Energy-momentum tensor for the electromagnetic field in a dispersive medium as an application of Noether theorem
EEnergy-momentum tensor for the electromagneticfield in a dispersive medium
Carlos Heredia ∗ and Josep Llosa † Facultat de Física (FQA and ICC)Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Catalonia, SpainMarch 2, 2020
Abstract
On the basis of a non-local Lagrangian for Maxwell equations in a dispersive medium, the energy-momentum tensor of the field is derived. The general result is then specialized to the case of a fieldwith slowly varying amplitude on a rapidly oscillating carrier. As a side result, we develop thevariational methods to derive the field equations and prove the Noether theorem for a non-local La-grangian.
The electromagnetic field produced by a distribution of free charge and current in a material medium isruled by Maxwell equations ∇ · B = 0 , ∇ × E + ∂ t B = 0 (1)and ∇ · D = 4 π ρ , ∇ × H − ∂ t D = 4 π j (2)where E and H respectively are the electric and magnetic fields, D is the electric displacement, B ismagnetic induction and ρ and j are the free charge and current densities (unrationalized Gaussian unitswith c = 1 have been assumed). E and B are the physical magnitudes as they manifest in the Lorentzforce on a test charge. The above system (1-2) does not determine E and B if only the distributions ofcharge and current are known, because the number of unknowns largely exceeds the number of equa-tions. This hindrance is circumvented by specifying the nature of the material medium, i. e. givingthe constitutive equations, a set of phenomenological relations connecting D and H with the physicalvariables E and B .For isotropic non-dispersive linear media the constitutive equations are D = ε E and H = µ − B ,where ε and µ respectively are the dielectric and magnetic constants. This is the case most considered ∗ e-mail address: [email protected] † e-mail address: [email protected] a r X i v : . [ phy s i c s . c l a ss - ph ] F e b n textbooks [1] and also in the seminal paper [2] where Minkowski set up the relativistic electrody-namics in material media and, particularly, derived his (non-simetric) energy-momentum tensor for theelectromagnetic field. Vacuum is a particular case with ε = ε and µ = µ .However, in natural media ε and µ are not constant and generally depend on frequency and wave-length. We then speak of dispersive media. For a plane monochromatic wave, E ( q , ω ) e i ( q · x − ωt ) and B ( q , ω ) e i ( q · x − ωt ) , the response of the medium is a displacement D ( q , ω ) e i ( q · x − ωt ) and a magneticfield H ( q , ω ) e i ( q · x − ωt ) , with D ( q , ω ) = ε ( q , ω ) E ( q , ω ) , H ( q , ω ) = µ − ( q , ω ) B ( q , ω ) (3)By Fourier transform we express a general electromagnetic field, E ( x , t ) and B ( x , t ) as a superpositionof plane monochromatic waves, each of them producing an electric displacement and magnetic field like(3). By the convolution theorem [3], the superposition of all of them results in D = (2 π ) − ˜ ε ∗ E , H = (2 π ) − (cid:103) µ − ∗ B (4)where ˜ ε and (cid:103) µ − are connected with the Fourier transforms of ε and µ − , that is ˜ ε ( y ) = (2 π ) − (cid:90) d k ε ( k ) e ik b y b , ε ( k ) = (2 π ) − (cid:90) d y ˜ ε ( y ) e − ik b y b (5)where k b = ( q , ω ) .Complemented with these constitutive relations, the system (1-2) allows to determine the electro-magnetic fields, E and B —hence the Lorentz force on any test charge— provided that we know: (a) thedistribution of free charge and current, (b) the nature of the medium specified by the dielectric and mag-netic function, ε ( q , ω ) and µ ( q , ω ) , and (c) the suitable boundary conditions for such a partial differentialsystem.Without leaving the mathematical framework described so far we can modify the variables of ourproblem and their interpretation by replacing D and H by some new variables that describe the collectivebehavior of the elementary charges in the material medium, namely the polarization and magnetizationdensities [4] D = E + 4 π P , H = B − π M (6)Using this, the inhomogeneous pair of Maxwell equations (2) becomes ∇ · E = 4 π ( ρ + ρ b ) , ∇ × B − ∂ t E = 4 π ( j + j b ) (7)where ρ b = −∇ · P and j b = ∇ × M + ∂ t P (8)respectively, the density of bound charge and bound current (in contrast with the free charge distributions ρ and j ).For an isotropic dispersive medium the definitions (6) and the equations (4) imply that P = ˜ χ e ∗ E , M = ˜ χ m ∗ B (9)where ˜ χ e = 14 π [˜ ε − δ ( x ) δ ( t )] and ˜ χ m = 14 π (cid:104) δ ( x ) δ ( t ) − (cid:103) µ − − µ − (cid:105) E = E + E ind , B = B + B ind , i. e. the sum of two contributions: whereas E and B are the solution of Maxwell equations in vacuumfor the distribution of free charges and currents, E ind and B ind are the solution of Maxwell equationsin vacuum for the distribution of bound charges and currents. The interpretation that follows is: freecharges and currents produce the electromagnetic field, E and B , which polarizes the medium. Thispolarization implies a distribution of bound charges and currents which in turn produce the inducedelectromagnetic field, E ind and B ind . The latter is physically indistinguishable of the primary field andonly the total field, E and B , manifests trough the total Lorentz force on a test charge.As far as the resolution of