | )! Z + ∞−∞ dp π e − ip ( t b − t a ) e il ( ϕ b − ϕ a ) × Z ∞ dλe i λ ( p − m − | G | [2 n +1+ | l |− κ ( l + s )] ) × (cid:26)(cid:18)(cid:16) − iκσ z b + σ ∂∂z b + iσ lz b (cid:17) e isϕ b + m + p s √ | G | (cid:19) χ s χ + s e iκ | G | λs + e iκ | G | λs χ s χ + s (cid:18) e − isϕ a (cid:16) − iκσ z a − σ ∂∂z a + iσ lz a (cid:17) + m + p s √ | G | (cid:19)(cid:27) × e − z b z | l | b L | l | n (cid:0) z b (cid:1) e − z a z | l | a L | l | n (cid:0) z a (cid:1) . (62)In order to symmetrize the angular part we make the shift l → l − s + 12 (63)8hich implies S c ( x b , x a ) = − i | G | √ | G | π ∞ X n =0 + ∞ X l = −∞ X s = ± | n | ! ( | n | + | l − s +12 | ) ! Z + ∞−∞ dp π e − ip ( t b − t a ) e il ( ϕ b − ϕ a ) e − i ϕb − ϕa × Z ∞ dλe i λ ( p − m − | G | [ n +1+ | l − s +12 | − κ ( l + s − )]) × (cid:26)(cid:18)(cid:18) κsz b + ∂∂z b − s l − s +12 z b (cid:19) χ − s χ + s e is ϕb + ϕa + m + p s √ | G | e − is ϕb − ϕa χ s χ + s (cid:19) + (cid:18)(cid:18) − κsz a − ∂∂z a + s l − s +12 z a (cid:19) χ s χ + − s e − is ϕb + ϕa + χ s χ + s m + p s √ | G | e − is ϕb − ϕa (cid:19)(cid:27) × e − z b z | l − s +12 | b L | l − s +12 | n (cid:0) z b (cid:1) e − z a z | l − s +12 | a L | l − s +12 | n (cid:0) z a (cid:1) . (64)Introducing F κ ′ n,l,s ( z ) = r ( n )! ( n + κ ′ ( l − s +12 )) ! e − z z κ ′ ( l − s +12 ) L κ ′ ( l − s +12 ) n (cid:0) z (cid:1) (65)where κ ′ = sgn (cid:0) l − s +12 (cid:1) and the momentum π κs ( z ) = ddz + κsz − s l − s +12 z (66)we write S c ( x b , x a ) as S c ( x b , x a ) = − i | G | √ | G | π + ∞ X n =0 + ∞ X l = −∞ X s = ± Z + ∞−∞ dp π e − ip ( t b − t a ) e − z b e − z a e il ( ϕ b − ϕ a ) e − i ϕb − ϕa × X s = ± Z ∞ dλe − i λ ( p − m − | G | [ n +1+ | l − s +12 | − κ ( l + s − )]) (67) × (cid:26)(cid:18) e is ϕb + ϕa π κs ( z b ) χ − s χ + s + e − is ϕb − ϕa m + p s √ | G | χ s χ + s (cid:19) F κ ′ n,l,s ( z b ) F κ ′ n,l,s ( z a )+ (cid:18) − e − is ϕb + ϕa π κs ( z a ) χ s χ + − s + e − is ϕb − ϕa m + p s √ | G | χ s χ + s (cid:19) F κ ′ n,l,s ( z b ) F κ ′ n,l,s ( z a ) (cid:27) . Integrating over λ to get the expression (cid:20) i (cid:18) p − m − | G | (cid:20) n + 1 + (cid:12)(cid:12)(cid:12)(cid:12) l − s + 12 (cid:12)(cid:12)(cid:12)(cid:12) − κ (cid:18) l + s − (cid:19)(cid:21)(cid:19)(cid:21) − (68)which has the poles p = ± q m + 2 | G | (cid:2) n + 1 + (cid:12)(cid:12) l − s +12 (cid:12)(cid:12) − κ (cid:0) l + s − (cid:1)(cid:3) = ± E n,l,s . (69)After using the residue theorem at pole p , we find Z + ∞−∞ f ( p ) dp π e − ip ( t b − t a ) p − E n,l,s = − i X ε = ± f ( εE n,l,s ) e − iεE n,l,s ( t b − t a ) E n,l,s Θ ( ε ( t b − t a )) (70)9here Θ( x ) is the Heaviside step function. Since ε = ± s = ±
