Lagrangian and Hamiltonian Formulation of Classical Electrodynamics without Potentials
aa r X i v : . [ phy s i c s . c l a ss - ph ] J a n Lagrangian and Hamiltonian Formulation ofClassical Electrodynamics without Potentials
Dan N. VollickIrving K. Barber School of Arts and SciencesUniversity of British Columbia Okanagan3333 University WayKelowna, B.C.CanadaV1V 1V7
Abstract
In the standard Lagrangian and Hamiltonian approach to Maxwell’s theory the poten-tials A µ are taken as the dynamical variables. In this paper I take the electric field ~E andthe magnetic field ~B as the the dynamical variables. I find a Lagrangian that gives thedynamical Maxwell equations and include the constraint equations by using Lagrangemultipliers. In passing to the Hamiltonian one finds that the canonical momenta ~ Π E and ~ Π B are constrained giving 6 second class constraints at each point in space. Gauss’slaw and ~ ∇ · ~B = 0 can than be added in as additional constraints. There are now 8second class constraints, leaving 4 phase space degrees of freedom. The Dirac bracket isthen introduced and is calculated for the field variables and their conjugate momenta.1 Introduction
In the standard Lagrangian and Hamiltonian approach to Maxwell’s theory the poten-tials A µ are taken as the dynamical variables (see for example [1, 2]). The Lagrangian L = F µν F µν , where F µν = ∂ µ A ν − ∂ ν A µ , gives Gauss’s law and Maxwell’s version ofAmpere’s law without sources while, F [ µν,α ] = 0 gives Faraday’s law and ~ ∇ · ~B = 0.The canonical momenta are given by Π µ = F µ , so that one runs into a problem in theHamiltonian approach since Π = 0. This problem can be dealt with by using Dirac’sapproach [3, 4] for systems with constraints. The consistency of the primary constraintΠ = 0 implies a secondary constraint ∂ k Π k = 0. These two constraints are first classand together generate a set of transformations that includes the standard gauge trans-formations.In this paper I find a Lagrangian with ~E and ~B as the dynamical variables that givesthe dynamical Maxwell equations. The constraint equations can be included by addingthem to this Lagrangian using Lagrange multipliers. The canonical momenta ~ Π E and ~ Π B are constrained giving 6 second class constraints at each point in space. Gauss’slaw and ~ ∇ · ~B = 0 can than be added in as additional constraints. There are now 8second class constraints at each point in space leaving 4 phase space degrees of freedom.The Dirac bracket is then introduced and is calculated for the field variables and theirconjugate momenta. Maxwell’s equations can be divided into the dynamical equations (with c = 1) ~ ∇ × ~E = − ∂ ~B∂t ~ ∇ × ~B = ∂ ~E∂t + 4 πc ~j (1)and the constraint equations ~ ∇ · ~E = 4 πρ ~ ∇ · ~B = 0 . (2)In this section I will find a Lagrangian that involves ~E and ~B and their first derivativesthat gives the dynamical equations of motion. The constraint equations can then beincluded using Lagrange multipliers.Consider the Lagrangian L D = a ~E · ∂ ~B∂t + b ~B · ∂ ~E∂t + c ~E · (cid:16) ~ ∇ × ~B (cid:17) + d ~E · (cid:16) ~ ∇ × ~E (cid:17) + e ~B · (cid:16) ~ ∇ × ~B (cid:17) , (3)where a, b, c, d and e are constants. I have not included a term of the form ~B · (cid:16) ~ ∇ × ~E (cid:17) since ~B · (cid:16) ~ ∇ × ~E (cid:17) = ~E · (cid:16) ~ ∇ × ~B (cid:17) + ~ ∇ · ( ~E × ~B ). The Euler-Lagrange equations ( µ =2 , , , k = 1 , , ∂∂x µ ∂L D ∂ ( ∂ µ E k ) ! − ∂L D ∂E k = 0 (4)give ( b − a ) ∂ ~B∂t − c ( ~ ∇ × ~B ) − d ( ~ ∇ × ~E ) = 0 . (5)To obtain Faraday’s law we require that c = 0 and a − b = 2 d , (6)with a − b = 0. The Euler-Lagrange equations ∂∂x µ ∂L D ∂ ( ∂ µ B k ) ! − ∂L D ∂B k = 0 (7)give ( a − b ) ∂ ~E∂t − e ( ~ ∇ × ~B ) = 0 . (8)To obtain Ampere’s law without sources we require that a − b = 2 e . (9)From this we can also conclude that d = e = ( a − b ). Thus, the Lagrangian L D = f ~E · ~ ∇ × ~E + ∂ ~B∂t + ~B · ~ ∇ × ~B − ∂ ~E∂t (10)gives the dynamical Maxwell equations for arbitrary f = ( a − b ) = 0. I also used therelationship ~E · ∂ ~B∂t = − ~B · ∂ ~E∂t + ∂ ( ~E · ~B ) ∂t to derive (10). Throughout the rest of the paperI will take f = 1.Sources can be included by adding L S = − π~j · ~B (11)to L D .Now consider the constraint equations. One approach is to add them to to the dy-namical equations outside the action principle. Another approach is to include them inthe action with the use of Lagrange multipliers. However, as we shall see below, addi-tional constraints have to be imposed on the Lagrange multipliers to obtain Maxwell’stheory. In this approach L C = λ (cid:16) ~ ∇ · ~E − πρ (cid:17) + λ (cid:16) ~ ∇ · ~B (cid:17) (12)3s added to L D + L S , where λ ( ~x, t ) and λ ( ~x, t ) are Lagrange multipliers (or auxiliaryfields). The addition of this term modifies the dynamical equations of motion. Theybecome ~ ∇ × ~E = − ∂ ~B∂t + ~ ∇ λ (13)and ~ ∇ × ~B = ∂ ~E∂t + 4 π~j + ~ ∇ λ . (14)The additional terms involving λ and λ act as electric and magnetic current sourcesfor the fields. This approach yields Maxwell’s equations if and only if these sourcesvanish. Taking the divergence of both sides of these equations, and assuming chargeconservation, gives ∇ λ = ∇ λ = 0 . (15)These equations do not fix λ or λ , so some additional conditions are required. We arefree to choose conditions that set ~ ∇ λ = ~ ∇ λ = 0, so that the equations of motion areMaxwell’s equations. One possibility is to require that λ and λ vanish uniformly as r → ∞ . This implies that λ = λ = 0. Another possibility is to require that λ and λ are bounded everywhere. According to Liouville’s theorem any harmonic function χ ( ~x )that is bounded on all of R must be a constant. Since λ and λ are functions of ~x and t we will have λ = C ( t ) and λ = C ( t ) implying that ~ ∇ λ = ~ ∇ λ = 0.The properties of the Lagrangian and constraints under Lorentz transformations arediscussed in Appendix I. It is shown that the Lagrangian and constraints can be writtenas L α = α µ h F µν ∂ β G νβ − G µν ∂ β F νβ + 8 πG µβ j β i , (16) α µ [ ∂ α F αµ + j µ ] = 0 and α µ [ ∂ α G αµ ] = 0 , (17)where F µν is the electromagnetic field strength tensor, G µν = ǫ µναβ F αβ is its dual, j µ is the current density and α µ = (1 , ~ L α and the constraints giveMaxwell’s equations in all inertial frames for any constant time-like four vector α µ . Thisgives a manifestly covariant formulation of the theory.There have been previous papers that have discussed electrodynamics without po-tentials. The approaches taken in these papers differ from the approach taken here. In[5] Infeld and Plebanski formulate a Lagrangian approach to classical electrodynamicswithout the use of potentials. Their action is S = Z [Ω ( F ) + A α F αβ ; β ] √ gd x (18)where F = − F αβ F αβ and the A α are Lagrange multipliers. The field equations thatfollow from this action are F αβ ; β = 0 (19)and Ω F F αβ = A β,α − A α,β , (20)4here Ω F = d Ω dF . These are the field equations equations for non-linear electrodynamicsintroduced by Born [6] and by Born and Infeld [7]. Maxwell’s equations are obtainedwhen Ω = F . This approach is quite different from the one taken here. In the ap-proach taken by Infeld and Plebanski the field equations ∂ α F αµ = 0, which contain oneconstraint equation and and one (vector) dynamical equation, are enforced by Lagrangemultipliers. Also, from (20) it can be seen that the Lagrange