Solution of the Dirac equation in presence of an uniform magnetic field
aa r X i v : . [ h e p - t h ] A ug Solution of the Dirac equation in presence of anuniform magnetic field
Kaushik Bhattacharya ∗ Instituto de Ciencias Nucleares,Universidad Nacional Autonoma de Mexico,Circuito Exterior, C.U., A. Postal 70-543, C.P. 04510 Mexico D.F.,Mexico.
February 1, 2008
Abstract
In this work we discuss the properties of the solutions of the Dirac equationin presence of an uniform background magnetic field. In particular we focus onthe nature of the solutions, their ortho-normality properties and how these solu-tions depend on the choice of the vector potential giving rise to the magnetic field.We explicitly calculate the spin-sum of the solutions and using it we calculate thepropagator of the electron in presence of an uniform background magnetic field.
Calculations of elementary particle decays and scattering cross-sections in presence ofa background magnetic field are commonly found in literature [1, 2, 3, 4, 5]. Thesecalculations became more important after it was understood that the neutron star corescan sustain magnetic fields of the order of 10 G or more. These realistic fields may bevery complicated in their structure but for simplicity many of the times we assume thesefields to be uniform. The advantage of an uniform magnetic field is that in presence ofthis field the Dirac equation can be exactly solved. Once the the Dirac equation is exactlysolved then we can proceed to quantize those solutions and calculate elementary particledecays and scattering cross-sections in presence of the background magnetic field. In thisarticle we will solve the Dirac equation in a background magnetic field and discuss aboutthe nature of the solutions. We will quantize the fermionic theory in presence of a magnetic ∗ e-mail addresses:[email protected], In this article we will assume that the uniform classical background magnetic field isalong the z -direction of the co-ordinate axis. The background gauge fields giving rise toa magnetic field along the z -direction, of magnitude B , can be fixed in many ways: A = A y B = A z B = 0 , A x B = − y B + b . (1)or A = A x B = A z B = 0 , A y B = x B + c . (2)or A = A z B = 0 , A y B = 12 x B + d , A x B = − y B + g , (3)where b , c , d and g are constants. Here A µ B designates that the gauge field is a classicalbackground field and not a quantized dynamical field. In the above equations x , y are justcoordinates and not 4-vectors. In this article we will assume that the gauge configurationas given in Eq. (1) with b = 0. More over in this article we will be employing theDirac-Pauli representation of the Dirac matrices. The Dirac equation for a particle of mass m and charge eQ , in presence of a magneticfield is given by: i ∂ψ∂t = H B ψ , (4)where the Dirac Hamiltonian in presence of a magnetic field is given by: H B = α · Π + βm . (5)Here Π µ is the kinematic momentum of the charged fermion. In our convention, e is thepositive unit of charge, taken as usual to be equal to the proton charge. From Eq. (4) wecan infer that for the stationary states, we can write: ψ = e − iEt φχ ! , (6)2here φ and χ are 2-component objects. With this notation, we can write Eq. (4) as:( E − m ) φ = σ · ( − i ∇ − eQ A ) χ , (7)( E + m ) χ = σ · ( − i ∇ − eQ A ) φ . (8)Eliminating χ , we obtain( E − m ) φ = h σ · ( − i ∇ − eQ A ) i φ . (9)With our choice of the vector potential, Eq. (9) reduces to the form( E − m ) φ = h − ∇ + ( eQ B ) y − eQ B (2 iy ∂∂x + σ ) i φ . (10)Here σ is the diagonal Pauli matrix. Noticing that the co-ordinates x and z do notappear in the equation except through the derivatives, we can write the solutions as φ = e i p · X \ y f ( y ) , (11)where f ( y ) is a 2-component matrix which depends only on the y -coordinate, and possiblysome momentum components, as we will see shortly. We have also introduced the notation X for the spatial co-ordinates (in order to distinguish it from x , which is one of thecomponents of X ), and X \ y for the vector X with its y -component set equal to zero. Inother words, p · X \ y ≡ p x x + p z z , where p x and p z denote the eigenvalues of momentum inthe x and z directions. There will be two independent solutions for f ( y ), which can be taken, without anyloss of generality, to be the eigenstates of σ with eigenvalues s = ±
1. This means thatwe choose the two independent solutions in the form f + ( y ) = F + ( y )0 ! , f − ( y ) = F − ( y ) ! . (12)Since σ f s = sf s , the differential equations satisfied by F s is d F s dy − ( eQ B y + p x ) F s + ( E − m − p z + eQ B s ) F s = 0 , (13)which is obtained from Eq. (10). The solution is obtained by using the dimensionlessvariable ξ = q e | Q |B y + p x eQ B ! , (14)which transforms Eq. (13) to the form " d dξ − ξ + a s F s = 0 , (15) It is to be understood that whenever we write the spatial component of any vector with a letteredsubscript, it would imply the corresponding contravariant component of the relevant 4-vector. a s = E − m − p z + eQ B se | Q |B . (16)This is a special form of Hermite’s equation, and the solutions exist provided a s = 2 ν + 1for ν = 0 , , , · · · . This provides the energy eigenvalues E = m + p z + (2 ν + 1) e | Q |B − eQ B s , (17)and the solutions for F s are N ν e − ξ / H ν ( ξ ) ≡ I ν ( ξ ) , (18)where H ν are Hermite polynomials of order ν , and N ν are normalizations which we taketo be N ν = q e | Q |B ν ! 2 ν √ π / . (19)With our choice, the functions I ν satisfy the completeness relation X ν I ν ( ξ ) I ν ( ξ ⋆ ) = q e | Q |B δ ( ξ − ξ ⋆ ) = δ ( y − y ⋆ ) , (20)where ξ ⋆ is obtained by replacing y by y ⋆ in Eq. (14).So far, Q was arbitrary. We now specialize to the case of electrons, for which Q = − E n = m + p z + 2 ne B , (21)which is the relativistic form of Landau energy levels. The solutions are two fold degener-ate in general: for s = 1, ν = n − s = − ν = n . In the case of n = 0, from Eq.(17) we see that for Q = − ν = − (1 + s ), and as ν cannot be negative s = −
1. Thusthe n = 0 state is not degenerate. The solutions can have positive or negative energies.We will denote the positive square root of the right side by E n . Representing the solutioncorresponding to this n -th Landau level by a superscript n , we can then write for thepositive energy solutions, f ( n )+ ( y ) = I n − ( ξ )0 ! , f ( n ) − ( y ) = I n ( ξ ) ! . (22)For n = 0, the solution f + does not exist. We will consistently incorporate this fact bydefining I − ( y ) = 0 , (23)in addition to the definition of I n in Eq. (18) for non-negative integers n .4he solutions in Eq. (22) determine the upper components of the spinors through Eq.(11). The lower components, denoted by χ earlier, can be solved using Eq. (8), and finallythe positive energy solutions of the Dirac equation can be written as e − ip · X \ y U s ( y, n, p \ y ) , (24)where X µ denotes the space-time coordinate. And U s are given by [6, 7, 8] U + ( y, n, p \ y ) = I n − ( ξ )0 p z E n + m I n − ( ξ ) − √ ne B E n + m I n ( ξ ) , U − ( y, n, p \ y ) = I n ( ξ ) − √ ne B E n + m I n − ( ξ ) − p z E n + m I n ( ξ ) . (25)For the case of positrons which are positively charged negative energy solutions of theDirac equation we have to put Q = − n = 0 solution must only have the s = 1 component. Although the dispersion relation of the electrons and positrons becomedifferent in presence of a magnetic field but they can be written in a unique form as givenin Eq. (21), the difference shows up in the spin of the zeroth Landau level state. A similarprocedure, as used for solving for the positive energy spinors, can be adopted to solve forthe negative energy spinors and the solutions are: e ip · X \ y V s ( y, n, p \ y ) , (26)where V − ( y, n, p \ y ) = p z E n + m I n − ( e ξ ) √ ne B E n + m I n ( e ξ ) I n − ( e ξ )0 , V + ( y, n, p \ y ) = √ ne B E n + m I n − ( e ξ ) − p z E n + m I n ( e ξ )0 I n ( e ξ ) . (27)where e ξ is obtained from ξ by changing the sign of the p x -term. These solutions areeigenstates of Π x and Π z but not of Π y . As Π x and Π y do not commute we cannot havesimultaneous eigenstates of both.The solutions of the Dirac equation in presence of a magnetic field are exact solutionsand not perturbative excitations around the free Dirac equation solutions, which is evidentfrom Eq. (16). Consequently we cannot put B → z direction,and we can at best gauge transform these solutions to obtain equivalent solutions in abackground magnetic field. The choice of the background gauge does not permit us toobtain the free solutions in any limit as the free solutions belong to another gauge orbit,namely the pure gauge solutions. It is previously stated that the n = 0 solution is non-degenerate and in this state wehave only one solution available for the positive energy and one for the negative energy.They are the s = − s = 1 for the negative energystate, which is evident from Eq. (25) and Eq. (27). Only in the n = 0 state the wavefunctions are eigenstates of Σ z , where Σ z = iγ γ , and for all other higher Landau statesthe solutions do not have any definite Σ z eigenvalue. In actual calculations when thestrength of the magnetic field is high we require to work with the n = 0 solutions. We canroughly estimate the magnitude of the magnetic field suitable for the n = 0 approximation.Suppose we know the typical electron energy in a system is E and the magnitude of themagnetic field is B from experimental observations. If it happens that 2 ne B > E − m for any positive value of n then from the dispersion relation in Eq. (21) we see that p z has to be negative, which is impossible. Consequently when ever 2 e B is greater thanthe square of the typical electron energy of the system minus the rest mass square of theelectron then we have only the n = 0 level contributing to the energy levels and only thosecorresponding wave functions must be used in calculating the other details of the system.As an example if the typical electron energy of the system is of the order of 1MeV thenfor magnetic field magnitude greater than 10 Gauss we must only have the n = 0 levelcontributions in the energy. For lower magnitude of the magnetic field the other Landaulevels will start to contribute in the electron energy. For a fixed energy of the electronand for very low magnetic field magnitude we will have many possible Landau levels. Using the relation Z ∞−∞ I n ( a ) I m ( a ) da = √ e B δ n,m , (28)where δ n,m = 1 when n = m and zero otherwise and a is dimensionless we can calculatethe ortho-normality of the spinors. The ortho-normality of the spinors in the present casehas to be modified as the spinors have explicit co-ordinate dependencies. Using Eq. (28)it can be shown in a straight forward fashion that, Z ∞−∞ dy U † s ( y, n, p \ y ) U s ′ ( y, m, p \ y ) = Z ∞−∞ dy V † s ( y, n, p \ y ) V s ′ ( y, m, p \ y ) = δ n,m δ s,s ′ E n E n + m , (29)6nd Z ∞−∞ dy U † s ( y, n, p \ y ) V s ′ ( y, m, − p \ y ) = Z ∞−∞ dy V † s ( y, n, − p \ y ) U s ′ ( y, m, p \ y ) = 0 . (30)Except the integration over y and the appearance of the Landau levels the above