1, one can easily check the identity P s = ± f s P ε = ± g ε = P s = ± f s ( g s + g − s ) . After rearranging different terms, we obtain S c ( x b , x a ) = i π q | G | ∞ X n =0 + ∞ X l = −∞ X s = ± E n,l,s e i ( l − / ϕ b − ϕ a ) × (cid:26) e − isE n,l,s ( t b − t a ) Θ ( s ( t b − t a )) (cid:20)(cid:18) e is ϕb + ϕa π κs ( z b ) χ − s χ + s + e − is ϕb − ϕa m + E n,l,s √ | G | χ s χ + s (cid:19) − (cid:18) e − is ϕb + ϕa π κs ( z a ) χ s χ + − s − e − is ϕb − ϕa m + E n,l,s √ | G | χ s χ + s (cid:19)(cid:21) (71)+ e isE n,l,s ( t b − t a ) Θ ( − s ( t b − t a )) (cid:20)(cid:18) e is ϕb + ϕa π κs ( z b ) χ − s χ + s + e − is ϕb − ϕa m − E n,l,s √ | G | χ s χ + s (cid:19) − (cid:18) e − is ϕb + ϕa π κs ( z a ) χ s χ + − s − e − is ϕb − ϕa m − E n,l,s √ | G | χ s χ + s (cid:19)(cid:21)(cid:27) F κ ′ n,l,s ( z b ) F κ ′ n,l,s ( z a ) . To go further, we proceed by using the following mapping s → s ′ = − s, n → n ′ = n − κ ′ + κ s = ( n − κs, if κ ′ = κ or κ ′ = 0 n, if κ ′ = − κ (72)only for terms containing Θ ( − s ( t b − t a )) in (71) to the dummy variables n and s , which does notalter the integer nature of n . By taking into consideration the energy invariance under (72), the Greenfunction takes the following form S c ( x b , x a ) = i π q | G | ∞ X n =0 + ∞ X l = −∞ X s = ± E n,l,s e i ( l − / ϕ b − ϕ a ) e − isE n,l,s ( t b − t a ) Θ ( s ( t b − t a )) × (cid:26)(cid:20)(cid:18) e is ϕb + ϕa π κs ( z b ) χ s ′ χ + s + e − is ϕb − ϕa m + E n,l,s √ | G | χ s χ + s (cid:19) (73)+ (cid:18) − e − is ϕb + ϕa π κs ( z a ) χ s χ + s ′ + e − is ϕb − ϕa m + E n,l,s √ | G | χ s χ + s (cid:19)(cid:21) F κ ′ n,l,s ( z b ) F κ ′ n,l,s ( z a )+ (cid:20)(cid:18) e − is ϕb + ϕa π κs ′ ( z b ) χ s χ + s ′ + e is ϕb − ϕa m − E n,l,s √ | G | χ s ′ χ + s ′ (cid:19) + (cid:18) − e is ϕb + ϕa π κs ′ ( z a ) χ s ′ χ + s + e is ϕb − ϕa m − E n,l,s √ | G | χ s ′ χ + s ′ (cid:19)(cid:21) F κ ′ n ′ ,l,s ′ ( z b ) F κ ′ n ′ ,l,s ′ ( z a ) (cid:27) . By virtue of Rodrigues’ formula L αn ( x ) = 1 n ! e x x − α d n dx n (cid:0) e − x x n + α (cid:1) (74)we deduce after verification for all numbers ( κ , κ ′ , s ) the interesting properties π κs ( z ) F κ ′ n,l,s ( z ) = − s r E n,l,s − m | G | F κ ′ n ′ ,l,s ′ ( z ) (75) π κs ′ ( z ) F κ ′ n ′ ,l,s ′ ( z ) = s r E n,l,s − m | G | F κ ′ n,l,s ( z ) . (76)10hese give S c ( x b , x a ) = i | G | π + ∞ X n =0 + ∞ X l = −∞ X s = ± E n,l,s e i ( l − / ϕ b − ϕ a ) e − isE n,l,s ( t b − t a ) Θ ( s ( t b − t a )) × n e − is ϕb − ϕa ( E n,l,s + m ) F κ ′ n,l,s ( z b ) F κ ′ n,l,s ( z a ) χ s χ + s + e is ϕb − ϕa ( E n,l,s − m ) F κ ′ n ′ ,l,s ′ ( z b ) F κ ′ n ′ ,l,s ′ ( z a ) χ s ′ χ + s ′ − s q E n,l,s − m h e − is ϕb + ϕa F κ ′ n,l,s ( z b ) F κ ′ n ′ ,l,s ′ ( z a ) χ s χ + s ′ + e is ϕb + ϕa F κ ′ n ′ ,l,s ′ ( z b ) F κ ′ n,l,s ( z a ) χ s ′ χ + s io sσ (77)which can be simplified to end up with S c ( x b , x a ) = i + ∞ X n =0 + ∞ X l = −∞ X s = ± Ψ κ,κ ′ n,l,s ( r b , ϕ b ; t b ) (cid:16) Ψ κ,κ ′ n,l,s ( r a , ϕ a ; t a ) (cid:17) + σ s Θ ( s ( t b − t a )) (78)where the normalized wave functions are given byΨ κ,κ ′ n,l,s ( r, ϕ ; t ) = q | G | π e − i E n,l,s t e ilϕ e − i σ ϕ (cid:20)r E n,l,s + sm E