multiplier A α is really thevector potential. In the approach taken here only the constraint equations are enforcedby Lagrange multipliers and there is no quantity which plays the role of the vectorpotential. The Hamiltonian formalism is also not investigated in [5].In [8] Fiutak and Zukowski take the action to be (in their notation) W = − Z f µν f µν d x (21)where f µν is required to satisfy the “equations of constraint” ∂ α f βγ + ∂ γ f αβ + ∂ β f γα = 0 . (22)They are, therefore, assuming half of Maxwell’s equations at this point (one constraintequation and and one vector dynamical equation). The action is varied with respect to f µν giving δ W = − Z f µν δ f µν d x = 0 (23)The δ f µν are not independent but must satisfy ∂ α δ f βγ + ∂ γ δ f αβ + ∂ β δ f γα = 0 . (24)These constraints are taken into account by taking δ f µν to be of the form δ f µν = ∂ µ ψ ν − ∂ ν ψ ν (25)Fiutak and Zukowski then state “Note that all we need is the form of the variation (25),and this implies that the electromagnetic potentials are not introduced.” (I have changedthe equation number to match the number here). This approach is also mentioned in[9]. In [10] Strazhev and Shkol’nikov develop classical and quantum theories of the elec-tromagnetic field without potentials. They start with the action (in their notation) W = − Z Φ µν d x where Φ µν = ∂ [ µ Z F | ρ | ν ] f ρ ( x − x ′ ) dx ′ , (26) f µ ( x ) = 12 Z ∞ [ δ ( x − ξ ) − δ ( x + ξ )] dξ µ (27)and ξ µ = ξ µ ( η ) is a spacelike path with ξ µ (0) = 0 and ξ µ ( η ) → + ∞ as η → + ∞ . Thedynamical variables are taken to be F µν ( x ) and they are taken to satisfy ∂ ν ˜ F µν = 05here ˜ F µν is the dual of F µν . They have, therefore, assumed half of Maxwell’s equationsby imposing this condition on ˜ F µν . They then add R λ µ ∂ ν ˜ F µν d x to the action and state“In order to avoid misunderstandings we stress that the Lagrange multipliers λ µ are notvaried, since the relations (5) are, by definition, known independently of the variationalprocedure. Hence the λ µ cannot be treated as auxiliary dynamical variables.” Here (5) isthe equation ∂ ν ˜ F µν = 0. The variation of the action with respect to F µν using ∂ ν ˜ F µν = 0does not give the remaining Maxwell equations but instead gives Z dx ′ [ ∂ ′ ν F αν ( x ′ )] f β ( x − x ′ ) − ∂ ′ ν F βν ( x ′ ) f α ( x − x ′ )] + 12 ǫ µναβ ∂ µ λ ν = 0 . (28)This appears to be very different than the remaining set of Maxwell equations: ∂ ν F µν =0. Maxwell’s equations can be obtained by differentiating (28) with respect to x β . How-ever (28) cannot be expected to be equivalent to Maxwell’s theory, since we need todifferentiate (28) to obtain Maxwell’s equations. For example, it is well known that onecan obtain wave equations for the electric and magnetic field by differentiating Maxwell’sequations. However, not all solutions to the wave equations will satisfy Maxwell’s equa-tions. The solution space of these wave equations is larger than that of Maxwell’sequations. The Hamiltonian formalism is not investigated in this paper.In [11] Mandelstam develops a quantum theory of the electromagnetic field coupled toa complex scalar field without potentials. He considers the electromagnetic field coupledto a complex scalar field and takes the gauge invariant variables Φ( x, P ), Φ ∗ ( x, P ) and F µν ( x ) are taken as the field variables whereΦ( x, P ) = φ ( x ) exp (cid:26) − ie Z x −∞ dξ µ A µ ( ξ ) (cid:27) (29)and the integral is taken over the spacelike path P . The Lagrangian is given by L = − ∂ µ Φ ∗ ∂ µ Φ − m Φ ∗ Φ − { F µν } (30)where ∂ µ Φ is the gauge invariant derivative of Φ. This approach differs significantlyfrom the approach taken in this paper.
