Modified Dirac equation with Lorentz invariance violation and its solutions for particles in an external magnetic field S. I. Kruglov 1 Department of Chemical and Physical Sciences, University of Toronto at Mississauga, 3 1 3359 Mississauga Rd. North, Mississauga, Ontario, Canada L5L 1C6 0 2 n Abstract a J The second-order modified Dirac equation leading to the modified 2 dispersion relation due to the Lorentz invariance violation corrections ] is suggested. The equation is formulated in the 16-component first- c q orderform. Ihaveobtained theprojection matrixextracting solutions - r of the equation with definite spin projections which can be considered g as the density matrix for pure spin states. Exact solutions of the [ equation are found for particles in the external constant and uniform 3 magnetic field. The synchrotron radiation radius within the novel v 9 modified Dirac equation is estimated. 0 5 0 1 Introduction . 0 1 2 The possibility of the spontaneous violation of Lorentz and CPT symmetries 1 in the framework of the string theory was proposed in [1]. Thus, the Stan- : v dard Model was extended and the Lorentz invariance violation (LIV) can be i X described by the effective field theory [2]. From experiment, bounds on LIV r coefficients within the effective field theory were obtained [3]. Different mod- a els of LIV in the photon sector [4] and fermion sector [5] were investigated (there are many other publications). At present energies stringy effects are suppressed by the Planck scale M = 1.22 1019 GeV and there are not P × signs yet of LIV in experiments. It should be mentioned that any LIV mod- els modify dispersion relations. It was mentioned in [6], [7] that quantum gravity corrections can lead to deformed dispersion relation: p2 = p2 +m2 (Lp )αp2, (1) 0 0 − 1E-mail: [email protected] 1 where the speed of light in vacuum c = 1, p is an energy and p is a mo- 0 mentum of a particle and L can be considered as “minimal length” which is −1 of the order of the Plank length L = M . The subluminal propagation of P P particles corresponds to L > 0 at α = 1. The last term in Eq.(1) violates the Lorentz invariance. The modified dispersion relation (1) with the parameter α = 1 was introduced in the framework of space-time foam Liouville-string models [8]. The wave equation for spinless particles with the dispersion equa- tion (1) for α = 1 was considered in [9], [10]. From the analysis of the Crab Nebula synchrotron radiation some constrains on the parameters L and α were made [11], [9]. In this paper, I postulate the modified Dirac equation for particles with spin-1/2 leading to the deformed dispersion relation (1) with α = 2. This modified Dirac equation can be considered within effective field theory with LIV in flat space-time. The Euclidean metric and the system of units h¯ = c = 1 is used. Greek letters run 1,2,3,4 and Latin letters run 1,2,3. 2 Modified equation for spin-1/2 particles I suggest the modified Dirac equation for spin-1/2 particles of the second order: (γ ∂ +m iLγ ∂ γ ∂ )ψ(x) = 0, (2) µ µ 4 t i i − where ∂ = ∂/∂x = (∂/∂x ,∂/(i∂t)), x = t is a time and notations as µ µ i 0 in [12] are used. Eq.(2) is covariant under the rotational group but the additional term in Dirac equation (2) violates the invariance under the boost transformations. Thus, the Lorentzsymmetry is broken andonecanconsider Eq.