n,l,s F κ ′ n,l,s ( z ) χ s − s r E n,l,s − sm E n,l,s F κ ′ n − κ ′ + κ s,l, − s ( z ) χ − s (cid:21) (79)and the corresponding eigenvalues take the form E n,l,s = sE n,l,s = s q m + 2 | G | (cid:2) n + 1 + κ ′ (cid:0) l − s +12 (cid:1) − κ (cid:0) l + s − (cid:1)(cid:3) . (80)In compact form, we have the solutions of the energy spectrumΨ n,l,s ( r, ϕ ; t ) = q | G | π e − is E n,l,s t e ilϕ e − i σ ϕ (81) × (cid:20)r E n,l,s + sm E n,l,s F n,l,s ( z ) χ s + r | G | E n,l,s ( E n,l,s + sm ) π s ( z ) F n,l,s ( z ) χ − s (cid:21) E n,l,s = s r m + 2 | G | h n + 1 + (cid:12)(cid:12) l − s +12 (cid:12)(cid:12) − G | G | (cid:0) l + s − (cid:1)i (82)with the quantities κ = sgn ( G ) = G | G | = 0, κ ′ = sgn (cid:0) l − s +12 (cid:1) , s = sgn ( E n,l,s ) and the variable z = p | G | r . It is interesting to note that our obtained energy spectrum is completely in agreementwith that derived quantum mechanically in [15]Note that by putting ω = 0, which gives G = − mω c , and making the substitution l −→ l + s +12 in(82) we end up with the energy E s = ± = m + 2 mω c [2 n + 1 + | l | + l + s ] . (83)This can be compared to the energy for a massive fermion in a constant magnetic field in 3+1-dimensional space-time [24]. It is given by E λ = ± − p z = m + 2 mκ [2 n + | l | + l − λ + 1] (84)which shows that it is in agreement with our derived result (83).11 Hidden symmetries
We show that there are nontrivial hidden symmetries in our solutions of the energy spectrum. Asfar as the eigenvalues are concerned, we notice that by considering three configurations of quantumnumber ( n, l, s ), the energy remains invariant absolutely. These are listed in the following tableSign s Quantum number l Quantum number n Energy E n,l,s s −→ − s l −→ l − s n −→ n − κs E n − κs,l − s, − s = −E n,l,s s −→ − s l −→ l + s n −→ n − κ ′ s E n − κ ′ s,l + s, − s = −E n,l,s s −→ − s l −→ l n −→ n − κ ′ + κ s E n − κ ′ + κ s,l, − s = −E n,l,s Table 1:
Table summarizes different symmetries of our system according to changes in terms of thequantum numbers ( n, l, s ).Furthermore, similar attention reed to be paid to the eigenspinors Ψ n,l,s ( r, ϕ ; t ) and underline theirbasic features. Let us focus particularly on the last configuration and express Ψ n,l,s ( r, ϕ ; t ) asΨ n,l,s ( r, ϕ ; t ) = u n,l,s ( r, ϕ ; t ) χ s + v n,l,s ( r, ϕ ; t ) χ − s (85)where u n,l,s and v n,l,s are two-component defined by u n,l,s ( r, ϕ ; t ) = e − i E n,l,s t q | G | π e ilϕ e − i s +12 ϕ r E n,l,s + sm E n,l,s F κ ′ n,l,s ( z ) (86) v n,l,s ( r, ϕ ; t ) = − se − i E n,l,s t q | G | π e ilϕ e − i − s +12 ϕ r E n,l,s − sm E n,l,s F κ ′ n − κ ′ + κ s,l, − s ( z ) . (87)Now using the third configuration in Table 1 to show there is a relation between v n,l,s and u n,l,s . Thisis given by v n − κ ′ + κ s,l, − s ( r, ϕ ; − t ) = s s E n,l,s − sm E n,l,s + sm u n,l,s ( r, ϕ ; t ) (88)which is in agreement with [23]. According to the above results, we