In this section I will consider the free electromagnetic field (i.e. ρ = 0 and ~j = 0).The Hamiltonian for the dynamical equations will be derived first and the constraintequations will then be included. The canonical momenta are given by ~ Π E = ∂L D ∂ ˙ ~E = − ~B (31)and ~ Π B = ∂L D ∂ ˙ ~B = ~E. (32)6e therefore have the primary constraints ~φ = ~ Π E + ~B ≈ and ~φ = ~ Π B − ~E ≈ , (33)where ≈ denotes a weak equality which can only be imposed after the Poisson bracketshave been evaluated. These constraints satisfy { φ n ( ~x ) , φ m ( ~y ) } = { φ n ( ~x ) , φ m ( ~y ) } = 0 (34)and { φ n ( ~x ) , φ m ( ~y ) } = 2 δ mn δ ( ~x − ~y ) . (35)Thus, these constraints are second class.The canonical Hamiltonian H C = R d x [ ~ Π E · ˙ ~E + ~ Π B · ˙ ~B − L D ] is given by H C = Z d x (cid:20) ~φ · ˙ ~E + ~φ · ˙ ~B − ~E · (cid:16) ~ ∇ × ~E (cid:17) − ~B · (cid:16) ~ ∇ × ~B (cid:17)(cid:21) (36)and the total Hamiltonian is given by H T = − Z d x h ~E · (cid:16) ~ ∇ × ~E (cid:17) + ~B · (cid:16) ~ ∇ × ~B (cid:17) − ~u · ~φ − ~u · ~φ i , (37)where ~u and ~u are undetermined parameters.For consistency we require that˙ ~φ = { ~φ , H T } ≈ and ˙ ~φ = { ~φ , H T } ≈ . (38)These two equations give ~u = ~ ∇ × ~B (39)and ~u = − ~ ∇ × ~E . (40)Substituting these expressions for ~u and ~u into (37) gives H T = Z d x h ~ Π E · (cid:16) ~ ∇ × ~B (cid:17) − ~ Π B · (cid:16) ~ ∇ × ~E (cid:17)i . (41)It is easy to see that ˙ ~E = { ~E, H T } = ~ ∇ × ~B and ˙ ~B = { ~B, H T } = − ~ ∇ × ~E .The constraints φ = ~ ∇ · ~E ≈ and φ = ~ ∇ · ~B ≈ { φ ( x ) , ~φ ( y ) } = ~ ∇ x δ ( ~x − ~y ) , { φ ( x ) , ~φ ( y ) } = 0 , (43)7nd { φ ( x ) , ~φ ( y ) } = 0 , { φ ( x ) , ~φ ( y ) } = ~ ∇ x δ ( ~x − ~y ) . (44)the constraints { ~χ , ~χ , χ , χ } are, therefore, a set of second class constraints.Systems involving second class constraints can be treated using the Dirac bracket.Before introducing the Dirac bracket it will be convenient to relabel the constraints asfollows η = φ x , η = φ y , η = φ z , η = φ x (45)and η = φ y , η = φ z , η = φ , η = φ (46)and to define the matrix ∆ m~x,n~y = { η m ( ~x ) , η n ( ~y ) } . (47)The Dirac bracket is defined as { A, B } D = { A, B } − Σ m,n Z d xd y { A, η m ( ~x ) } C m~x,n~y { η n ( ~y ) , B } , (48)where C m~x ; n~y is the inverse of ∆ m~x,n~y :Σ k Z d z C m~x,k~z ∆ k~z,n~y = δ mn δ ( ~x − ~y ) . (49)The non-zero elements of C m~x,n~y are given by (see Appendix II) C ~x, ~y = − δ ( ~x − ~y ) + 18 π | ~x − ~y | − x − y ) π | ~x − ~y | , (50) C ~x, ~y = − δ ( ~x − ~y ) + 18 π | ~x − ~y | − x − y ) π | ~x − ~y | , (51) C ~x, ~y = − δ ( ~x − ~y ) + 18 π | ~x − ~y | − x − y ) π | ~x − ~y | , (52) C ~x, ~y = C ~x, ~y = − x − y )( x − y )8 π | ~x − ~y | , (53) C ~x, ~y = C ~x, ~y = − x − y )( x − y )8 π | ~x − ~y | , (54) C ~x, ~y = C ~x, ~y = − x − y )( x − y )8 π | ~x − ~y | , (55) C ~x, ~y = C ~x, ~y = ( x − y )4 π | ~x − ~y | , (56) C ~x, ~y = C ~x, ~y = ( x − y )4 π | ~x − ~y | , (57)8 ~x, ~y = C ~x, ~y = ( x − y )4 π | ~x − ~y | , (58) C ~x, ~y = 12 π | ~x − ~y | . (59)plus terms related by antisymmetry: C m~x ; n~y = − C n~y ; m~x . The advantage of using Diracbrackets instead of Poisson brackets is that the constraints can be set to zero strongly.The Dirac brackets involving ~E and ~B are { E i ( ~x ) , E j ( ~y ) } D = { B i ( ~x ) , B j ( ~y ) } D = 0 (60)and { E i ( ~x ) , Π Ej ( ~y ) } D = 12 " δ ( ~x − ~y ) δ ij + ∂ ∂x i ∂x j π | ~x − ~y | ! . (61)It is interesting to note that in the standard approach in the Coulomb gauge one finds[12] that A i ( ~x ) and Π j ( ~x ) satisfy the same Dirac bracket up to a factor of 1/2. The otherDirac brackets involving the momenta can be found from these using the constraints (i.e. { E i ( ~x ) , B j ( ~y ) } D = −{ E i ( ~x ) , Π Ej ( ~y ) } D ).The constraints ~ ∇ · ~E = 0 and ~ ∇ · ~B = 0 imply that the longitudinal components of ~E and ~B vanish. The constraints ~φ = 0 and ~φ = 0 then imply that the longitudinalcomponents of ~ Π E and ~ Π B vanish. The four independent degrees of freedom can thereforebe taken to be the transverse components of ~E and ~ Π E .One could also have started with L D + L C and obtained similar results with twoadditional constraints, Π λ ≈ λ ≈ In this paper I found a Lagrangian that gives the dynamical Maxwell equations andincluded the constraint equations using Lagrange multipliers. In the Hamiltonian for-malism one finds that the canonical momenta ~ Π E and ~ Π B are constrained giving 6 secondclass constraints at each point in space. Gauss’s law and ~ ∇ · ~B = 0 can than be addedin as additional constraints giving a total of 8 constraints. This leaves 4 independentdegrees of freedom, as expected. Acknowledgements
This research was supported by the Natural Sciences and Engineering Research Councilof Canada. 9
Appendix I