(2) as an effective wave equation with LIV introducing preferred frame effects. To obtain the dispersion relation, we investigate the plane-wave solution for positive energy ψ(x) = ψ(p)exp[i(px p x )] to Eq.(2). Then 0 0 − Eq.(2) reads (ipˆ+m iLp γ p¯)ψ(p) = 0, (3) 0 4 − where pˆ = γ p , p¯ = γ p . One can verify, with the help of the γ-matrix µ µ i i algebra, that the matrix Λ = ipˆ+m iLp γ p¯ (4) 0 4 − obeys the equation as follows: 2 2 2 2 2 (Λ m) +p L p p = 0, (5) 0 − − 2 where p2 = p2 p2. The matrix equation (3) possesses non-trivial solutions 0 − if detΛ = 0 which is equivalent that some of the eigenvalues λ of the matrix Λ equal zero. Then the requirement λ = 0 leads to the modified dispersion relation: 2 2 2 2 2 2 p = p +m L p p . (6) 0 0 − Equation (6) corresponds to Eq.(1) for α = 2. Thus, Eq.(2) postulated realizes the modified dispersion relation (6). One can treat Eq.(2) as an effective equation for spin-1/2 particles with LIV parameter L of the order to the Plank length. 3 Wave equation in the first-order 16-component form Now we present the second-order equation (2) in the 16-component first- order form which is convenient for different calculations. Let us introduce, following the method of [13] (see also [14]) the system of first order equations equivalences to Eq.(2) (γ ∂ +m)ψ(x) iLmγ ∂ γ ψ (x) = 0, ∂ ψ(x) = mψ (x). (7) µ µ 4 t i i i i − We define the 16-component wave function ψ(x) Ψ(x) = Ψ (x) = , (8) { A } ψi(x) ! so thatΨ (x) = ψ(x), Ψ (x) = (1/m)∂ ψ(x). Using theelements oftheentire 0 i i matrix algebra εA,B, with matrix elements and products of matrices [14] εM,N = δ δ , εM,AεB,N = δ εM,N, (9) MA NB AB AB (cid:16) (cid:17) where A,B,M,N = (0,i), the system of equations (7), with taking into consideration Eq.(8), can be written in the 16-component matrix form ε0,0 γ +mLδ ε0,i γ γ δ εi,0 I ∂ µ 4µ 4 i µi 4 µ ⊗ ⊗ − ⊗ (cid:20)(cid:16) (cid:17) (10) 3 +m ε0,0 +εi,i I Ψ(x) = 0, 4 ⊗ (cid:16) (cid:17) (cid:21) where I is unit 4 4-matrix, the is the direct product of matrices, and 4 × ⊗ we imply the summation over all repeated indices. One can introduce the 16 16-matrices × Γ = ε0,0 γ +mLδ ε0,i γ γ δ εi,0 I , I = ε0,0 +εi,i I , (11) µ µ 4µ 4 i µi 4 16 4 ⊗ ⊗ − ⊗ ⊗ (cid:16) (cid:17) whereI isunit16 16-matrix. ThenEq.(10)becomesthefirst-order16 16- 16 × × component wave equation (Γ ∂ +m)Ψ(x) = 0. (12) µ µ From Eq.(11) we have generalized Dirac matrices as follows: Γ = ε0,0 γ εm,0 I , Γ = ε0,0 γ +mLε0,i γ γ . (13) m m 4 4 4 4 i ⊗ − ⊗ ⊗ ⊗ The rotational group generators in the 16-dimension representation space are given by 1 J = (εm,n εn,m) I +I (γ γ γ γ ). (14) mn 4 4 m n n m − ⊗ ⊗ 4 − The wave equation (12) is form-invariant under the rotation group trans- formations because matrices (13) obey the commutation relations as follows [15]: [Γ ,J ] = δ Γ δ Γ , [Γ ,J ] = 0. (15) a mn am n an m 4 mn − For the positive energies Ψ(x) = Ψ(p)exp[i(px p x )] and Eq.(12) becomes 0 0 − (ipˇ+m)Ψ(p) = 0, (16) where pˇ= Γ p . One can verify that the matrix pˇobeys the equation µ µ 5 2 3 2 2 2 pˇ p pˇ +(mL) p p pˇ= 0, (17) 4 − where p = ip . Introducing the matrix of Eq.(16) 4 0 Σ = ipˇ+m, (18) and using Eqs.(6),(17), we obtain the matrix equation 2 2 2 Σ(Σ m)(Σ 2m) (Σ m) +p +m = 0. (19) − − − h i 4 With the help of Eq.(19), we find solutions to Eq.(16) in the form of the projection matrix 2 2 2 Π = N (Σ m)(Σ 2m) (Σ m) +p +m . (20) − − − h i Indeed, it follows fromEq.(19) that (ipˇ+m)Π = 0. Therefore, every column of the matrix Π is the solution to Eq.(16). The normalization constant N can be find from the requirement [16]: 2 Π = Π. (21) From Eqs.(19)(20), after some calculations, one obtains the normalization constant 1 N = . (22) 2m2(p2 +2m2) WetreatthematrixΠasadensitymatrixwhichdescribes impurespinstates. To extract pure spin states, one may introduce the spin projection operator i σ = ǫ p J , (23) p −2 p abc a bc | | where p = p2, which obeys the minimal matrix equation as follows [17]: | | i q 1 9 2 2 σ σ = 0. (24) p − 4 p − 4 (cid:18) (cid:19)(cid:18) (cid:19) With the help of the method [16], we find from Eq.(24) the projection oper- ators extracting spin projections 1/2: ± 1 1 9 2 P±1/2 = ∓2 σp ± 2 σp − 4 . (25) (cid:18) (cid:19)(cid:18) (cid:19) Then the projection operator extracting pure spin states with positive energy is ∆ = ΠP , (26) ±1/2 ±1/2 obeying∆2±1/2 = ∆±1/2. Thus, ∆±1/2 maybeexplored incalculationsofsome processes including electrons with taking into consideration LIV parameter L. 5 4 Spin-1/2 particle in an external magnetic field Let us consider a particle in an external uniform and static magnetic field along the x axis, H = (0,0,H). Then the 4-potential can be chosen as 3 A = (0,Hx ,0,0). (27) µ 1 The electromagnetic interaction of particles in Eq.(2) can be introduced by the standard substitution ∂ D = ∂ ieA . For the electrons e = e , µ µ µ µ 0 → − − e > 0 and Eq.(2) becomes 0 γ ∂ +ie Hx γ iγ 1+Lγ ∂ m m 0 1 2 4 m m − (cid:20) (cid:18) (28) +ie HLx γ ∂ +m Ψ(x) = 0. 0 1 2 t (cid:19) (cid:21) One can look for a solution to Eq.(28), as for Dirac equation [12], [18], in the form 1 Ψ(x) = exp[i(p x +p x p t)]Ψ(x ), (29) 2 2 3 3 0 1 √L L − 2 3 where p = 2πn /L , p = 2πn /L ; n and n are integer quantum numbers. 2 2 2 3 3 3 2 3 Taking into account Eq.(29), equation (28) becomes (γ +iLp α )∂ +i[γ p +γ p +iLp (α p +α p )] 1 0 1 1 2 2 3 3 0 2 2 3 3 (cid:26) (30) +ie Hx (γ +iLp α )+m Ψ(x ) = p γ Ψ(x ), 0 1 2 0 2 1 0 4 1 (cid:27) where we have introduced matrices as follows [12]: 0 σ I 0 α = iγ γ = k , γ = 2 , (31) k 4 k σ 4 0 I k ! 2 ! − and σ are the Pauli matrices. Replacing the bispinor wave function k ϕ(x ) 1 Ψ(x ) = , (32) 1 χ(x ) 1 ! 6 into Eq.(30) with the help of Eq.(31), we obtain the system of equations (1 Lp )( iσ ∂ +p σ +p σ +e Hx σ )χ = (p m)ϕ, 0 1 1 2 2 3 3 0 1 2 0 − − − (33) (1+Lp )(iσ ∂ p σ p σ e Hx σ )ϕ = (p +m)χ. 0 1 1 2 2 3 3 0 1 2 0 − − − − From Eqs.(33) one finds the equation for ϕ: 2 2 2 2 2 1 L p (iσ ∂ p σ p σ e Hx σ ) +m p ϕ(x ) = 0. (34) 0 1 1 2 2 3 3 0 1 2 0 1 − − − − − h(cid:16) (cid:17) i To take into consideration the spin projection of an electron, we require that the ϕ is the eigenfunction of the operator of the spin projection on the magnetic field direction σ [12]: 3 σ ϕ(x ,µ) = µϕ(x ,µ), (35) 3 1 1 where µ = 1. Then Eq.(34), with the help of Eq.(35) and the properties of ± Pauli’s matrices, becomes p2 m2 2 2 2 0 ∂ +p +(p +e Hx ) +e Hµ ϕ(x ,µ) = − ϕ(x ,µ). (36) − 1 3 2 0 1 0 1 1 L2p2 1 0 h i − After introducing new variable p 2 ξ = e H x + 0 1 e H q (cid:18) 0 (cid:19) Eq.(36) takes the form of the equation for the harmonic oscillator: d2 1 p2 m2 2 0 2 +ξ ϕ(ξ,µ) = − p e Hµ ϕ(ξ,µ). (37) −dξ2 ! e0H 1 L2p20 − 3 − 0 ! − It should be mentioned that p defines the coordinate of the orbit center [18] 2 x = p /(e H) (ξ = 0). The requirement that ϕ(ξ,µ) 0 at ξ 1 2 0 − → → ±∞ gives 1 p2 m2 0 2 − p e Hµ = 2n+1, (38) e0H 1 L2p20 − 3 − 0 ! − where n = 0,1,2,... is the principal quantum number. From Eq.