conclude that the obtained energyspectrum is invariant with respect to the symmetries listed above. This invariance can be used to dealwith some issues related to the present system. Let us consider the density of current ~J for our system and investigate its basic features. Indeed, fromour results we can end up with the form ~J = i h σ ~σ i . (89)For this calculation, we use the spinor wavefunction obtained above. This gives a null value in Cartesiancoordinates, i.e. J x = J y = 0, which of course, is expected since there is no net charge drift in oursystem. 12s a reassuring exercise, we calculate the same current ~J but in cylindrical coordinates. Doingthis process to obtain two components J r = ~J . ˆ r = i h σ ~σ · ˆ r i , J θ = ~J · ˆ θ = i D σ ~σ. ˆ θ E . (90)We can show that the radial component is null, i.e. ¯ J n,l,sr = 0. For the angular one, we use thedefinition to obtain ¯ J n,l,sϕ = Z + ∞ Z π Ψ + n,l,s ( r, ϕ ) σ e iσ ϕ Ψ n,l,s ( r, ϕ ) rdrdϕ (91)which can be written as¯ J n,l,sϕ = 2 | G | r | G |E n,l,s Z + ∞ F n,l,s ( z ) π κs ( z ) F n,l,s ( z ) rdr (92)= 2 | G | r | G |E n,l,s Z + ∞ F n,l,s ( z ) (cid:18) ddz + κsz − s l − s +12 z (cid:19) F n,l,s ( z ) rdr. Using the relation [21] Z + ∞ x α − e − x L γm ( x ) L λn ( x ) dx = (1+ γ ) m ( λ − α +1) n Γ( α ) m ! n ! 3 F ( − m, α, α − λ ; 1 + γ, α − λ − n ; 1) (93)for Re ( α ) > , the well-known recurrence and derivative properties of generalized Laguerre functionsto obtain the final form for (92)¯ J n,l,sϕ = 2 s n ! √ | G | ( n + | l − s +12 | ) ! E n,l,s (cid:0)(cid:12)(cid:12) l − s +12 (cid:12)(cid:12) − s (cid:0) l − s +12 (cid:1)(cid:1) ( | l − s +12 | ) n (cid:16) (cid:17) n Γ (cid:16)(cid:12)(cid:12)(cid:12) l − s +12 (cid:12)(cid:12)(cid:12) + 12 (cid:17) ( n !) × F (cid:0) − n, (cid:12)(cid:12) l − s +12 (cid:12)(cid:12) + , ; 1 + (cid:12)(cid:12) l − s +12 (cid:12)(cid:12) , − n ; 1 (cid:1) − s n ! √ | G | ( n + | l − s +12 | ) ! E n,l,s ( | l − s +12 | ) n − (cid:16) − (cid:17) n Γ (cid:16)(cid:12)(cid:12)(cid:12) l − s +12 (cid:12)(cid:12)(cid:12) + 32 (cid:17) ( n − n ! × F (cid:18) − n + 1 , (cid:12)(cid:12) l − s +12 (cid:12)(cid:12) + , ; 2 + (cid:12)(cid:12)(cid:12)(cid:12) l − s + 12 (cid:12)(cid:12)(cid:12)(cid:12) , − n ; 1 (cid:19) (94)+2 s n ! √ | G | ( n + | l − s +12 | ) ! E n,l,s (cid:16) G | G | s − (cid:17) ( | l − s +12 | ) n (cid:16) − (cid:17) n Γ (cid:16)(cid:12)(cid:12)(cid:12) l − s +12 (cid:12)(cid:12)(cid:12) + 32 (cid:17) ( n !) × F (cid:0) − n, (cid:12)(cid:12) l − s +12 (cid:12)(cid:12) + , ; 1 + (cid:12)(cid:12) l − s +12 (cid:12)(cid:12) , − n ; 1 (cid:1) where F is the generalized hypergeometric function. This expression can be worked under differentassumptions to end up with a simple form and therefore make link with some physical phenomena.Let us illustrate (94) by summarizing some simple forms in Table 2Configuration Quantum numbers n, l, s Angular current ¯ J n,l,sϕ n = 0 , l = 0 , s = +1 (1 + 3 κ ) q π | G | m +4 | G | n = 0 , l = 0 , s = − − (1 + κ ) q π | G | m +2 | G | (1+ κ ) n = 0 , l = 1 , s = +1 (1 − κ ) q π | G | m +4 | G | n = 0 , l = 1 , s = − − (1 − κ ) q π | G | m +2 | G | (1 − κ ) Table illustrates some examples of the angular current for some quantum numbers values ( n, l, s ).From Table 2, we can immediately realize that there are some symmetries. Indeed, for the config-urations (1 ,
