The relativistic dynamics of oppositely charged two fermions interacting with external uniform magnetic field
aa r X i v : . [ phy s i c s . g e n - ph ] O c t Noname manuscript No. (will be inserted by the editor)
Relativistic dynamics of oppositely charged twofermions interacting with external uniform magneticfield
Abdullah Guvendi · Semra GurtasDogan
Received: date / Accepted: date
Abstract
We investigated the relativistic dynamics of oppositely charged twofermions interacting with an external uniform magnetic field. We chose the in-teraction of each fermion with the external magnetic field in the symmetricgauge, and obtained a precise solution of the corresponding fully-covarianttwo-body Dirac equation that derived from Quantum Electrodynamics viaAction principle. The dynamic symmetry of the system we deal with allowedus to determine the relativistic Landau levels of such a spinless compositesystem, without using any group theoretical method. As a result, we deter-mined the eigenfunctions and eigenvalues of the corresponding two-body DiracHamiltonian.
Keywords landau levels · two-body dirac equation · fermion-antifermionsystems · magnetic field Landau quantization is the quantization of cyclotron orbits of the motionof a charged particle moving in a magnetic field [1]. In the non-relativisticregime, Landau quantization was discussed for many different cases [2,3,4]. Inthe relativistic regime, Landau quantization firstly was discussed by Jackiw [5](see also [6]). Afterwards, many experimental and theoretical studies were con-ducted on this subject [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] (alsosee [23,24]). Today, it is though that the magnetic fields exist in all over thespacetime background and they magnetize the universe [25,26]. We do notknow yet exactly the origins of intra-cluster, galactic and cosmological mag-netic fields, but it is predicted that dynamo effects in turbulent fluids canexponentially amplify the seed fields [27]. Although it is not easy to fully
Medical Imaging Techniques, Simav Vocational School of Health Services, Kutahya HealthSciences University, TR- 43500 Kutahya, TurkeyE-mail: [email protected] · E-mail: [email protected] A Guvendi et al. explain the galactic and cluster magnetic fields with dynamo theory [28], itis suggested that these fields that permeate the universe may have arisen asa result of the compression of a primordial field [27,29]. Therefore, the de-termination of the magnetic field effects at all scales from planets to stars,from galaxies to galaxy clusters and even in the inner galactic environmentis still one of the important and active research fields [30,31]. Also, it hasbeen studied for many years that the structure of the vacuum is modified inthe presence of electromagnetic fields [32,33]. The modification of the vacuumcan cause some novel phenomena such as photon decays into electron-positiveelectron (positron) pair [34], the birefringence of a photon [33] and splittingof a photon [35]. It is natural to expect that these effects can occur, sincethe vacuum (in QED) is regarded as filled with electrons. Hence, they can re-act like an ordinary medium in the presence of external fields. We think thatmagnetic fields exist at almost every point in the universe [36] and they maybe responsible for many interesting physical effects. The effect illustrated inNambu-Jona-Lasinio model is that a constant magnetic field is a strong cata-lyst of a dynamical flavour symmetry breaking [37]. It is though that this pointmay be important in the phenomena like Hall effect in the condensed mattersystems [37]. The presence of magnetic fields can deeply affect the dynamicsof relativistic and semi-relativistic systems and hence the determination of thedynamics of systems in magnetic field is a subject of great interest in manyareas of physics [38]. The origin of the magnetic field differs from one systemto another, but in some cases we expose the systems to the effect of externalmagnetic fields in order to better understand the underlying physics of systemsor to determine the effect of the magnetic field on the systems. Of course, theproduction of magnetic fields can be a natural feature of the systems. In thispresent manuscript, we will investigate the relativistic dynamics of oppositelycharged two fermions interacting with external magnetic field, without detailydiscussing the origin of the external magnetic field.It is worth to mention that, in general, in the relativistic regime phe-nomenologically established one-time equations are used to describe the rel-ativistic dynamics of interacting particles. These phenomenologically estab-lished equations include free Hamiltonians for each particle plus interparticleinteraction potentials [39,40]. It is crucial to mention that there are main diffi-culties even in the determination of relativistic dynamics of two particles. Oneof them is that two-time problem appears in such problems in the relativisticframework (see [41]). Also, one of the other problems is total angular momen-tum of the interested composite structures [41]. In the literature, the acceptedfirst relativistic two-body equation was introduced by Breit [42]. The history ofrelativistic two-body equations goes long way back. More details about themcan be found in [41,43]. It is important to note that the equation introducedby Breit does not hold in every situation, due to the retardation effects. Betheand Salpeter introduced another formalism [44] to overcome this problem.The Bethe-Salpeter formalism provided a different approach to one-electronatom problems, but this formalism was not fully-acceptable for bound-states(see [41]). Barut and his collaborators have discussed whether it is possible wo fermions interacting with external uniform magnetic field 3 to obtain Poincar´e invariant many-body one-time equation [45,46] (also see[47]). They have shown us how is possible to derive a complete fully-covarianttwo-body Dirac equation [45]. This equation has been derived from QuantumElectrodynamics with the help of Action principle [45]. This equation is verysimilar to the former two-body equation introduced by Kemmer, Fermi andYang [39,40] and moreover it is in fully-covariant form (more details can befound in Refs. [45,46,47,48,49,50,51]). In 3 + 1 dimensions, the fully-covarianttwo-body equation gives a 16 ×
16 dimensional matrix equation including themost general electric and magnetic potentials. But, in 4-dimensions, the so-lution of this equation requires to be applied group theoretical methods [48]and the well-known energy spectrum for Hydrogen-like atoms can be obtainedonly via a perturbative solution of this equation [48,49,50]. Nevertheless, it isshowed that the fully-covariant two-body Dirac equation can be exactly solv-able for composite systems that have dynamical symmetries [51]. On the otherhand, one can see that it may not be possible to obtain a complete analyticalsolution for one-electron atom systems under the influence of external mag-netic field with the corresponding one-body Dirac equation [52] even in theabsence of the third spatial coordinate [53,54]. In the literature, one can seethat there are a few study based on relativistic dynamics of non-interacting(mutually) two particles in the presence of external magnetic field. Previously,the separation of center of mass motion coordinates of a two-body system(neutral) in homogeneous magnetic fields was introduced [55] and then thetheoretical foundations of two-body systems were studied in the presence ofboth homogeneous [56] and inhomogeneous [57] magnetic fields (also see [58,59]). In this present paper we hoped to fill this gap in the literature and weinvestigated the relativistic dynamics of oppositely charged two fermions inter-acting with an external uniform magnetic field, without considering any mutu-ally charge-charge interaction. At the beginning, we wrote the fully-covarianttwo body Dirac equation in 3-dimensional Minkowski spacetime [51] and wechose the interaction of each fermion with the external magnetic field in thesymmetric gauge, since this problem has 2 + 1-dimensional dynamical sym-metry [60]. The dynamic symmetry of the system allowed us to determinethe eigenfunctions and eigenvalues (in closed-form) of the corresponding two-body Dirac Hamiltonian for such a spinless composite system, without usingany group theoretical method.
In a general three dimensional spacetime background, the relativistic dy-namics of charged two fermions interacting with external uniform magneticfield can be investigated via the following fully-covariant two-body Dirac equa-tion, nh γ η (1) π (1) η + ib I i ⊗ γ (2) + γ (1) ⊗ h γ η (2) π (2) η + ib I io Ψ ( x , x ) = 0 , A Guvendi et al. π (1) η = (cid:18) ∂ (1) η + i e A (1) η ¯ hc − Γ (1) η (cid:19) , π (2) η = (cid:18) ∂ (2) η + i e A (2) η ¯ hc − Γ (2) η (cid:19) ,b = m c ¯ h , b = m c ¯ h , ( η = 0 , , . ) , (1)in which the superscripts (1) and (2) refer to the first fermion with the mass m and second fermion with mass m , respectively. In Eq. (1), I is 2 × Ψ ( x , x ) is the composite field that is constructedby a direct production ( ⊗ ) of arbitrary massive two Dirac fields as follows, Ψ ( x , x ) = Υ ( x ) ⊗ χ ( x ) , (2) e and e are charges of these particles, ¯ h is usual Planck constant, the letter c represents to the light speed, A η and Γ η correspond to the vector potentials andspinor connections, respectively. It has been showed that this two-body Diracequation is Poincar´e invariant [45,46,47] and one-time equation including spinalgebra spanned by direct (Kronocker) productions of the Dirac matrices. Eventhough Eq. (1) does not seem to be manifestly covariant at first look, the γ means γ η λ η in everywhere in this equation (note that λ η is a timelike vector λ η = (100) and more details about its structure can be found in Refs. [45,46,47,48,49,50,51]). As we mentioned in above, the problem we deal with can bestudied in 2 + 1 dimensional flat Minkowski spacetime background that canbe represented via the following line element, ds = c dt − dx − dy . (3)It is clear that the relativistic dynamics of the system we deal with doesnot changed by spinor connections in Eq. (1), since they vanish [61]. It isthought that there exist magnetic field at almost in every point in the uni-verse and the vacuum (in Quantum Electrodynamics) is filled by fermions(electron/positron). Hence, without considering any charge-charge interaction,we can think that the relativistic dynamics of a fermion-antifermion systemin the presence of external magnetic field thought to be important to betterunderstand the universe. Therefore, in the presence of the external uniformmagnetic field, the vector potentials in Eq. (1) can be chosen in the symmetricgauge as follows (without any charge-charge interaction term, see also [51]), A (1)0 = 0 , A (1)1 = − B y , A (1)2 = B x ,A (2)0 = 0 , A (2)1 = − B y , A (2)2 = B x , (4)in which B relates with the strength of external uniform magnetic field and x p , y p ( p = 1 , . ) pairs correspond to the spatial coordinates of the particles inthe spacetime background represented by Eq. (3). For the line element givenin Eq. (3), the Dirac matrices can be chosen as in the following [51], γ (1 , = σ z , γ (1 , = iσ x , γ (1 , = iσ y ,σ x = (cid:18) (cid:19) , σ y = (cid:18) − ii (cid:19) ,σ z = (cid:18) − (cid:19) , (5) wo fermions interacting with external uniform magnetic field 5 where σ x , σ y and σ z are the Pauli spin matrices. As is usual with two-body problems, the center of mass motion coordinatesand relative motion coordinates can be separated (covariantly) via help of thefollowing expressions [51], R η = M (cid:16) m x (1) η + m x (2) η (cid:17) , r η = x (1) η − x (2) η ,x (1) η = m M r η + R η , x (2) η = − m M r η + R η ,∂ (1) x η = ∂ r η + m M ∂ R η , ∂ (2) x η = − ∂ r η + m M ∂ R η ,∂ (1) x η + ∂ (2) x η = ∂ R η . (6)It is important to underline that the total energy of the system is determinedaccording to the proper time ( R ) of the system, since there is no relativetime difference between the first and second fermions. Now, we can assumethat the center of mass is located at the origin of the spacetime background( R = R = 0). At that rate, by substituting Eq. (4), Eq. (5) and Eq. (6) intoEq. (1) one can acquire the following matrix equation, (cid:0) γ ⊗ γ (cid:1) ∂ R Ψ + i m c ¯ h (cid:0) I ⊗ γ (cid:1) Ψ + i m c ¯ h (cid:0) γ ⊗ I (cid:1) Ψ + (cid:0) γ ⊗ γ (cid:1) (cid:0) ∂ x + m M ∂ X − iB m M y (cid:1) Ψ + (cid:0) γ ⊗ γ (cid:1) (cid:0) ∂ y + m M ∂ Y + iB m M x (cid:1) Ψ + (cid:0) γ ⊗ γ (cid:1) (cid:0) − ∂ x + m M ∂ X + iB m M y (cid:1) Ψ + (cid:0) γ ⊗ γ (cid:1) (cid:0) − ∂ y + m M ∂ Y − iB m M x (cid:1) Ψ = 0 ,B = e B hc , B = e B hc , (7)in which x, y and X, Y pairs are the spatial coordinates of the relative motionand center of mass motion, respectively. Provided that center of mass momen-tum is a constant of motion and the interaction is time-independent, we candefine the composite field Ψ as in the following, Ψ ( t, r , R ) = e − iωt e i K . R Ω ( r ) , Ω ( r ) = ψ ( r ) ψ ( r ) ψ ( r ) ψ ( r ) , (8)in which the ω is total frequency determined according to the proper time ofthe system and the K relates with the spatial momentum of the center of massmotion (¯ h K ). For the ansatz in Eq. (8) defined for a moving ( K = 0) systemformed by oppositely charged ( e = e, e = − e ) and arbitrary massive twofermions, by multiplying the Eq. (7) with γ ⊗ γ from left , one can obtain Here, (cid:0) γ ⊗ γ (cid:1) gives 4 × the following matrix equation, ε − M ∂ − − ∂ − − ∂ + ε − ∆m − ∂ − ∂ + ε + ∆m ∂ − ∂ + − ∂ + ε + M Ω ( r ) − µ Br − µ iK − µ Br + µ iK − − µ iK + µ Br − − µ iK + µ Br + Ω ( r ) − µ iK − µ Br − − µ iK + µ Br − µ Br + µ iK − µ Br + µ − iK + Ω ( r ) = 0 ,M = ( m + m ) c ¯ h , ∆m = ( m − m ) c ¯ h , B = eB hc ε = ωc , ∂ ± = ∂ x ± i∂ y , r ± = x ± iy,K ± = K x ± iK y , µ q = m q / ( m + m ) , ( q = 1 , . ) . (9)We can transform the spacetime background into the polar spacetimebackground in order to exploit the angular symmetry. We can do this by usingthe following expressions, ∂ ± = e ∓ iφ (cid:18) ∓ ir ∂ φ + ∂ r (cid:19) , x ± iy = re ± iφ , K ± = Ke ± iφ , in which ∂ − and ∂ + represent to the well-known spin lowering and spin raisingoperators, respectively. Now, we can write the matrix equation in Eq. (9) interms of polar coordinates ( r, φ ) only for the transformed spinor componentsthat are defined as follows, ψ ( r ) ψ ( r ) ψ ( r ) ψ ( r ) = ⇒ ψ ( r ) e i ( s − φ ψ ( r ) e isφ ψ ( r ) e isφ ψ ( r ) e i ( s +1) φ , as, εϕ ( r ) − M ϕ ( r ) + (cid:0) ∂ r + i ∆mM K (cid:1) ϕ ( r ) − Brϕ ( r ) = 0 ,εϕ ( r ) − M ϕ ( r ) + (cid:0) sr − B ∆mM r (cid:1) ϕ ( r ) − iKϕ ( r ) = 0 ,εϕ ( r ) − ∆mϕ ( r ) + (cid:0) sr − B ∆mM r (cid:1) ϕ ( r ) − (cid:0) r + 2 ∂ r + i ∆mM K (cid:1) ϕ ( r ) = 0 ,εϕ ( r ) − ∆mϕ ( r ) − Brϕ ( r ) + iKϕ ( r ) = 0 , (10) wo fermions interacting with external uniform magnetic field 7 in which, ϕ ( r ) = ψ ( r ) + ψ ( r ) , ϕ ( r ) = ψ ( r ) − ψ ( r ) ,ϕ ( r ) = ψ ( r ) − ψ ( r ) , ϕ ( r ) = ψ ( r ) + ψ ( r ) , and the letter s stands for the total spin of system formed by two fermions.There are two main difficulties in a complete analytical solution of this systemof coupled equations. The first difficulty is the mass difference between theparticles even for the static case ( K = 0). The second difficulty is the totalangular momentum of such a system as we mentioned before [41]. For such aspinless ( s = 0) system formed by equal massive ( m = m = m ) oppositelycharged two fermions, one can obtain the following system of coupled equationsby defining a dimensionless independent variable that reads z = B r , εϕ ( z ) − M ϕ ( z ) + 2 q zB B∂ z ϕ ( z ) − q zB Bϕ ( z ) = 0 ,εϕ ( z ) − M ϕ ( z ) = 0 ,εϕ ( z ) − q Bz ϕ ( z ) − q zB B∂ z ϕ ( z ) = 0 ,εφ ( z ) − q zB Bϕ ( z ) = 0 . (11)in the rest frame ( K = 0). Of course, the K = 0 case is relatively simple case,but any pairing effect becomes important in this case. One can solve the system of coupled equations in Eq. (11) in favour of ϕ ( z ) and arrive at the following 2 nd order wave equation, ∂ z ϕ ( z ) + 1 z ∂ z ϕ ( z ) − (cid:18) z + 1 z − ε − M Bz (cid:19) ϕ ( z ) = 0 , This equation can be reduced into the well-known shape of the Whittakerdifferential equation via defining the ansatz that reads ϕ ( z ) = √ z χ ( z ), ∂ z χ ( z ) + (cid:16) µz + − ν z − (cid:17) χ ( z ) = 0 , µ = ε − M B , ν = , (12)and the solution function of this wave equation is given in follows [62,63], χ ( z ) = QW µ,ν ( z )in which the Q is the normalization constant. The condition of the solutionfunction to be polynomial is given as follows [62,63],12 + ν − µ = − n, (13) A Guvendi et al. B - - E.. n = = = = = = = = Fig. 1
The dependence of total energy on the strength of external uniform magnetic field.Here, | e | = m = c = ¯ h = 1. in which the n is principal quantum number (non-negative integer). The ex-pression in Eq. (13) leads to the quantization condition for the formation ofsuch a spinless system. With the help of Eq. (12) and Eq. (13) one can acquirethe following non-perturbative spectrum in energy domain by assuming e < E = ± mc r − ω c ¯ hmc ( n + 1) , ω c = | e | B mc , (14)in which ω c is the well-known cyclotron frequency. Eq. (14) clearly gives therelativistic Landau levels of such a system (spinless) formed by oppositelycharged two fermions (non-interacting). Dependence of total energy ( E ) onthe strength of the external homogeneous magnetic field can be seen in Fig.1. Also, one can obtain all of the spinor components as follows, ϕ ( z ) ϕ ( z ) ϕ ( z ) ϕ ( z ) = Q W µ,ν ( z ) √ zMε √ z W µ,ν ( z ) ( z − µ ) √ Bεz W µ,ν ( z ) − √ Bεz W µ +1 ,ν ( z ) √ Bε W µ,ν ( z ) , (15)which, of course, satisfy the system of coupled equations given in Eq. (11). In this study, we investigated the relativistic dynamics of oppositely chargedtwo fermions interacting with an external homogeneous magnetic field. To ob-tain a non perturbative energy spectrum of such a system, we solved the cor-responding fully-covariant two-body Dirac equation in 2 + 1 dimensional flatMinkowski spacetime background, since this problem has 2 + 1-dimensional wo fermions interacting with external uniform magnetic field 9 dynamical symmetry. For a spinless and static system formed by oppositelycharged two fermions (non-interacting), the dynamic symmetry of the prob-lem we deal with allowed us to obtain the eigenfunctions and eigenvalues (inclosed-form) of the corresponding fully-covariant two-body Dirac Hamiltonian,without using any group theoretical method. As it expected, the obtained en-ergy spectrum (Eq. (14)) shows that the total energy ( E ) of the system closesto total rest mass energy (2 mc ) of the system when the external magneticfield is very weak ( ω c ¯ h ≪ mc ). It is clear that in Eq. (14), the term associ-ated with the external magnetic field does not vanish even for the ground state( n = 0) of such a system. Also, the total energy value of the system closes tothe zero when ω c ¯ h ≈ mc and n = 0. The non-perturbative energy spectrumin Eq. (14) also indicates that this system can decay when ω c ¯ h > mc in anyphysically possible quantum state. Acknowledgements
The authors thank Dr. Yusuf SUCU for suggestions and useful dis-cussions and anonymous referee for valuable comments and style suggestions.