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Proceedings of Institute of Mathematics of NAS of Ukraine 2002, Vol. 43, Part 2, 616–622

3 Equation for Particles of 2 with Anomalous Interaction

Alexander GALKIN

Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., 01601 Kyiv-4, Ukraine E-mail: [email protected]

3 We consider tensor-bispinor equation, which describes doublets of particles with spin 2 , nonzero and anomalous interaction with electromagnetic field. Using this equation 3 we find the energy levels of the particle with spin 2 in constant magnetic field, crossed electric and magnetic fields. We show that it is possible to introduce anomalous interaction in such way that energy levels are real for all values of magnetic field and arbitrary gyromagnetic ratio g.

1 Introduction

After the discovery of the acausality of the Rarita–Schwinger equation [1] various equations have 3 been adopted for the description of spin 2 particles. But these equations also have different defects. Some of these defects are acausal propagation of solutions, complex energies, incorrect value of the gyromagnetic ratio etc [2, 3, 4]. 3 In the present paper we propose the equation for doublets of massive particles with spin 2 interacting with external electromagnetic field, which do not have many of defects enumerated 3 above. In our case the of particle with spin 2 is described by irreducible anti- symmetric tensor-bispinor of rank 2. We generalize results [5, 6, 7] for the case of linear and quadratic anomalous interactions with electromagnetic field. Finally we consider the motion of 3 particle with spin 2 in crossed magnetic and electric fields.

2 Tensor-bispinor equation with minimal interaction in electromagnetic field

3 µν Here we describe particles with spin 2 in terms of irreducible -bispinor Ψ µν νµ µν µν µν (Ψ = −Ψ ). In the case of free particle, Ψ is a 24-component tensor-bispinor, Ψα = −Ψα , which has two tensorial indexes µ, ν =0, 1, 2, 3 and a spinorial index α =0, 1, 2, 3. We suppose µν Ψα satisfies the following covariant condition

µν γµγνΨ =0, (1) where γµ are Dirac matrices acting on index α which is omitted. Ψµν must also satisfy the

λ µν (γλp − m)Ψ =0, (2) ∂ λ − µν where pµ = i ∂xµ . Commuting γµγν and (γλp m) we come to the secondary constraint for Ψ µν pµγνΨ =0. (3)

It is possible to show that conditions (1), (3) reduce the number of independent components µν 3 of Ψ to 16. Equations (1)–(3) describe a doublet of particles with spin 2 and m [8, 9]. 3 Equation for Particles of Spin 2 with Anomalous Interaction 617

In order to introduce interaction with electromagnetic field, we rewrite system (1)–(3) as a single equation 1 (γ pλ − m)Ψµν + (pµγν − pνγµ)[γ ,γ ]Ψρσ λ 12 ρ σ 1 1 − [γµ,γν](p γ − p γ )Ψρσ + [γµ,γν]γ pλ[γ ,γ ]Ψρσ =0. (4) 12 ρ σ σ ρ 24 λ ρ σ The minimal interaction can be introduced in equation (4) in standard way

pµ −→ πµ = pµ − eAµ, (5) where Aµ is the vector-potential of electromagnetic field. As a result we obtain 1 (γ πλ − m)Ψµν + (πµγν − πνγµ)[γ ,γ ]Ψσρ λ 12 ρ σ 1 1 − [γµ,γν](π γ − π γ )Ψρσ + [γµ,γν]γ πλ[γ ,γ ]Ψρσ =0. (6) 12 ρ σ σ ρ 24 λ ρ σ

Contracting (6) with γµγν and with πµγν − πνγµ we come to the constraints ie γ γ Ψµν =0,πγ Ψµν = (F − γλγ F )Ψµν, (7) µ ν µ ν m µν ν µλ where Fµν is the tensor of electromagnetic field Fµν = i(pµAν − pνAµ). Substituting (7) in (6) we obtain an equation for Ψµν ie (γ πλ − m)Ψµν − (γµγν − γνγµ)(F − γσγ F )Ψρλ =0. (8) λ 6m ρλ λ ρσ Equation (8) is equivalent to introduced in [7]. It can be shown by using the substitution 1 i Ψab =  Φ(1) +Φ(2) , Ψ0c = Φ(2) − Φ(1) , (9) 2 abc c c 2 c c (1) (2) (a, b, c =1, 2, 3), Φc and Φc are bispinors.