3) one can establish the relation¯ J , , +1 ϕ ( κ ) = ¯ J , , +1 ϕ ( − κ ) (95)and the same between (2 ,
4) ¯ J , , − ϕ ( κ ) = ¯ J , , − ϕ ( − κ ) (96)under the change κ −→ − κ , with the parameter κ = sgn ( G ) = G | G | = 0. We expect to have othersymmetries can be found by choosing new configurations of the three quantum numbers. This isinteresting and could be used to formulate a theory to describe low dimensional systems withouthaving external excitations. To recover the nonrelativistic limit, we use the standard process. Indeed, by requiring the limit m −→ ∞ in the obtained solutions of the energy spectrum, one can see E n,l,s −→ sm (cid:18) | G | m h n + 1 + (cid:12)(cid:12) l − s +12 (cid:12)(cid:12) − G | G | (cid:0) l + s − (cid:1)i(cid:19) (97) r E n,l,s + sm E n,l,s −→ r E n,l,s − sm E n,l,s −→ . (99)These allow the Green function to behave as S c ( x b , x a ) −→ i | G | π + ∞ X n =0 + ∞ X l = −∞ X s = ± e i ( l − (1+ s ) / ϕ b − ϕ a ) e − i (cid:16) sm + s | G | m h n +1+ (cid:12)(cid:12)(cid:12) l − s +12 (cid:12)(cid:12)(cid:12) − G | G | (cid:16) l + s − (cid:17)i(cid:17) ( t b − t a ) × Θ ( s ( t b − t a )) F κ ′ n,l,s ( z b ) F κ ′ n,l,s ( z a ) χ s χ + s . (100)Now taking l −→ l + s +12 to write (100) as S c ( x b , x a ) −→ i | G | π + ∞ X n =0 + ∞ X l = −∞ e il ( ϕ b − ϕ a ) e − i (cid:16) m + | G | m h n +1+ | l |− G | G | ( l +1) i(cid:17) ( t b − t a ) Θ (( t b − t a )) × F n,l ( z b ) F n,l, ( z a ) χ +1 χ ++1 (101)+ i | G | π + ∞ X n =0 + ∞ X l = −∞ e il ( ϕ b − ϕ a ) e i (cid:16) m + | G | m h n +1+ | l |− G | G | ( l − i(cid:17) ( t b − t a ) Θ ( − ( t b − t a )) × F n,l ( z b ) F n,l ( z a ) χ − χ + − and therefore to end up with usual eigenfunctions for the nonrelativistic problemΨ n,l,s ( r, ϕ ; t ) → e − i (cid:16) m + | G | m h n +1+ | l |− G | G | ( l +1) i(cid:17) t Ψ NR ( r, ϕ )0 ! (102)14r in the form Ψ n,l,s ( r, ϕ ; t ) → e + i (cid:16) m + | G | m h n +1+ | l |− G | G | ( l − i(cid:17) t NR ( r, ϕ ) ! (103)where Ψ NR ( r, ϕ ) are the nonrelativistic functions defined byΨ NR ( r, ϕ ) = q | G | n ! π ( n + | l | )! e ilϕ (cid:16)p | G | r (cid:17) | l | e − | G | r L | l | n (cid:0) | G | r (cid:1) (104)and the corresponding nonrelativistic energy is E NR = | G | m (cid:20) n + 1 + | l | − G | G | ( l ± (cid:21) = E n,l,s − m m . (105)The same results can be found for one-relativistic particle living on the plane ( x ; y ) in presence ofa perpendicular magnetic field B described by the Pauli-Schr¨odinger Hamiltonian H P S = 12 m