Maxwell equations is concerned, this second view is not practical at all,however it will be helpful and illuminating to understand the exchange of energy and momentum betweenthe electromagnetic field and the electric charges, either free or bound.Poynting theorem [5] is about the energy exchange between the free charges and the electromagneticfield and it holds for non-dispersive media only. Its derivation follows from the scalar product of E andthe second equation (2). Then, a vector identity is invoked with the need of assuming that the dielectricand magnetic functions are constant. The quantity U = 18 π [ E · D + B · H ] and the Poynting vector S are respectively taken as the energy density of the electromagnetic field and the current density of energy.The main idea at the back of the theorem is that the increase of the field energy and the kinetic energy offree electric charges in a region in space is due to the energy flowing through its boundary. To obtain thelinear momentum balance, one can do similarly [5] although the procedure is much more elaborated.However the spacetime formalism introduced by Minkowski [2] is largely simpler. It treats the energyand momentum exchanges on the same foot and proceeds similarly as in the proof of Poynting theorem,combining some differential tensor identity and Maxwell equations.As previously mentioned, Poynting theorem does not hold for dispersive media. In our view this isdue to the fact that the energy-momentum balance should also include the energy and momentum storedin the “bound charges” which will depend, in the end, on the polarization and magnetization densities, P and M , and maybe on their derivatives as well.The plan of the paper is as follows. In Section 2 we outline the electrodynamics of Minkowski [2]for media with constant dielectric and magnetic functions. Furthermore, we use an action principle and aLagrangian for the electromagnetic field in the medium that, applying Noether theorem, yields a canon-ical conserved energy-momentum tensor. Finally, we find the associated Belinfante-Rosenfeld tensor—see for instance [15] and the outline in Appendix A3. The latter coincides with the non-symmetricenergy-momentum tensor obtained by Minkowski by merely elaborating from the field equations.In Section 3 we generalize Minkowski electrodynamics to dispersive media. This leads to a non-local Lagrangian density, i. e. it contains a convolution product whose value at the point x depends onthe values of the field at any point in spacetime. We then derive the field equations and apply Noethertheorem to obtain a conserved energy-momentum tensor.As we are aware that non-local Lagrangians are seldom found in textbooks, we devote AppendixA to outline the derivation of the field equations and Noether theorem for such a Lagrangian. Theprocedure is rather heuristic but it has been followed successfully in the past [7], [8], [9]. First thenon-local Lagrangian is converted into an infinite order Lagrangian —that depends on derivatives of the3eld of any order—, then the equations of motion and Noether theorem are derived [10] as though itwas an order- n Lagrangian but replacing n with infinite. The outcomes contain formal series that can besummed by the techniques displayed in Appendix B. In the spacetime formalism the coordinates in an inertial reference system are denoted as x = x , x = y , x = z , x = t ; (10)the electromagnetic field is represented by Faraday tensor F ab = − F ba , F = B , F = B , F = B , F i = E i (11)with a, b = 1 . . . and similarly the electric displacement D and the magnetic field H are representedby the displacement tensor H ab = − H ba , H = H , H = H , H = H , H i = − D i (12)Adopting the notation ∂ a = ∂∂x a , ∇ = ( ∂ , ∂ , ∂ ) , ∂ = ∂∂t , Maxwell equations (1-2) can be respectively written as ∂ a F bc + ∂ b F ca + ∂ c F ab = 0 and ∂ b H ab = J a (13)where J = j x , J = j y , J = j z , J = ρ is a 4-vector made of the free charge and current densities.This arrangement is specially suited to deal with coordinate transformations connecting two inertialsystems. Indeed, given two systems of coordinates ( x , x , x , x ) and ( x (cid:48) , x (cid:48) , x (cid:48) , x (cid:48) ) connectedby a Poincaré transformation x (cid:48) a = Λ ab x b + s b (where Λ ab is a Lorentz matrix and s b four constants),then we have that F (cid:48) ab = Λ ca Λ db F cd , H (cid:48) ab = Λ ac Λ bd H cd and J (cid:48) a = Λ ab J b (14)A relevant role is reserved to the Minkowski metric η ab = diag(1 1 1 − , and its inverse η ab = diag(1 1 1 − , such that η ac η cb = δ ab (the Einstein convention of summationover repeated indices is adopted). This is a version of Minkowski’s proposal in a notation closer to that used in present time textbooks [6] F ab = η ac F cb or F ab = η ac η bd F cd The first Maxwell equation (13) means that the Faraday tensor can be derived from a 4-potencial [6]it exists A a such that F ab = ∂ a A b − ∂ b A a (15)As pointed out before, the system (13) must be complemented with a set of constitutive relationsthat, for a non-dispersive homogeneous and isotropic medium, are D = ε E , H = µ − B However, as the Poincaré transformation (14) entangles the electric and magnetic parts, this simple formcannot be valid in all inertial frames but only in the proper inertial frame , i. e. that respect to which themedium is at rest.