(38) one obtains the energy quantization: 2 2 2 2 2 p = m + 1 L p p +e H(2n+1+µ) . (39) 0 0 3 0 − (cid:16) (cid:17)h i 7 Due to the condition (39) the solution to Eq.(37) is given by ξ2 ϕ(ξ) = Cexp H (ξ), (40) n − 2 ! where H (ξ) are the Hermite polynomials and C is the normalization con- n stant. Eq.(39) shows that the energy of the electron depends on the LIV parameter L. At L = 0 one arrives to the well-known result [12]. From (29), (32), (33), one may find the bispinor wave function Ψ(x). The normalization constant C is defined from the relation [12]: L3 L2 ∞ + dx dx Ψ (x)Ψ(x)dx = 1, (41) 3 2 1 0 0 −∞ Z Z Z where Ψ+(x) is the Hermitian conjugate function. One can apply Eqs.(29), (32), (40) to study the synchrotron radiation of electrons with modified dis- persion relation (39). Electrons in an external magnetic field moves in helical orbits and emit the synchrotron radiation and the frequency depends on the orbit radius [18]. Let us consider the high (ultra-relativistic) energies of electrons (p m). In the case p = 0 particles move in a plane and at 0 3 ≫ 2L2e Hn 1 Eq.(39) is approximated as 0 ≪ p 2e Hn √2L2(e Hn)3/2. (42) 0 0 0 ≈ − q The classical radius of the orbit is given by [18]: βp 0 R = , (43) e H 0 with β = v being the electron velocity. Substituting Eq.(42) into (43) at v 1, we obtain the orbit radius ≈ R R L2 2e Hn3/2, (44) 0 0 ≈ − q where 2n R (45) 0 ≈ se0H is the orbit radius in the framework of the Klein Gordon equation [18]. It − follows from Eq.(44) that LIV parameter L reduces the orbit radius. The 8 angular orbital frequency ω = v/R becomes greater due to the Lorentz- violating term. ToestimatethecontributionofLIVparameterLtoobservable, weusethe data from experiments with magnetic trapping of electrons [19]. Electrons in the Penning trap90%of the time are inthe cyclotron ground state n = 1 and inexternalmagneticfieldH = 6T.UsingtheHeaviside Lorentzsystemwith − the fine structure constant α = e2/(4π) and the SI units which are related to the energy units 1 T= 195.5 eV2, 1 m= 5.1 106 eV−1, we obtain the radius change at L = L from Eq.(44): ∆R = L×2√2e Hn3/2 10−60 m. Thus, P P 0 ≈ when the LIV parameter Lis equal to the Planck length L the change of the P radius of the electron orbit is extremely small and Penning trap experiments are not relevant. From the Crab Nebula data, one can use the possible energy of electrons p = 1 TeV, and the magnetic field H = 260 µG . We obtain the principal 0 quantum number n p2/(2e H) 3 1029, and the change of the syn- 0 0 chrotron radius at L≈= L , ∆R =≈L2√×2e Hn3/2 10−22 m. This value is P P 0 ≈ very small compared to the classical radius (45). If the LIV parameter L is much greater than the Planck length L , then the LIV effect has to be taken P into account. 5 Conclusion The wave equation for spin-1/2 particles (electrons) suggested leads to the modifieddispersionrelation(1)withα = 2whichwasdiscussedinthecontext of quantum gravity corrections in [9]. I have formulated the novel equation in the first-order form and found the minimal polynomial of matrices of the equation and the spin projection operators. 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