h ~σ. (cid:16) ~P − e ~A (cid:17)i (106)in the symmetric gauge ~A = B ( − y, x ), more detail can be found in [6]. This tells us that our findingsare interesting and allow to recover the well-known results for the present problem. It is interesting to examine the completeness relation associated to our solutions of the energy spec-trum. Indeed, the closure relation obeyed by eigenfunctions is given by + ∞ X n =0 + ∞ X l = −∞ X s = ± Ψ κ,κ ′ n,l,s ( x b ) (cid:16) Ψ κ,κ ′ n,l,s ( x a ) (cid:17) + = I × δ ( x b − x a ) . (107)Introducing I ( x b , x a ) = + ∞ X n =0 + ∞ X l = −∞ X s = ± Ψ κ,κ ′ n,l,s ( r b , ϕ b ) (cid:16) Ψ κ,κ ′ n,l,s ( r a , ϕ a ) (cid:17) + = q | G | π e ilϕ b (cid:20)r E n,l,s + sm E n,l,s F κ ′ n,l,s ( z ) χ s − s r E n,l,s − sm E n,l,s F κ ′ n − κ ′ + κ s,l, − s ( z ) χ − s (cid:21) × q | G | π (cid:20)r E n,l,s + sm E n,l,s F κ ′ n,l,s ( z ) χ + s − s r E n,l,s − sm E n,l,s F κ ′ n − κ ′ + κ s,l, − s ( z ) χ + − s (cid:21) = | G | π + ∞ X n =0 + ∞ X l = −∞ X s = ± n E n,l,s + sm E n,l,s F κ ′ n,l,s ( z b ) F κ ′ n,l,s ( z a ) e − i s +12 ϕ b e i s +12 ϕ a χ s χ + s + E n,l,s − sm E n,l,s F κ ′ n − κ ′ + κ s,l, − s ( z b ) F κ ′ n − κ ′ + κ s,l, − s ( z a ) e − i − s +12 ϕ b e i − s +12 ϕ a χ − s χ + − s (108) − s E n,l,s − sm E n,l,s E n,l,s + sm E n,l,s F κ ′ n − κ ′ + κ s,l, − s ( z b ) F κ ′ n,l,s ( z a ) e − i − s +12 ϕ b e i s +12 ϕ a χ − s χ + s − s E n,l,s − sm E n,l,s E n,l,s + sm E n,l,s F κ ′ n − κ ′ + κ s,l, − s ( z a ) F κ ′ n,l,s ( z b ) e − i s +12 ϕ b e i − s +12 ϕ a χ s χ + − s (cid:27) . s → − s and n → n − κ ′ + κ s , one can simplify I ( x b , x a ) as I ( x b , x a ) = | G | π + ∞ X n =0 + ∞ X l = −∞ X s = ± F κ ′ n,l,s ( z b ) F κ ′ n,l,s ( z a ) e − i s +12 ϕ b e i s +12 ϕ a χ s χ + s = | G | π + ∞ X n =0 + ∞ X l = −∞ X s = ± (cid:26) ( n )! ( n + κ ′ ( l − s +12 )) ! e − z b z κ ′ ( l − s +12 ) b L κ ′ ( l − s +12 ) n (cid:0) z b (cid:1) (109) × e − z a z κ ′ ( l − s +12 ) a L κ ′ ( l − s +12 ) n (cid:0) z a (cid:1) e − i s +12 ϕ b e i s +12 ϕ a χ s χ + s (cid:27) . Taking l → l + s +12 , I ( x b , x a ) becomes I ( x b , x a ) = 2 | G | + ∞ X n =0 ( n )!( n + κ ′ l )! e − z b z κ ′ lb L κ ′ ln (cid:0) z b (cid:1) e − z a z κ ′ la L κ ′ ln (cid:0) z a (cid:1) + ∞ X l = −∞ e il ( ϕ b − ϕ a ) π X s = ± χ s χ + s . (110)By making use of the known closure relation obeyed by the generalized Laguerre functions + ∞ X n =0 ( n )!( n + α )! e − ( ρ a + ρ b ) ρ α a ρ α b L αn ( ρ a ) L αn ( ρ b ) = δ ( ρ b − ρ a ) (111)to obtain I ( x b , x a ) = 2 | G | + ∞ X n =0 ( n )!