Electric and magnetic fields.
An inertial reference system is characteritzed by its proper velocity withrespect to the laboratory frame; this is a 4-vector u a = ( γ u , γ ) , where γ = (1 − v ) − / and v is thestandard 3-velocity. It is a timelike unit vector, that is u a u a = u a u b η ab = − . Given any skewsymmetric tensor as F ab , they exist E b and B d such that F ab = 2 u [ a E b ] + ˆ F ab , ˆ F ab = ε abcd u c B d , u a E a = u d B d = 0 (16)where the square bracket means antisymmetrization and ε abcd is the totally skewsymmetric Levi-Civitasymbol in 4 dimensions: ε abcd = − if abcd is an even permutation of 1234 if abcd is an odd permutation of 1234 if there is some repeated indexIt can be easily checked that E a = F ab u b and B d = 12 ε cdab u c F ab (17)where we have used that u a is a unit vector and that ε abcd ε mned = − δ abcmne = (cid:88) σ sign( σ ) δ σ a m δ σ b n δ σ c e , σ runs over the permutation group S The particular case u a = (0 , , , corresponds to the laboratory frame and the relations (11) and(16) yield E a = ( E , E , E , and B a = ( B , B , B , . This is why we respectively call E a and B d the electric field and the magnetic induction in the reference frame characterized by u a .5e can proceed similarly with the skewsymmetric displacement tensor H ab and have H ab = 2 u [ a D b ] + ˆ H ab , ˆ H ab = ε abcd u c H d , u a D a = u d H d = 0 (18) D a = H ab u b and H d = 12 ε cdab u c H ab (19)where D a and H d respectively stand for the electric displacement and the magnetic field in the referenceframe characterized by the 4-velocity u a .As the constitutive relations only hold in the proper reference frame, we have that D a = ε E a and H d = 1 µ B d provided that u a is the 4-velocity of the medium. Now, using (17) and (19), this amounts to H ab u b = ε F ab u b and ε cdab u c H ab = 1 µ ε cdab u c F ab whence, after a little algebra, it follows that for an isotropic medium H ab = M abcd F cd with M abcd = µ − ˆ η a [ c ˆ η d ] b + 2 ε u [ a ˆ η b ][ c u d ] (20)where ˆ η ab = η ab + u a u b is the projector onto the hyperplane orthogonal to u b . The coefficients M abcd present the obvious symmetries M abcd = − M bacd = − M abdc = M cdab (21) By a simple algebraic manipulation of Maxwell equations (13) Minkowski obtained a local conservationlaw, namely ∂ b (cid:16) F ac H bc (cid:17) = − F ac J c + ∂ b F ac H bc = − F ac J c + 12 H bc ( ∂ b F ac + ∂ c F ab )= − F ac J c + 12 H bc ∂ a F bc = − F ac J c + 14 ∂ a (cid:16) H bc F bc (cid:17) (22)where we have used the identity H bc ∂ a F bc = M bcmn F mn ∂ a F bc = 12 ∂ a (cid:16) M bcmn F mn F bc (cid:17) = 12 ∂ a (cid:16) H bc F bc (cid:17) and the fact that M bcmn is constant and presents the symmetries (21).Therefore the tensor Θ ab = F ac H bc − η ab H mn F mn (23) The derivation is quite similar to the proof of Poynting theorem ∂ b Θ ab = − F ac J c (24)If there are no free charges, this relation becomes a local conservation law for the tensor Θ ab , which iscalled Minkowski energy-momentum tensor . It is generally non-symmetric, except if εµ = 1 , in whichcase the tensor coefficient M abcd in (20) does not depend on u a . This non-symmetry is at the origin ofthe so called Abraham-Minkowski controversy [11]Contrarily, if there are free charges, the Lorentz force on the charges contained in the elementaryvolume d x is the result of the energy-momentum current flowing into it through its boundary F ac J c = − ∂ b Θ ab . The finding of the Minkowski energy-momentum tensor for non-dispersive media involves a certainamount of good luck. We shall now present an alternative derivation which is based in a Lagrangianformulation of Minkowski electrodynamics [12],[13] and Noether theorem [10]. The methodology israther routine and involves very little creativity, which makes it appropriate for a further extension to thegeneral case of