relationsclosely resemble the corresponding relations in free-space. The above relations fix thenormalization of the spinors. We will rederive the normalization constants of the spinorswhen we quantize the theory in section 4.Using now the solutions for the U and the V spinors from Eqs. (25) and (27), it isstraight forward to verify that, Z ∞−∞ dy X s (cid:16) U s ( y, n, p \ y ) N U N † s ( y, m, p \ y ) + V Ns ( y, n, − p \ y ) V N † s ( y, m, − p \ y ) (cid:17) = δ n,m (31)Here U Ns ( y, n, p \ y ) and V Ns ( y, n, − p \ y ) are the normalized spinors and is the unit 4 × X s (cid:16) U s ( y, n, p \ y ) U † s ( y ⋆ , n, p \ y ) + V s ( y, n, − p \ y ) V † s ( y ⋆ , n, − p \ y ) (cid:17) = p z + 2 ne B ( E n + m ) ! × diag h I n − ( ξ ) I n − ( ξ ⋆ ) , I n ( ξ ) I n ( ξ ⋆ ) , I n − ( ξ ) I n − ( ξ ⋆ ) , I n ( ξ ) I n ( ξ ⋆ ) i , (32)where ‘diag’ indicates a diagonal matrix with the specified entries, and ξ and ξ ⋆ involvethe same value of p x . A sum over the Landau levels for spinors situated at different y co-ordinates gives, ∞ X n =0 X s (cid:16) U Ns ( y, n, p \ y ) U N † s ( y ⋆ , n, p \ y ) + V Ns ( y, n, − p \ y ) V N † s ( y ⋆ , n, − p \ y ) (cid:17) = δ ( y − y ⋆ ) , (33)where we have used the result of Eq. (20). The two equations in Eq. (31) and Eq. (33)stands for the completeness relations for the spinors in the present case. In this section we derive the spin-sum P s U s ( y, n, p \ y ) U s ( y ⋆ , n, p \ y ) of the solutions of theDirac equation in presence of a magnetic field. The two spinors in the above sum can havetwo different position coordinates in general and so their spatial dependence is explicitlyshown to be different. From the nature of the solutions as given in Eq. (25) we see that P s U s ( y, n, p \ y ) U s ( y ⋆ , n, p \ y ) can be written as: P U ( y, y ⋆ , n, p \ y ) ≡ X s U s ( y, n, p \ y ) U s ( y ⋆ , n, p \ y ) = 1 E n + m n X i,j = n − I i ( ξ ) I j ( ξ ∗ ) T i,j (34)7he spin-sum of the product of the spinors, P s U s ( y, n, p \ y ) U s ( y ⋆ , n, p \ y ) will give rise to a4 × I i ( ξ ) I j ( ξ ∗ ), where i, j runs from n − , n .If these terms as I i ( ξ ) I j ( ξ ∗ ) are taken as common factors then the whole 4 × I i ( ξ ) I j ( ξ ∗ ) timesthe corresponding 4 × T i,j .Using the dispersion relation E n = p z + m + 2 ne B , T n,n can be written as [7], T n,n = E n + m ) 0 p z − p z − ( E n − m ) . (35)In the 2 × T n,n = E n (1 − σ ) 00 − (1 − σ ) ! + p z (1 − σ ) − (1 − σ ) 0 ! + m (1 − σ ) 00 (1 − σ ) ! , (36)where σ is the third Pauli matrix. In the 4 × T n,n = 12 [ m (1 − Σ z ) + E n ( γ + γ γ ) − p z ( γ γ + γ )] , = 12 [ m (1 − Σ z ) + / p k + e / p k γ ] , (37)where σ z = iγ γ . In the last equation / p k = p γ + p γ and e / p k = p γ + p γ and γ = iγ γ γ γ . In our case p = E n .In a similar way T n − ,n − can be written as: T n − ,n − = ( E n + m ) 0 − p z
00 0 0 0 p z − ( E n − m ) 00 0 0 0 . (38)In the 2 × T n − ,n − = E n (1 + σ ) 00 − (1 + σ ) ! + p z − (1 + σ ) (1 + σ ) 0 ! + m (1 + σ ) 00 (1 + σ ) ! . (39)In the 4 × T n − ,n − = 12 [ m (1 + Σ z ) + E n ( γ − γ γ ) + p z ( γ γ − γ )] , = 12 [ m (1 + Σ z ) + / p k − e / p k γ ] . (40)8rom the matrix multiplication in the left hand side of Eq. (34) it can be seen that T n − ,n is given as, T n − ,n = √ ne B − . (41)In the 2 × T n − ,n = √ ne B ( σ + iσ ) − ( σ + iσ ) 0 ! . (42)Here σ and σ are the first two Pauli matrices. When converted back to the 4 × T n − ,n = − √ ne B ( γ + iγ ) . (43)Similarly T n,n − is given by, T n,n − = √ ne B − . (44)In the 2 × T n,n − = √ ne B ( σ − iσ ) − ( σ − iσ ) 0 ! , (45)which when converted back to the 4 × T n − ,n = − √ ne B ( γ − iγ ) . (46)Supplying the values of T i,j s from Eq. (37), Eq. (40), Eq. (43) and