In this appendix I examine the transformation properties of L = L D + L S and the con-straints under Lorentz transformations and find covariant expressions for these quanti-ties. For notational simplicity define ~a = ∂ ~E∂t − ~ ∇ × ~B + 4 π~j (62)and ~b = ∂ ~B∂t + ~ ∇ × ~E (63)Now consider the constraints. They can be written as ~ ∇ · ~E − πρ = ∂ α F α + 4 πj = 0 (64)and ~ ∇ · ~B = ∂ α G α = 0 , (65)where F µν is the electromagnetic field strength tensor, G µν = ǫ µναβ F αβ is its dual and j µ is the current density. Consider the constraints in the inertial frame ¯ S . Under theLorentz transformation Λ νµ from ¯ S to the inertial frame S ¯ ∂ α ¯ F α + 4 π ¯ j = Λ µ ( ∂ α F αµ + 4 πj µ ) = 0 (66)and ¯ ∂ α ¯ G α = Λ µ ∂ α G αµ = 0 . (67)The constraints in S are thenΛ µ ( ∂ α F αµ + 4 πj µ ) = 0 and Λ µ ∂ α G αµ = 0 . (68)They can also be written as ~ ∇ · ~E − πρ + ~ Λ · ~a = 0 (69)and ~ ∇ · ~B + ~ Λ · ~b = 0 , (70)where ~ Λ k = Λ k Λ .The Lagrangian L = L D + L S can be written as L = F µ ∂ α G µα − G µ ∂ α F µα + 8 πG µ j µ . (71)Consider the Lagrangian ¯ L = ¯ L d + ¯ L S in the inertial frame ¯ S . Under the Lorentztransformation Λ νµ from ¯ S to the inertial frame S ¯ F µ ¯ ∂ α ¯ G µα − ¯ G µ ∂ α ¯ F µα + 8 π ¯ G µ ¯ j µ = Λ ν ( F νµ ∂ α G µα − G νµ ∂ α F µα + 8 πG νµ j µ ) . (72)10he Lagrangian in S is then given by. L = Λ µ ( F µν ∂ α G να − G µν ∂ α F να + 8 πG µν j ν ) . (73)Expressing this Lagrangian in terms of ~E and ~B gives L = Λ n ~b · ~E − (cid:16) ~a + 4 π~j (cid:17) · ~B + ~ Λ · h ~E ( ~ ∇ · ~B ) − ~B ( ~ ∇ · ~E ) + ~B × ~b + ~E × (cid:16) ~a + 4 π~j (cid:17) + 8 πρ ~B io . (74)The field equations, using the constraints (69) and (70), generated by this Lagrangianare ~a + ~ Λ × ~b − ( ~a · ~ Λ) ~ Λ = 0 (75)and ~b − ~ Λ × ~a − ( ~b · ~ Λ) ~ Λ = 0 . (76)Taking the inner product of both sides of these equations with ~ Λ gives(1 − Λ )( ~ Λ · ~a ) = 0 and (1 − Λ )( ~ Λ · ~b ) = 0 , (77)where Λ = ~ Λ · ~ Λ. Thus, for Λ = 1, which will be the case for Lorentz transformations,we have ~ Λ · ~a = ~ Λ · ~b = 0 Substituting (76) into (75) and using ~ Λ · ~a = ~ Λ · ~b = 0 gives ~a = ∂ ~E∂t − ~ ∇ × ~B + 4 π~j = 0 . (78)A similar argument gives ~b = ∂ ~B∂t + ~ ∇ × ~E = 0 . (79)In the above derivation of Maxwell’s equations I used | ~ Λ | 6 = 1, which is equivalent to − (Λ ) + Σ k Λ k Λ k = 0, and Λ = 0, both of which are true for Lorentz transforma-tions. It is easy to see that the Lagrangian and constraints will still produce Maxwell’sequations if Λ µ in (68) and (73) is replaced by any constant four-vector α µ satisfying α µ α µ = 0 and α = 0. The condition α µ α µ = 0 constrains α µ to be non-null. If α µ is a space-like four vector there exists an inertial frame in which α = 0. Therefore, ifthe Lagrangian and constraints are to produce Maxwell’s theory in all inertial