3 Anomalous interaction

µν ρσ µν ρσ δε In this section we generalize equation (6) by adding the terms Tρσ Ψ and Tρσ Tδε Ψ [8, 9], which are linear and quadratic in F µν correspondingly, i.e. consider both minimal and anomalous interactions [2] 1 1 (γ πλ − m)Ψµν + (πµγν − πνγµ)[γ ,γ ]Ψρσ − [γ ,γ ](π γ − π γ )Ψρσ λ 12 ρ σ 12 µ ν ρ σ σ ρ 1 + [γ ,γ ]γ πλ[γ ,γ ]Ψρσ + T µνΨρσ + T˜µνT˜ρσΨδ =0. (10) 24 µ ν λ ρ σ ρσ ρσ δ We suppose the following relations are satisfied µν ˜µν γµTρσ =0,γµTρσ =0. (11) It is possible to show [8, 9] that (11) are the necessary and sufficient conditions to obtain consistent equation (3) whose solutions propagate with the velocity less then velocity of light. µν µνρσ Using F ,  , gµν and γµ one can construct the basis antisymmetric tensor-bispinors linear in F µν: µν ν µ − µ ν − ν µ µ ν T1 ρσ = Fρ γ γσ Fρ γ γσ Fσ γ γρ + Fσ γ γρ, 618 A. Galkin