Eq. (46) to Eq. (34)we get the result: P U ( y, y ⋆ , n, p \ y ) ≡ X s U s ( y, n, p \ y ) U s ( y ⋆ , n, p \ y ) = 1( E n + m ) S U ( y, y ⋆ , n, p \ y ) , (47)where, S U ( y, y ⋆ , n, p \ y ) = 12 (cid:20) n m (1 + Σ z ) + / p k − e / p k γ o I n − ( ξ ) I n − ( ξ ⋆ )+ n m (1 − Σ z ) + / p k + e / p k γ o I n ( ξ ) I n ( ξ ⋆ ) − √ ne B ( γ − iγ ) I n ( ξ ) I n − ( ξ ⋆ ) − √ ne B ( γ + iγ ) I n − ( ξ ) I n ( ξ ⋆ ) (cid:21) . (48)9imilarly, the spin sum for the V -spinors can also be calculated, and we obtain: P V ( y, y ⋆ , n, p \ y ) ≡ X s V s ( y, n, p \ y ) V s ( y, n, p \ y ) = 1( E n + m ) S V ( y, y ⋆ , n, p \ y ) , (49)where, S V ( y, y ⋆ , n, p \ y ) = 12 " n − m (1 + Σ z ) + / p k − e / p k γ o I n − ( e ξ ) I n − ( e ξ ⋆ )+ n − m (1 − Σ z ) + / p k + e / p k γ o I n ( e ξ ) I n ( e ξ ⋆ )+ √ ne B ( γ − iγ ) I n ( e ξ ) I n − ( e ξ ⋆ ) + √ ne B ( γ + iγ ) I n − ( e ξ ) I n ( e ξ ⋆ ) . (50)One important property of the above spin-sums is that, P U ( y, y ⋆ , n, p \ y ) = − P V ( y, y ⋆ , n, − p \ y ) , (51)which is similar to the result in vacuum. In this section we will use the spin-sum results in writing the electron propagator inpresence of an external uniform magnetic field. But before doing so we will first writedown the QED Lagrangian for the electron in presence of a background magnetic field.In presence of a background magnetic field we can decompose the photon field asfollows: A µ ( x ) = A µ D ( x ) + A µ B ( x ) , (52)where A µ D ( x ) is the dynamical photon field which will be quantized and A µ B ( x ) is theclassical background field which gives rise to the magnetic field. If the uniform backgroundclassical magnetic field is called B then we must have: B = ∇ × A B ( x ) , (53)where A µ B ( x ) = (0 , A B ( x )). In presence of the background magnetic field we can alsowrite the field strength tensor as: F µν ( x ) = F µν D ( x ) + F µν B , (54)where F µν D ( x ) = ∂ µ A ν D ( x ) − ∂ ν A µ D ( x ) and F ij B = ∂ i A j B ( x ) − ∂ j A i B ( x ) is a constant as givenin Eq. (53). 10he QED Lagrangian can be written as: L = ψ ( iγ µ D µ − m ) ψ − F µν F µν , (55)where D µ = ∂ µ − ieA µ is the covariant derivative of the fermion fields. The QED La-grangian can also be written as: L = ψ [ γ µ Π µ − m ] ψ + eψγ µ ψA µ D − F µν F µν , (56)where Π µ = i∂ µ + eA µ B is the kinetic momentum of the fermions in presence of the back-ground field. The first term of the Lagrangian contains no dynamical photon dependencebut it depends upon the background magnetic field through Π and this part of the La-grangian gives rise to the Hamiltonian of the electron in presence of the magnetic fieldused in Eq. (4). The equation of motion which we obtain from the first term of the aboveLagrangian is in fact the Dirac equation in presence of a magnetic field which we solved insection 2. Consequently the most important effect of the background magnetic field is tomodify the solutions of the Dirac equation. The interaction term of electrons and photonsremains the same as in normal QED. The free fermionic part of the Lagrangian in Eq. (56)is also important for the definition of the propagator of the electron and in the next partof this section we will find out the expression of the electron propagator in presence ofthe background magnetic field. Before we calculate the electron propagator we quantizethe theory. The photons do not interact with the magnetic field and consequently theirquantization procedure is the same as in normal QED. Since we have found the solutions to the Dirac equation, we can now use them to constructthe fermion field operator in the second quantized version. For this, we write ψ ( X ) = X s = ± ∞ X n =0 Z dp x dp