frames ofreference α µ must be a time-like four-vector. Thus, the Lagrangian L α = α µ h F µν ∂ β G νβ − G µν ∂ β F νβ + 8 πG µβ j β i (80)and the constraints α µ [ ∂ β F βµ + 4 πj µ ] = 0 and α µ [ ∂ β G βµ ] = 0 (81)are Lorentz scalars and give Maxwell equations for any constant time-like four-vector α µ . For simplicity I have imposed the constraints outside the variational principle sothat the Lagrange multipliers do not appear in the equations for the fields.11 Appendix II
In this appendix I derive a few of the C i~x,j~y to illustrate how the calculations are done. Iwill assume that C n~x,n~y = 0 (no sum on n ) and that C m~x,n~y is a function of ( x − y , x − y , x − y ). These assumptions will be justified by the success of the calculations. Someof the equations for C i~x,j~y are differential equations, so boundary conditions need to beimposed. I will take these conditions to be C i~x,j~y → as | ~x − ~y | → ∞ . (82)Define P i~x,j~y by P m~x,n~y = Σ k Z d z C m~x,k~z ∆ k~z,n~y . (83)Consider m = 7 and n = 1 P ~x, ~y = C ~x, ~y = 0 (84)and m = 1 and n = 4 P ~x, ~y = − ∂∂y C ~x, ~y = 0 . (85)This implies that C ~x, ~y is independent of x − y . The boundary condition (82) thengives C ~x, ~y = 0.Now consider P ~x, ~y = − C ~x, ~y − ∂∂y C ~x, ~y = δ ( ~x − ~y ) , (86) P ~x, ~y = − C ~x, ~y − ∂∂y C ~x, ~y = 0 , (87) P ~x, ~y = − C ~x, ~y − ∂∂y C ~x, ~y = 0 , (88)and P ~x, ~y = − ∂∂y C ~x, ~y − ∂∂y C ~x, ~y − ∂∂y C ~x, ~y = 0 . (89)Substituting (86), (87) and (88) into (89) gives ∇ y C ~x, ~y = − ∂∂y δ ( ~x − ~y ) . (90)This equation has the solution C ~x, ~y = ( x − y )4 π | ~x − ~y | . (91)Equations (87) and (88) now give C ~x, ~y = − x − y )( x − y )8 π | ~x − ~y | (92)and C ~x, ~y = − x − y )( x − y )8 π | ~x − ~y | . (93)12 eferences [1] J.D. Jackson, Classical electrodynamics (3rd edition, Wiley, 1999).[2] L.D. Landau and E.M. Lifshitz, The classical theory of fields (Pergamon, Oxford,1962).[3] P.A.M. Dirac, Lectures on Quantum Mechanics (Dover Publications, Mineola, NewYork, 1964), pp 1-24.[4] P.A.M. Dirac, Can. J. Math. 2, 129 (1950).[5] L. Infeld and J. Plebanski, Proc. Roy. Soc. A222, 224 (1954)[6] M. Born, Proc. Roy. Soc. A143, 410 (1934)[7] M. Born and L. Infeld, Proc. Roy. Soc. A144, 425 (1934)[8] J. Fiutak and M. Zukowski, J. Phys. A14, 3229 (1981)[9] V. I. Ogievetskii and I. V. Polubarinov, Zh. Eksp. Teor. Fiz. 43, 1365 (1962), JETP16, 969 (1963)[10] V. I. Strazhev and P. L. Shkol’nikov, Sov. Phys. J. 32, 378 (1989)[11] S. Mandelstam, Annals Phys. 19, 1 (1962)[12] Steven Weinberg,