( n + κ ′ l )! e − z b z κ ′ lb L κ ′ ln (cid:0) z b (cid:1) e − z a z κ ′ la L κ ′ ln (cid:0) z a (cid:1) + ∞ X l = −∞ e il ( ϕ b − ϕ a ) π X s = ± χ s χ + s = 2 | G | δ (cid:0) z b − z a (cid:1) δ ( ϕ b − ϕ a ) I × = 2 | G | δ (cid:0) | G | r b − | G | r a (cid:1) δ ( ϕ b − ϕ a ) I × (112)Now considering the properties of the Dirac delta function δ (cid:0) | G | r b − | G | r a (cid:1) = δ ( r b − r a )2 | G | √ r b r a (113)to finally get I ( x b , x a ) = δ ( r b − r a ) √ r b r a δ ( ϕ b − ϕ a ) I × = I × δ ( x b − x a ) (114)which proves the completeness of the Dirac oscillator eigenfunctions obtained above.We close by noting that choosing the energy sign equal to that of the quantum number s , i.e.sgn ( E n,l,s ) = sgn ( s ), is required by the supersymmetric nature of our spinors solution of the Diracequation. This together with the quantification of the radial variable suggest to take the quantumnumber n belongs to N . However, in [25] the relation sgn ( E n,l,s ) = sgn ( n ) is made by hand andtherefore n must be in Z to prove the completeness of the Dirac oscillator in 3-dimensional. We have established a new symmetric expression to solve the problem of the relativistic confiningfermion interacting with the constant magnetic field within the framework supersymmetric represen-tation path integrals of Fradkin and Gitman [19]. This new symmetric form for global path integral16epresent the first attempt to find a general method to extract spinors in certain elegance. The energyspectrum is derived from the spectral decomposition of the causal Green function and have found tobe in agreement with that obtained [15].The solutions of the energy spectrum are obtained to be dependent on different physical parametersand quantum numbers. By inspecting the basic features of such solutions, we have showed there oursystem is hidden some interesting symmetries. Indeed, by considering three different configurationsof the quantum numbers, we have noticed that the energy remained invariant. This properties couldbe used to systemically establish a theory based on such symmetries to deal with some physicalphenomena like the quantum Hall effect [14].Subsequently, we have focused on two important issues related to our system. The first one is thedensity of current where the radial part was found to be null, while the angular one was not and wasexpressed in terms of the generalized hypergeometric function. The second issue is the nonrelativisticlimit that has been studied by using the standard method and therefore the corresponding solutionsof the energy spectrum were recovered in accordance with [15].To close, let us notice that the advantage of using the path integral techniques for the systemunder consideration is to extract the corresponding spinors from the spectral decomposition of theGreen function in the simple and compact forms compared to those obtained in [15]. These formsmake them suitable for variational calculations such that the currant density, the nonrelativistic limitand etc.
Acknowledgments
A.M and A.J acknowledge the financial support from King Faisal University. The present work wasdone under Project Number 140233, ‘Path Integral Techniques for Interacting Dirac Particles’. A.Mand A.J are very grateful to T. Sbeouelji for numerous helpful discussions and assistant.