z πD h f s ( n, p \ y ) e − ip · X \ y U s ( y, n, p \ y ) + b f † s ( n, p \ y ) e ip · X \ y V s ( y, n, p \ y ) i . (57)Here, f s ( n, p \ y ) is the annihilation operator for the fermion, and b f † s ( n, p \ y ) is the creationoperator for the antifermion in the n -th Landau level with given values of p x and p z . It isto be noted that the wave-functions of the electron used in Eq. (57) are not free-particlesolutions and they never tend to the free-particle solutions in any limit. As the thepositive and negative frequency parts of the solutions are as free-particles so the notionof a particle and anti-particle is unambiguous in the present circumstance. The creationand annihilation operators satisfy the anti-commutation relations h f s ( n, p \ y ) , f † s ′ ( n ′ , p ′ \ y ) i + = δ ss ′ δ nn ′ δ ( p x − p ′ x ) δ ( p z − p ′ z ) , (58)and a similar one with the operators b f and b f † , all other anti-commutators being zero. Thequantity D appearing in Eq. (57) depends on the normalization of the spinor solutions, and11n this section we will rederive the normalization of the spinors calculated in subsection2.3. The factor of 2 π multiplying D helps to associate D with the normalization constantfound in subsection 2.3. Once we have chosen the spinor normalization, the factor D appearing in Eq. (57) is however fixed, and it can be determined from the equal timeanti-commutation relation h ψ ( X ) , ψ † ( X ⋆ ) i + = δ ( X − X ⋆ ) . (59)Plugging in the expression given in Eq. (57) to the left side of this equation and using theanti-commutation relations of Eq. (58), we obtain h ψ ( X ) , ψ † ( X ⋆ ) i + = X s X n Z dp x dp z (2 πD ) (cid:16) e − ip x ( x − x ⋆ ) e − ip z ( z − z ⋆ ) U s ( y, n, p \ y ) U † s ( y ⋆ , n, p \ y )+ e ip x ( x − x ⋆ ) e ip z ( z − z ⋆ ) V s ( y, n, p \ y ) V † s ( y ⋆ , n, p \ y ) (cid:17) . (60)Changing the signs of the dummy integration variables p x and p z in the second term, wecan rewrite it as h ψ ( X ) , ψ † ( X ⋆ ) i + = X s X n Z dp x dp z (2 πD ) e − ip x ( x − x ⋆ ) e − ip z ( z − z ⋆ ) (cid:16) U s ( y, n, p \ y ) U † s ( y ⋆ , n, p \ y )+ V s ( y, n, − p \ y ) V † s ( y ⋆ , n, − p \ y ) (cid:17) . (61)At this stage, we can perform the sum over n in Eq. (61) using Eq. (32) and Eq. (20)which gives the δ -function of the y -coordinate and perform the integrations over p x and p z to recover the δ -functions for the other two coordinates as well, provided2 E n E n + m πD ) = 1(2 π ) . (62)In this way we get back the same value of the normalization of the spinors which weobtained in subsection 2.3. Putting the solution for D , we can rewrite Eq. (57) as ψ ( X ) = X s = ± ∞ X n =0 Z dp x dp z π s E n + m E n × h f s ( n, p \ y ) e − ip · X \ y U s ( y, n, p \ y ) + b f † s ( n, p \ y ) e ip · X \ y V s ( y, n, p \ y ) i . (63)The one-fermion states are defined as | n, p \ y , s i = Cf † s ( n, p \ y ) | i . (64)The normalization constant C is determined by the condition that the one-particle statesshould be orthonormal. For this, we need to define the theory in a finite but large regionwhose dimensions are L x , L y and L z along the three spatial axes. This gives C = 2 π √ L x L z . (65)Next we calculate the electron propagator in presence of an uniform background magneticfield. 12 .2 The electron propagator The electron propagator is given by, iS BF ( X − X ⋆ ) αβ = h | T { ψ ( X ) α ψ ( X ⋆ ) β }| i = θ ( t − t ⋆ ) h | ψ ( X ) α ψ ( X ⋆ ) β | i − θ ( t ⋆ − t ) h | ψ ( X ⋆ ) β ψ ( X ) α | i , (66)where T is the time-ordered product and θ ( λ ) is the step-function which is unity when λ ≥ θ ( λ ) = i Z ∞−∞ dω π e − iλω ω − iǫ , (67)where ǫ is an infinitesimal parameter. Using Eq. (63) we can write, h | ψ ( X ) α ψ ( X ⋆ ) β }| i = X s = ± ∞ X n =0 Z dp x dp z (2 π ) (cid:18) E n + m E n (cid:19) e − ip · ( X \ y − X ⋆ \ y ) × U s,α ( y, n, p \ y ) U s,β ( y ⋆ , n, p \ y ) , (68)and using Eq. (47) and suppressing the spinor indices the above equation can also bewritten as, h | ψ ( X ) ψ ( X ⋆ ) | i = ∞ X n =0 Z dp x dp z (2 π ) (cid:18) E n + m E n (cid:19) e − ip · ( X \ y − X ⋆ \ y ) P U ( y, y ⋆ , n, p \ y ) . (69)In a similar way it can be shown that, h | ψ ( X ⋆ ) ψ ( X ) | i = ∞ X n =0 Z dp x dp z (2 π ) (cid:18) E n + m E n (cid:19) e ip · ( X \ y − X ⋆ \ y ) P V ( y, y ⋆ , n, p \ y ) , (70)where P V is given in Eq. (49). Using the above results in Eq. (66) and utilizing the formof the θ -function in Eq. (67) we can write, iS BF ( X − X ⋆ ) = i ∞ X n =0 Z dp x dp z dω (2 π ) (cid:18) E n + m E n (cid:19) × " e − iω ( t − t ⋆ ) − ip · ( X \ y − X ⋆ \ y ) ω − iǫ P U ( y, y ⋆ , n, p \ y ) − e iω ( t − t ⋆ )+ ip · ( X \ y − X ⋆ \ y ) ω − iǫ P V ( y, y ⋆ , n, p \ y ) . (71)Changing the signs of p x and p z in the second term of the integrand and using Eq. (51)we get, iS BF ( X − X ⋆ ) = i ∞ X n =0 Z dp x dp z dω (2 π ) e i p · ( X \ y − X ⋆ \ y ) (cid:18) E n + m E n (cid:19) P U ( y, y ⋆ , n, p \ y ) × " e − i ( ω + E n )( t − t ⋆ ) ω − iǫ + e i ( ω + E n )( t − t ⋆ ) ω − iǫ . (72)13ow appropriately doing the integration over ω we get, iS BF ( X − X ⋆ ) = i ∞ X n =0 Z dp x dp z dp (2 π ) e − ip · ( X \ y − X ⋆ \ y ) S U ( y, y ⋆ , n, p \ y ) p − p z − m − ne B − iǫ , = i ∞ X n =0 Z dp x dp z dp (2 π ) e − ip · ( X \ y − X ⋆ \ y ) S U ( y, y ⋆ , n, p \ y ) p k − m − ne B − iǫ , (73)where S U ( y, y ⋆ , n, p \ y ) is given by Eq. (48) and p k = p − p z . It is to be noted that the poleof the propagator is now dependent on the Landau levels as it should be in an uniformbackground magnetic field. The form of the propagator suggests that it is not translationinvariant and so it cannot be written down completely in Fourier space. As we have solved the Dirac equation in presence of a uniform background magnetic fieldusing a particular gauge, as given in Eq. (1) with b = 0, the solutions are dependent on thegauge choice. The spinor solutions are themselves not physical observables and so they canbe gauge dependent. But not all the results discussed in this article are gauge dependent.The energy of the electron as given in Eq. (17) is not a gauge dependent quantity, anygauge we choose we will get the same dispersion relation of the electrons. The special formsof the ortho-normality relations as given in section 2.3 are gauge dependent as the resultscontain the functions which has p x which is not a gauge invariant quantity. The spin-sumalso depends on the particular gauge we work in and the above results will be different if wehad chosen another gauge to represent the magnetic field. But actual calculations yieldingphysical quantities like scattering cross-section or decay rates must be independent of thechoice of the background gauge field. We can see the gauge invariance of the physicalquantities in a heuristic way. If we had chosen the gauge specified in Eq. (2) with c = 0instead of the gauge in Eq. (1) with b = 0 then the solutions of the Dirac equations asspecified in Eq. (25) and Eq. (27) should have been the same except all the y should bereplaced by x and p x should be replaced by p y inside the spinors and the free-particlepart should contain e ip · X \ x instead of e ip · X \ y . A similar replacement should yield the newspin-sums and the propagator. Consequently the quantities calculated in these two gaugesdiffer by the way we name the x and y coordinate axes. But in calculations of scatteringcross-sections and decay-rates we always have integrations over x, y, z coordinates at eachvertex and consequently the end results will not depend upon which gauge we startedwith.The above discussion highlights the fact that most of the quantities calculated in thisarticle using the exact solutions in presence of the magnetic field rely heavily on our choiceof the vector potential. All the solutions of the Dirac equation in presence of a uniformmagnetic field along the z direction obtained by using various vector potentials will bedifferent but are related by gauge transformations. It is to be noted that the free Diracsolutions can also be gauge rotated where the gauge fields are pure gauge configurations.As there is no connection between the gauge configurations giving rise to a magnetic field14long the z direction and pure gauge fields so we do not get back the free Dirac solutionsas a limit of the exact solutions in a magnetic field. In this article we solved the Dirac equation in presence of a background uniform magneticfield specified by a particular vector potential. The dispersion relation of the electron isseen to change from its form in the vacuum and we see the emergence of Landau levelsdesignating the quantized nature of the transverse motion of the electrons. The solutionsof the Dirac equation are dependent on the Landau levels, the energy of the electron isseen to be degenerate except the lowest Landau level energy. It is seen that there is no wayto get back the free Dirac solutions from the exact solutions in presence of the magneticfield by letting the field strength to go to zero in the solutions, a fact which is related tothe gauge invariance of the system. Using the appropriate spinors in a magnetic field theortho-normality and completeness of the spinors were worked out in section 2.3, whichclosely resembles the corresponding results in vacuum. The spin-sum of the solutions arederived explicitly using the exact solutions of the Dirac equation in a magnetic field. Thetheory is quantized and with the quantum field operators the propagator of the electronin presence of an uniform background magnetic field is calculated. Some thing similar toour derivation of the electron propagator was presented in [9] where the authors workedin the chiral representation of the Dirac gamma matrices. But the presentation of theexpression of the propagator was not compact and nor the authors in [9] calculate thespin-sum explicitly. As most of the quantities calculated in this article depend on thechoice of the vector potential giving rise to the magnetic field so the gauge invariance ofthe calculations become less transparent. In the penultimate section we discuss about thegauge invariance of the calculations in presence of a magnetic field and show that althoughthe spin-sums and propagators may not be gauge invariant but physical quantities likescattering-cross sections and decay rates of elementary particles in presence of a magneticfield can be gauge invariant.
References [1] L. Fassio-Canuto, Phys. Rev. , 2141 (1969).[2] J. J. Matese and R. F. O’Connell, Phys. Rev. , 1289 (1969).[3] J. J. Matese and R. F. O’Connell, Astroph. Jour. , 451 (1970).[4] O. F. Dorofeev, V. N. Rodionov and I. M. Ternov, JETP Lett. , 917 (1984) [PismaZh. Eksp. Teor. Fiz. , 159 (1984)].[5] D. A. Dicus, W. W. Repko and T. M. Tinsley, arXiv:0704.1695 [hep-ph].[6] K. Bhattacharya and P. B. Pal, Pramana , 1041 (2004) [arXiv:hep-ph/0209053].157] K. Bhattacharya, Ph. D. Thesis. arXiv:hep-ph/0407099.[8] K. Bhattacharya and P. B. Pal, published in Indian National Science Academyproceedings
70, A , 145, (2004). arXiv:hep-ph/0212118. hep-ph/0212118;[9] M. Kobayashi and M. Sakamoto, Prog. Theor. Phys.70