µν µ ν − ν µ − µ ν ν µ T2 ρσ = Fρ δσ Fρ δσ Fσ δρ + Fσ δρ , µν ν λ µ − µ λ ν − ν λ µ µ λ ν T3 ρσ = γ γ Fρλδσ γ γ Fρλδσ γ γ Fσλδρ + γ γ Fσλδρ λ ν µ − λ µ ν λ µ ν − λ ν µ + γργ Fλ δσ γργ Fλ δσ + γσγ Fλ δρ γσγ Fλ δρ , µν µ ν − ν µ αβ T4 ρσ =(δσ δρ δσδρ )γαγβF , µν ˜µ ν − ˜ν µ − ˜µ ν ˜ν µ T5 ρσ = γ4(Fρ δσ Fρ δσ Fσ δρ + Fσ δρ ), µν µ αν − ν αµ αµν − αµν T6 ρσ = γ4(Fα ρσ Fα ρσ + Fαρ σ Fασ ρ), µν µ ν − ν µ µν − T7 ρσ =(γ γ γ γ )Fρσ + F (γργσ γσγρ), µν ν λ µ − µ λ ν − ν λ µ µ λ ν T8 ρσ = γ γ Fρλδσ γ γ Fρλδσ γ γ Fσλδρ + γ γ Fσλδρ − λ ν µ λ µ ν − λ µ ν λ ν µ γργ Fλ δσ + γργ Fλ δσ γσγ Fλ δρ + γσγ Fλ δρ , µν µ αν − ν αµ − αµν αµν T9 ρσ = γ4(Fα  ρσ Fα  ρσ Fαρ σ + Fασ ρ), µν µ ν − ν µ − µν − T10 ρσ =(γ γ γ γ )Fρσ F (γργσ γσγρ), ˜µν 1 µν ρσ where F = 2 ρσ F . µν µν Then the general form of Tρσ and T˜ρσ is the following 10 10 µν µν ˜µν ˜µν Tρσ = αiTiρσ, Tρσ = α˜iTiρσ, i=1 i=1 where αi,˜αi are arbitrary constants. Using (11) and asking for existence of real Lagrangian correspondingly to (3) 1 1  = Ψ¯ (γ πλ − m)Ψµν + Ψ¯ (πµγν − πνγµ)[γ ,γ ]Ψρσ − Ψ¯ [γ ,γ ] µν λ 12 µν ρ σ 12 µν µ ν 1 × (π γ − π γ )Ψρσ + Ψ¯ [γ ,γ ]γ πλ[γ ,γ ]Ψρσ + Ψ¯ T µνΨρσ + Ψ¯ T˜µνT˜ρσΨδ ρ σ σ ρ 24 µν µ ν λ ρ σ µν ρσ µν ρσ δ we come to the conditions λ λ α1 = α6 = −α9 = ,α2 =2λ, α3 = α8 = ,α4 = α5 = α7 = α10 =0, 2 4 where λ is an arbitrary constant. The analogous relations are valid forα ˜i. Substituting (9) into (3) we can express (3) in the Dirac-like form [7] µ e i µν µν 2 (1) Γµπ − m + (1 − iΓ4) (g − 2)[Γµ, Γν]+gτµν F + g1(SµνF ) Ψ =0, (12) 4m 4 µ e i µν µν 2 (2) Γ π − m + (1 + iΓ4) (g − 2)[Γ , Γ ]+gτ F + g1(S F ) Ψ =0, (13) µ 4m 4 µ ν µν µν T T 2 λ λ˜2 (1) (1) (1) (1) (2) (2) (2) (2) where g = 3 1 − 2 , g1 = 9 ,Ψ = Φ1 , Φ2 , Φ3 ,Ψ = Φ1 , Φ2 , Φ3 , Sµν = i 4 [Γµ, Γν]+τµν and τµν satisfy the relations τab = abcτc; τ0a = iτa, τaτa = τ(τ +1);[τa,τb]= iabcτc, a, b, c =1, 2, 3. Matrices Γµ and τa can be represented in the following forms; Γµ = γµ⊗I3, τa = I4⊗τˆa, symbol ⊗ denotes the direct product of matrices,τ ˆa are 3×3 matrices, realizing the representation D(3) of the algebra AO(3), I3 and I4 are the unit 3 × 3and4× 4 matrices correspondingly. In the representation (9) constraints (7) are reduced to the forms

µ µν (1) (1) [(Γµπ + m)(1 + iΓ4)(SµνS − 3)]Ψ =24mΨ , (14) µ µν (2) (2) [(Γµπ + m)(1 − iΓ4)(SµνS − 3)]Ψ =24mΨ . (15) 3 Equation for Particles of Spin 2 with Anomalous Interaction 619

We see that the value g = 2 corresponds to the most simple form of equations (12)–(13). Moreover, using Foldy–Wouthysen transformation [7] it can be shown that the Hamiltonian of the equation (12) or (13) in quasiclassical approximation is Hermitian, when g =2.

3 4 Particle with spin 2 in homogeneous magnetic field

3 Now let us use proposed equations to solve the problem of motion of charged particle with spin 2 in constant magnetic field. We start with equation (12) which can be written in the following equivalent form µ 2 eg µν eg1 µν 2 (1) π π − m + S F + (S F ) Ψ+ =0, (16) µ 2 µν 2 µν µν (1) (SµνS − 15)Ψ+ =0, (17)

(1) 1 µ (1) (1) (1) (1) Ψ− = Γ π Ψ+ , Ψ =Ψ+ +Ψ− (18) m µ (the similar equation can be obtained for (13)). The tensor F µν corresponding to the constant and homogeneous magnetic field can be chosen in the form

F0a = F23 = −F32 = F31 = −F13 =0,a=1, 2, 3,F12 = −F21 = H3 = H, H ≥ 0, where H is the strength of the magnetic field. The solution of equation (17), (18) is of the form   (1)  Φ 3   2   ˆ   0  (1)   Ψ+ =  1 2 (1)  , (19) ε + 3 Saπa Φ 3  m 2   3  2 2 (1) − 3 Ka πaΦ 3 m 2 where 3 3 3 2 9 2 2 2 3 K3 = δmm − m ; K1 ± i K2 = ±δm±m ∓ m(m ∓ 1) ± 3m, mm 4 mm mm 2

3 1 1 3 T (1) m, m = − 2 , − 2 , 2 , 2 ; 0=(0ˆ , 0) ,Φ3 is a 4 component spinor which satisfies the equation 2 2 2 2 2 2 (1) 2 2 (1) p + e H x2 − eH gS3 +2g1S3 H +2x2p1 Φ 3 = ε − m Φ 3 . (20) 2 2

3 So the problem of describing the motion of particle with spin 2 reduces to the solution of equation (20). 3 2 3 1 1 3 Using the eigenvectors Ων of S3 (ν = − 2 , − 2 , 2 , 2 are eigenvalues of S3)wecan represent Φ 3 in the form 2

3 2 3 3 (1) 2 2 Φ 3 = exp(ip1x1 + ip3x3) fν (x2)Ων , (21) 2 =− 3 ν 2 620 A. Galkin

3 3 3 2 2 2 here fν (x2) are unknown functions, Ων are 4 components spinor eigenvectors of S3 and Ων are 4 components         1 0 0 0 3   3   3   3   2  0  2  1  2  0  2  0  Ω− 3 =   , Ω− 1 =   , Ω 1 =   , Ω 3 =   . 2 0 2 0 2 1 2 0 0 0 0 1

3 2 Substituting (21) into (20) we come to the equation for fν d2 3 3 − + y2 f 2 (y)=ηf 2 (y), dy2 ν ν

√1 1 2 2 2 where y = (eHx2 − p1)andη = (ε − m − p3 + eH(νg +2g1νH)). It is the equation eH eH 3 2 for the harmonics oscillator. So, requiring that function fν → 0 when x2 →±∞, we obtain the condition for η

η =2n +1,n=0, 1, 2, 3,...,

3 then the energy levels for the particle with spin 2 in constant magnetic field can be written in the following form

2 2 2 ε = m + p3 + eH(2n +1− ν(g +2g1νH)),n=0, 1, 2, 3,.... (22)

3 2 Function fν has the form 3 − − 2 eHx2 p1) eHx2 p1 fν (x2) = exp − hn √ , 2eH eH

Hn(y) where hn(y)= ||Hn(y)|| , Hn(y) are Hermitian polynomials. 2 We note that for the case of anomalous interaction linear in F µν (i.e. for g1 =0)ε can be 2 negative, provided n = p3 =0,(νg − 1)eH < m . Thus the difficulty with complex energies 3 indicated earlier for spin-1 equation [7] appears also for spin 2 equation [9]. However, this difficulty is overcome introducing anomalous interaction quadratic in F µν (i.e. choosing g1 =0 in (16)), namely

(3g − 2)2e g1 ≤− . (23) 72m2

3 5 The charged particle with spin 2 in electric and magnetic fields

In this case, as it was shown in [10, 11], we can confine our attention to the parallel and orthogonal configurations of electric and magnetic fields (all others configurations can be obtained ones mentioned in the above using ) a) E ||H . For constant, uniform E and H directed along z axis we may choose E =(0, 0,E), H =(0, 0,H), Aµ =(x3E,x2H, 0, 0) and E = H. After substituting (19) into (16), equation (16) takes the form 2 2 2 2 2 (p0 − ex3E) − (p1 − ex2H) − p2 − p3 − m 2 2 (1) +egS3(H − iE)+2eg1S3 (H − iE) Φ 3 =0. (24) 2 3 Equation for Particles of Spin 2 with Anomalous Interaction 621

(1) Choosing Φ 3 in the form [10, 11] 2

3 2 3 (1) 2 Φ 3 = exp(ip1x1 − εx0)f(x2) gν(x3)Ων , (25) 2 =− 3 ν 2 where f(x2)andgν(x3) are unknown functions, we can decompose equation (24) into two sepa- (1) rate equations for f(x2)andgν(x3). After solving these equations, Φ 3 takes the form [10, 11] 2 2 (1) (p1 + ex2H) Φ 3 = exp(ip1x2 − εx0) exp − hn(x2) 2 2eH 3 iz2 2 3 × exp Gj (−iδ , −iz2)Ω 2 ,j=1, 2, (26) 2 ν ν ν =− 3 ν 2

Hn(x2) √ 1 where p1,ε = const, hn(x2)= || ( )|| , Hn(x2) are Hermitian polynomials, z = (ε − Hn x2 |eH| 2 2 2 m −eνg(H−iE)−2eν g1(H−iE) 1 2 ex2E), δ = − (2n +1), n =0, 1, 2, 3,..., G −iδ , −iz = ν |eH| √ ν ν 1 − 1 − 2 2 − − 2 1 − 1 3 2 2 F 4 (1 iδν), 2 , iz and Gν iδν, iz = F 4 (1 iδν)+ 2 , 2 ,iz iz . F is the conf- luent hypergeometric function. So, in this case, the energy levels are not quantized. b) E ⊥H . Setting E =(0,E,0), H =(0, 0,H), Aµ =(x2E,x2H, 0, 0) we obtain following (1) equation for Φ 3 instead of (24) 2 2 2 2 2 2 (p0 − ex2E) − (p1 − ex2H) − p2 − p3 − m 2 (1) + eg(S3H − iS2E)+2eg1(S3H − iS2E) Φ 3 =0. (27) 2 (1) Representing Φ 3 in the form 3

3 2 3 (1) 2 Φ 3 = exp(ip1x1 + ip3x3 − iεx0) Pν(x2)Ων , (28) 2 =− 3 ν 2 substituting (28) into (5) and using transformation Pˆν = Uνν Pν (x2) we come to the equation 2 2 2 d 2 2 −1 − − − − − ˆ    ˆ  (ε ex2E) (p1 ex2H) + 2 p3 m Pν(x2)=eUνν Λν ν Uνν Pν (x2). (29) dx2 2 −1  −  −     where Λνν = eg(S3H iS2E)νν +2eg1(S3H iS2E)νν and Uνν Λν ν Uνν = λνδνν . The solution of equation (29) has the following form

E = H : Pˆν(x2)=Φ(α − 2eHγ(p1 − ε)x2), (30) − 1 2 2 2 2 2 2 2 3 where Φ is Airy function, α = p3 + p1 + m − ε γ and γ = 4e h (p1 − ε) . So, the energy levels are not quantized 2 iz j 2 E = H : Pˆ (x2) = exp G (−ia , −iz ),j=1, 2. (31) ν 2 ν ν Here 2 2 2 2 2 p + p + m − ε + eλ (p1H − εE) a = − 3 1 ν + ,η= E2 − H2, ν eη eη3 622 A. Galkin 2 2 √ p1H − εE λ = −igνη − 2g1ν η ,z= eη x2 + . ν eη2 When E>H, iz2 of (31) becomes purely imaginary. So the energy levels are not quantized. In the case E

3 Thus we present the equation for particles with spin 2 interacting with electromagnetic field, which is casual (it can be proved using approach proposed in [7]), i.e. their solutions are propa- gated with the velocity smaller than the light velocity. We also find the solutions of this equation in constant magnetic field and in static electric field inclined at an arbitrary angle to a static magnetic field. As it was shown above the corresponding constant g1 canbechoseninsuchform 3 in which the energies levels of charged particle with spin 2 in constant magnetic field are not complex for arbitrary values H and g.

Acknowledgements I would like to thank Prof. A. Nikitin for useful discussion and suggestions.

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