dispersive media, as the one we shall endeavour in Section 3.The configuration space variables are the 4-potencial components A b and the Lagrangian density inthe absence of free charges is L = 14 M abcd F ab F cd , F ab = A b ; a − A a ; b (25)where a ‘semi-colon’ ; means “partial derivative” and M abcd is constant for non-dispersive media andis given by (20).We shall need ∂ L ∂A a ; b = M abmn F nm = − H ab , ∂ L ∂A a = 0 and the field equations ∂ L ∂A a − ∂ b (cid:18) ∂ L ∂A a ; b (cid:19) = 0 are ∂ b H ab = 0 (26)As the Lagrangian (25) is invariant under spacetime translations, by Noether theorem —see Ap-pendix A2 for an outline— it has associated four conserved currents which conform the canonical energy-momentum tensor that, according to equation (87) in appendix A2 (with n = 1 ), is T ac = H ab A b ; c − H mn F mn δ ac , ∂ a T ac = 0 (27)Besides of being non-symmetric (when both indices are raised), it is gauge dependent due to the occur-rence of A b ; c in the first term.The angular momentum current that Noether theorem associates to infinitesimal Lorentz transforma-tions —see equation (88) in Appendix A2— is J bca = x [ c T ba ] + S bca where S bca = − A [ c H ba ] (28)7s the spin current. However, as L is not Lorentz invariant —because it contains the 4-vector u a throughthe dielectric tensor M abcd —, the angular momentum current is not locally conserved, ∂ b J bca (cid:54) = 0 .Then applying the symmetrization technique [6], [14] described in Appendix A3 —see equations(90) and (91)— we obtain the so called Belinfante-Rosenfeld energy-momentum tensor Θ ca = T ca + ∂ b W bac , where W cab = 12 (cid:16) S cab + S cba − S abc (cid:17) , (29)that is W cab = H ca A b and Θ ca = H ab F cb − η ca H mn F mn , which recovers the Minkowski energy-momentum tensor (23), [13], [12]. The fact that W bac = − W abc implies that the local conservation ∂ a Θ ba = 0 is a straight consequence of ∂ a T ba = 0 .Of course it is not symmetric, and it does not have to. Recall that the Belinfante tensor Θ ca issymmetric if the angular momentum current J bca is conserved [6], which would follow from Noethertheorem and the Lorentz invariance of the Lagrangian. Now the Lagrangian (25) is not Lorentz invari-ant, as commented above, because the dielectric tensor M abcd privileges the time vector u a , whichbreaks boost invariance. As a matter of fact the Lagrangian (25) is invariant under the Lorentz subgroupthat preserves, u a , and it can be easily checked that the part of Θ ca that is orthogonal to u b is indeedsymmetric. When dealing with homogeneous isotropic dispersive media, the simple constitutive relations for con-stant ε and µ must be replaced with the convolutions (4) or, in tensor spacetime form, the relations (20)are superseeded by H ab = ˜ M abcd ∗ F cd with ˜ M abcd = (2 π ) − (cid:104) ˜ m ( x ) ˆ η a [ c ˆ η d ] b + 2˜ ε ( x ) u [ a ˆ η b ][ c u d ] (cid:105) (30)where ˜ m and ˜ ε are the Fourier transforms of µ − and ε .Deriving the conservation equations for some energy-momentum current of the field in the Minkowskiway, as in Section 2.1, involves a trial and error game with an uncertain outcome. Alternatively we shallgo for an extension of the method applied in Section 2.2, namely (a) proposing a Lagrangian densityfrom which the field equations are derived, then (b) obtaining the canonical energy-momentum and an-gular momentum currents by application of Noether theorem and finally (c) applying the symetrizationtechnique [15], [6], [14] to derive a Belinfante-Rosenfeld energy-momentum tensor.The constitutive relations (30) are non-local and so are the field equations, therefore we postulate thenon-local action integral S = (cid:90) d x (cid:90) d y
14 ˜ M abcd ( x − y ) F ab ( x ) F cd ( y ) (31)where ˜ M abcd ( x ) has to present the symmetries ˜ M abcd ( − x ) = ˜ M cdab ( x ) and ˜ M abcd ( x ) = − ˜ M bacd ( x ) = − ