Solution of the κ-deformed Dirac equation with vector and scalar interactions in the context of spin and pseudospin symmetries
aa r X i v : . [ h e p - t h ] A p r Solution of the κ -deformed Dirac equation with scalar, vector and tensor interactionsin the context of pseudospin and spin symmetries C. F. Farias ∗ and Edilberto O. Silva † Departamento de Física, Universidade Federal do Maranhão, 65085-580 São Luís, Maranhão, Brazil (Dated: September 18, 2018)The deformed Dirac equation invariant under the κ -Poincaré-Hopf quantum algebra in the contextof minimum, vector and scalar couplings under spin and pseudospin symmetric limits is considered.Expressions for the energy eigenvalues and wave functions are determined for both symmetry limits.We verify that the energies and wave functions of the particle are modified by the deformationparameter. PACS numbers: 03.65.Ge, 03.65.Pm, 11.30.Cp, 71.70.Di
I. INTRODUCTION
Quantum deformations based on the κ -Poincaré-Hopfalgebra constitute an important branch of research thatenables us to address problems in condensed matter andhigh energy physics through field equations. These fieldequations were first presented in Ref. [1], where a newreal quantum Poincaré algebra with standard real struc-ture, obtained by contraction of U q ( O (3 , . The re-sulting algebra of this contraction is a standard real Hopfalgebra and depends on a dimension-full parameter κ in-stead of q . Since then, the algebraic structure of the κ -deformed Poincaré algebra has been investigated inten-sively and have become a theoretical field of increasinginterest [2–21] . Through the field equations from the κ -Poincaré algebra ( κ -Dirac equation [22–24]), we canstudy the physical implications of the quantum deforma-tion parameter κ in relativistic and nonrelativistic quan-tum systems. In this context, we highlight the study ofrelativistic Landau levels [18], the Aharonov-Bohm effecttaking into account spin effects [17], the Dirac oscillator[25, 26] and the integer quantum Hall effect [27].When we want to study the relativistic quantum dy-namics of particles with spin, we must obviously considerthe presence of external fields, which include the vectorand scalar fields. The inclusion of vector and scalar po-tentials in the Dirac equation reveals interesting prop-erties of symmetries in nuclear theory. The first contri-butions in this subject revealed the existence of SU (2) symmetries, which are known in the literature as pseu-dospin and spin symmetries [28, 29]. Some investiga-tions have been made in this scenario in order to give ameaning to these symmetries. However, it was only ina work by Ginocchio, that pseudospin symmetry was re-vealed. He verified that pseudospin symmetry in nucleicould arise from nucleons moving in a relativistic meanfield, which has an attractive scalar and repulsive vectorpotential nearly equal in magnitude [30] (for a more de-tailed description see Ref. [31]). Spin and pseudo-spin ∗ cff[email protected] † [email protected] symmetries in the Dirac equation have been studied un-der different aspects in recent years (see Refs. [32, 33]).Some studies have been developed taking into accountthe spin and pseudospin symmetry limits to study rela-tivistic dynamics of physical systems interacting with aclass of potentials [34–41].The present work is proposed to investigate the κ -deformed Dirac equation derived in Refs. [23] in the con-text of minimum, vector and scalar couplings under spinand pseudospin symmetric limits.The structure of the paper is as follows. In Sec. II,we present the κ -deformed Dirac equation with couplingsfrom which we derive the κ -deformed Pauli-Dirac equa-tion, by using the usual procedure that consists of squar-ing the κ -deformed Dirac equation. In Sec. III, we con-sider the equation of Pauli and establish the spin andpseudospin symmetries limits. As an application, we con-sider the particle interacting with an uniform magneticfield in the z -direction in two different physical situations:(i) particle interacting with a harmonic oscillator and (ii)particle interacting with a linear potential. We obtain ex-pressions for the energy eigenvalues and wave functionsin both limits. In Sec. IV, we present our comments andconclusions. II. THE κ -DEFORMED DIRAC EQUATIONWITH COUPLINGS We begin with the deformed Dirac equation invariantunder the κ -deformed Poincaré quantum algebra [22, 23] (cid:26) ( γ p − γ i p i ) + 12 ε (cid:2) γ (cid:0) p − p i p i (cid:1) − M p (cid:3)(cid:27) ψ = M ψ. (1)The interactions can be performed through the followingprescriptions [42]: p i → p i − eA i , (2) E → E − ν ( r ) , (3) M → M + w ( r ) . (4)As we are interested in a planar dynamics, i.e., when thethird directions of the fields involved are zero, we choosethe following representation for the gamma matrices [43]: γ = σ , (5) α = γ γ = σ , (6) α = γ γ = sσ , (7)where the parameter s , which has a value of twice the spinvalue, can be introduced to characterizing the two spinstates, with s = +1 for spin "up" and s = − for spin"down". In the above representation, the κ -deformedDirac including the interactions can be written as [ α · ( p − e A ) + γ ( M + w ( r ))] ψ − [ E − ν ( r )] ψ + ε es ( σ · B ) ψ + γ (( α · p ) w ( r )) ψ + M γ ( α · p ) ψ ]+ ε γ w ( r ) ( α · p ) ψ − γ e ( α · A ) M ψ ] − ε γ e ( α · A ) w ( r ) ψ ] = 0 . (8)Let us now determine the Dirac equation in its quadraticform. This can be accomplished by applying the operator α · ( p − e A ) + γ ( M + w ( r )) + [ E − ν ( r )]+ ε es ( σ · B ) + γ (( α · p ) w ( r )) + M γ ( α · p )]+ ε γ w ( r ) ( α · p ) − γ e ( α · A ) M − γ e ( α · A ) w ( r )] (9)in Eq. (8). The result is the κ -deformed Dirac-Pauliequation ( p − e A ) ψ + α · [ p ν ( r )] ψ − γ α · [ p w ( r )] ψ + [ M + w ( r )] ψ − [ E − ν ( r )] ψ − esσ z Bψ − ε (cid:8) γ (cid:2) p w ( r ) (cid:3) + γ [( α · p ) w ( r )] [( α · p )] (cid:9) ψ + ε { isγ [ σ · [( p w ( r )) × p ] − eσ · ( p w ( r )) × A ] } ψ + ε { eγ [( α · p ) w ( r )] [( α · A )] − w ( r ) [( α · p ) w ( r )] } ψ + ε { M esB + 2 w ( r ) esB − M [( α · p ) w ( r )] } ψ + ε { M γ [( α · p ) ν ( r )] + γ w ( r ) [( α · p ) ν ( r )] } ψ = 0 . (10)In order to apply this equation to some physical system,we need to choose a representation for the vector poten-tial A and the scalar potentials w ( r ) and ν ( r ) . For cer-tain particular choices of these quantities, we can studythe physical implications of quantum deformation on theproperties of various physical systems of interest.For the field configuration, we consider a constant mag-netic field along the z -direction (in cylindrical coordi-nates), B = B ˆz , which is obtained from the vector po-tential (in the Landau gauge) [44], A = Br ˆ ϕ. (11) In this configuration, Eq. (10) reads X + ε Y = 0 , (12)with X = − ∂ ψ∂r − r ∂ψ∂r − r ∂ ψ∂ϕ + ieB ∂ψ∂ϕ + 14 e B r ψ + [ M + w ( r )] ψ − [ E − ν ( r )] ψ − esσ z Bψ + i (cid:20) ∂w ( r ) ∂r (cid:21) γ α r ψ − i (cid:20) ∂ν ( r ) ∂r (cid:21) α r ψ, (13)and Y = γ (cid:20) ∂ w ( r ) ∂r + 1 r ∂w ( r ) ∂r (cid:21) ψ − γ (cid:20) ∂w ( r ) ∂r (cid:21) ∂ψ∂r − is (cid:20) r ∂w ( r ) ∂r (cid:21) ∂ψ∂ϕ − is (cid:20) r ∂w ( r ) ∂r (cid:21) ∂ψ∂ϕ + es (cid:20) ∂w ( r ) ∂r (cid:21) Br ψ − es (cid:20) ∂w ( r ) ∂r (cid:21) Br ψ + iw ( r ) α r (cid:20) ∂w ( r ) ∂r (cid:21) ψ − iw ( r ) γ r (cid:20) ∂ν ( r ) ∂r (cid:21) ψ + iM α r (cid:20) ∂w ( r ) ∂r (cid:21) ψ − iM γ r (cid:20) ∂ν ( r ) ∂r (cid:21) ψ + 2 M esBψ + 2 w ( r ) esBψ (14)where the matrices (5)-(7) are now given in cylindricalcoordinates, γ r = iσ ϕ , γ ϕ = − isσ r , with [45] α r = γ γ r = (cid:18) e − isϕ e isϕ (cid:19) , (15) α ϕ = γ γ ϕ = (cid:18) − ie − isϕ ie isϕ (cid:19) , (16) γ = σ z = (cid:18) − (cid:19) . (17)For convenience, we will attribute expressions to func-tions ν ( r ) and w ( r ) in Eq. (12) only in the next sec-tion, when we treat analysis of spin and pseudo-spinsymmetries. We will argue after that only some partic-ular choices for these functions will lead to a differentialequation that admits an exact solution. III. SYMMETRIES LIMITS
To implement the spin and pseudospin symmetrieslimits, we make in Eq. (12) the requirement that w ( r ) = ± ν ( r ) , where the plus(minus) signal refers tospin(pseudo-spin) symmetry, respectively [30]. Next, byusing ψ = ( ψ + , ψ − ) T , the first and second lines in Eq.(12 ) can be written in a simple form, which allows us tosolve them separately. Furthermore, as mentioned above,we need to choose a representation for the radial function ν ( r ) . We give a representation in terms of cylindricallysymmetric scalar potentials which lead to results well-known in the literature. A. Particle interacting with a harmonic oscillator
Because of applications to various physical systems, weconsider the potential of a harmonic oscillator, ν ( r ) = ar , where a is a constant. By adopting solutions of theform ψ ± = (cid:18) P m f + ( r ) e imϕ i P m f − ( r ) e i ( m + s ) ϕ (cid:19) , (18)we arrive at radial equations d f ± ( r ) dr + (cid:18) r + εar (cid:19) df ± ( r ) dr − ( m ± ) r f ± ( r ) − (cid:0) ω ± (cid:1) r f + ( r ) + k ± f + ( r ) = 0 , (19)where k + = E − M + ( m + s ) eB − ε (2 a + 3 sma + M esB ) , k − = E − M + eB ( m + s ) + esB − ε [2 a − sa ( m + s ) + M esB ] , ( ω + ) = e B / M + E ) a − εesaB/ , ( ω − ) = e B / E − M ) a − εesaB/ , m + = m and m − = m + s .By using solutions of the form f ± ( ρ ) = e − ( κ ± +1 ) ρ ρ | m ± | F ± ( ρ ) , ρ = ω ± r (20)where κ ± = εa/ ω ± , Eq. (19) becomes ρ d F ± dρ + (cid:0) (cid:12)(cid:12) m ± (cid:12)(cid:12) − ρ (cid:1) dF ± dρ − (cid:20) (cid:0) (cid:12)(cid:12) m ± (cid:12)(cid:12) + κ ± (cid:1) − k ± ω + (cid:21) F ± = 0 . (21)Equation (21) is of the confluent hypergeometric equa-tion type and its solution is given in terms of the Kum-mer functions. In this manner, the general solution forEq. (19) is given by [46] f ± ( ρ ) = c e − ( κ ± ) ρ ρ | m ± |× M (cid:18) (cid:0) (cid:12)(cid:12) m ± (cid:12)(cid:12) + κ ± (cid:1) − k ± ω ± , (cid:12)(cid:12) m ± (cid:12)(cid:12) , ρ (cid:19) + c e − ( κ ± ) ρ ρ − | m ± |× M (cid:18) (cid:0) − (cid:12)(cid:12) m ± (cid:12)(cid:12) + κ ± (cid:1) − k ± ω ± , − (cid:12)(cid:12) m ± (cid:12)(cid:12) , ρ (cid:19) , (22)where M are the Kummer functions. In particular, when (1 + | m ± | + κ ± ) / − k ± / ω ± = − n , with n = 0 , , , ... ,the function M becomes a polynomial in ρ of degree notexceeding n . From this condition, we extract the ener-gies for the spin and pseudospin symmetries limits, given respectively by E − M = 2 r e B M + E ) a − εesaB × (2 n + | m | + 1) − eB ( m + s )+ ε (3 a + 3 sma + M esB ) , (23) E − M = 2 r e B E − M ) a + 12 εesaB × (2 n + 1 + | m + s | ) − eB ( m + s ) − esB − ε [ a − sa ( m + s ) + M esB ] . (24)These energies are a relativistic generalization of the Lan-dau levels in the context of quantum deformation. When a and ε are null, we obtain E − M = eB [2 n + 1 + | m | − m − s ] , (25) E − M = eB [2 n + 1 + | m + s | − ( m + s ) − s ] . (26)which are the usual relativistic Landau levels with theinclusion of the element of spin. B. Particle interacting with a linear potential
Let us consider the case where the particle interactswith a linear potential, ar . In this case, we make w ( r ) = ν ( r ) = ar (where a is a positive constant) inEq. (12) to the limits of spin and pseudo-spin symme-tries and proceed as before.In the case of the spin symmetry limit, the resultingequation is given by d f ( r ) dr + (cid:18) r + εa (cid:19) df ( r ) dr − ( m ± ) r f ( r ) − ω r f ( r ) − µ ± rf ( r ) − k ± r f ( r ) + l ± f ( r ) = 0 , (27)with m + = m , m − = m + s , ω = eB/ , µ + =2 ( E + M ) a + 3 εaesB/ , µ − = 2 ( E − M ) a − εaesB/ , k + = εa (1 + 3 sm ) / , k − = εa [1 − s ( m + s )] , l + = E − M + eB ( m + s ) − εM esB and l − = E − M + eB ( m + s ) + esB (1 − εM ) . In Eq. (27), the +( − ) singrefer to spin and pseudo-spin symmetries, respectively.By performing the variable change, x = √ ωr , Eq. (27)assumes the form d f ( x ) dx + (cid:18) x + 12 κ (cid:19) df ( x ) dx − m x f ( x ) − x f ( x ) − a ± L xf ( x ) − a ± C x f ( x ) + l ± ω f ( x ) = 0 , (28)where we have defined the parameters κ = εa/ √ ω , a ± L = µ ± /ω √ ω e a ± C = k ± / √ ω . Note that the choice w ( r ) = ν ( r ) = ar induces a Coulomb-like interactionin the resulting eigenvalue equation. The origin of thisCoulomb potential is due purely to the quantum defor-mation and boundary symmetries involved.Equation (28) is of the Heun equation type, which is ahomogeneous, linear, second-order, differential equationdefined in the complex plane. This equation can be putinto its canonical form using the solution f ( x ) = x | m ± | e − x e − ( a ± L + κ ) x y ± ( x ) , (29)where y + satisfies the biconfluent Heun differential equa-tion y ′′ ± + (cid:20) α ± + 1 x − x − β ± (cid:21) y ′± + n γ ± − α ± − − x (cid:2) β ± (cid:0) α ± + 1 (cid:1) + δ ± (cid:3) o y ± = 0 , (30)with α ± = 2 | m ± | , β ± = a ± L , γ ± = ( β ± ) / l ± /ω and δ ± = κ/ a ± C . Equation (30) has a regular singularityat x = 0 , and an irregular singularity at ∞ of rank .Usually, the solution of this equation is given in terms oftwo linearly independent solutions as y + ( x ) = N (cid:0) α ± , β ± , γ ± , δ ± ; x (cid:1) + x − α ± N (cid:0) − α ± , β ± , γ ± , δ ± ; x (cid:1) , (31)where (assuming that α ± is not a negative integer) N (cid:0) α ± , β ± , γ ± , δ ± ; x (cid:1) = ∞ X q =0 A + q ( α ± , β ± , γ ± , δ ± )(1 + α ± ) q x q q ! (32)are the Heun functions. After the insertion of this solu-tion into Eq. (30), we find ( q > ) A = 1 , (33) A ± = 12 (cid:2) δ ± + β ± (cid:0) α ± (cid:1)(cid:3) , (34) A ± q +2 = (cid:26) ( q + 1) β ± + 12 (cid:2) δ ± + β ± (cid:0) α ± (cid:1)(cid:3)(cid:27) A ± q +1 − ( q + 1) (cid:0) q + 1 + α ± (cid:1) (cid:2) γ ± − α ± − − q (cid:3) A ± q , (35) (cid:0) α ± (cid:1) q = Γ ( q + α ± + 1)Γ ( α ± + 1) , q = 0 , , , , . . . . (36)From the recursion relation (35), the function N ( α ± , β ± , γ ± , δ ± ; x ) becomes a polynomial of de-gree n if and only if the two following conditions areimposed: γ ± − α ± − n, n = 0 , , , . . . (37) A ± n +1 = 0 , (38)where n is a positive integer. In this case, the ( n + 1) thcoefficient in the series expansion is a polynomial of de-gree n in δ ± . When δ ± is a root of this polynomial, the n + 1 th and subsequent coefficients cancel and the seriestruncates, resulting in a polynomial form of degree n for N ( α ± , β ± , γ ± , δ ± ; x ) . From condition (37), we extractthe energies at the spin and pseudo symmetries limit,given respectively by: E nm − M = 2 ω ( n + | m | + 1) − a ω ( E nm + M ) − a ω ( E nm + M ) εs + 2 ω [ εM s − ( m + s )] , (39) E nm − M = 2 ω ( n + | m + s | + 1) − a ω ( E nm − M ) + 3 a ω ( E nm − M ) εs − ω [ s (1 − εM ) + ( m + s )] . (40)The equations (39) and (40) can not represent the spec-trum for the system in question. The energy of a physicalsystem must be a function involving all the parameterspresent in the equation of motion. As expected, from thecondition (37) alone one cannot a priori derive the energyof the system. In the case of the energies above, this isjustified by the absence of the parameters a ± C . Moreover,it also does not provide the energy spectrum for all val-ues of n . On the other hand, by analyzing more carefullythe condition (38), we see that it admits a natural ap-plication of (37), so that it is a necessary and sufficientcondition for the derivation of the energy of the particle.Let us consider the solution (32) up to second-order in x of the expansion, N (cid:0) α ± , β ± , γ ± , δ ± ; x (cid:1) = A (1 + α ± ) + A ± (1 + α ± ) x + A ± (1 + α ± ) x
2! + . . . . (41)By using the relation of recurrence (35) and Eqs. (33)-(34), the coefficient above A ± can be determined. If wewant to truncate solution (41) in x , we must impose that A ± = 0 through the condition (38); when we truncatein x , we make A ± = 0 , and so on. For each of thesecases, we have an associated energy. Thus, for A ± = 0 ,it means that we are investigating the particular solutionfor n = 0 . Then, from Eq. (34), we have (cid:0) δ ± + β ± ˜ m ± (cid:1) = 0 , (42)where ˜ m ± = 1 + α ± . Solving (42) for E , we find theenergies corresponding to the spin and pseudo-spin sym-metries. They can be written explicitly as E m + M = − εω | m | ) (cid:20) s ( | m | + m ) + 32 ( s + 1) (cid:21) , (43) E m − M = εω | m + s | ) × (cid:20) s ( | m + s | + m + s ) + 32 ( s − (cid:21) . (44)These energies, after imposing ε = 0 , lead to the groundstate, E m = − M and E m = M . Analogously, for A ± =0 , we get ( n = 1 ) β ± (cid:0) δ ± + β ± ˜ m ± (cid:1) + 14 (cid:0) δ ± + β ± ˜ m ± (cid:1) − m ± = 0 . (45)In this relation, we can observe that the only parame-ter that depends on the energy E is the parameter a ± L (through the parameter β ± ). So, we only need to solveit for a ± L . The result is given by E m + M = ± ωa s ω | m | + 3 − εω (cid:20) (1 + | m | ) [1 + 2 (1 + 3 sm )](1 + 2 | m | ) (3 + 2 | m | ) + 32 s (cid:21) , (46)for the spin symmetry limit, and E m − M = ± ωa s ω | m + s |− εω (cid:20) | m + s | ) (1 − s ( m + s ))(1 + 2 | m + s | ) (3 + 2 | m + s | ) − s (cid:21) , (47)for the pseudo-spin symmetry limit. If ε = 0 in Eqs.(46)-(47), we find E m + M = ± ωa s ω | m | , (48) E m − M = ± ωa s ω | m + s | . (49)The energies obtained from condition (37) (Eqs. (39)-(40)) together with those obtained from (38) (Eqs. (43)-(44) and (46)-(47)) specify the energy eigenvalues forthe system governed by equation (27). However, we canverify that these energies are connected to each otherthrough the parameters ω and a , so that we have a con-straint on the energies. In particular, if we solve Eqs.(46)-(47) for ω and replace them in (39)-(40), we willhave expressions for the energies involving the quantities α ± , β ± , γ ± , δ ± , which contains the Coulomb potentialcoupling constant a ± C , the mass of the particle M , theeffective angular momentum m ± and the frequency ω .For a specific physical system described by Eq. (27),its corresponding energy spectrum are the modified Lan-dau levels. In the absence of magnetic field, the energyeigenvalues are equivalent to those of a planar harmonicoscillator being corrected only by the parameter of quan-tum deformation. Because the spectrum of the systemhas the generalized form of the spectrum of a relativisticoscillator is more convenient to fix the frequency the fre-quency ω to give the energies corresponding to each valueof n . It is an immediate calculation to solve the equa-tions Eqs. (43)-(44) and (46)-(47) for ω . For each specific frequency, ω m , ω , we have the following energies: E m − M = 2 ω m ( | m | + 1) − a ω m ( E m + M ) − a ω m ( E m + M ) εs + 2 ω m [ εM s − ( m + s )] , (50) E m − M = 2 ω m ( | m + s | + 1) − a ω m ( E m − M ) + 3 a ω m ( E m − M ) εs − ω m [ s (1 − εM ) + ( m + s )] , (51)and E m − M = 2 ω m ( | m | + 2) − a ω m ( E m + M ) − a ω m ( E m + M ) εs + 2 ω m [ εM s − ( m + s )] , (52) E m − M = 2 ω m ( | m + s | + 2) − a ω m ( E m − M ) + 3 a ω m ( E m − M ) εs − ω m [ s (1 − εM ) + ( m + s )] , (53)To determine the energies corresponding to n =2 , , , . . . , we must make use of the above recipe. How-ever, the polynomials of degree n ≥ resulting fromcondition (38), in general, only some roots are physicallyacceptable. IV. CONCLUSIONS
We have studied the relativistic quantum dynamics of aspin- / charged particle with minimal, vector and scalarcouplings in the quantum deformed framework generatedby the κ -Poincaré-Hopf algebra. The problem have beenformulated using the κ -deformed Dirac equation in twodimensions. The κ -deformed Pauli equation was derivedto study the dynamics of the system taking into accountthe spin and pseudospin symmetries limits. For the κ -deformed Dirac-Pauli equation obtained (Eq. (12)), wehave argued that only particular choices of radial func-tion ν ( r ) lead to exactly solvable differential equations.We have considered the case where the particle inter-acts with an uniform magnetic field, a planar harmonicoscillator and a linear potential. We have verified thatthe linear potential leads to a Coulomb-type term in the κ -deformed sector of the radial equation. The resultingequation obtained is a Heun-type differential equation.Analytical solutions for both spin and pseudospin sym-metries limits enabled us to obtain expressions for theenergy eigenvalues (through the use of the Eqs. (37) and(38)) and wave functions. Because of the limitations im-posed by the condition (38), we have derived expressionsfor the energies corresponding only to n = 0 (Eqs. (50)-(51)) and n = 1 (Eqs. (52)-(53)). We have shown thatthe presence of the spin element in the equation of motionintroduces a correction in the expressions for the boundstate energy and wave functions. ACKNOWLEDGMENTS
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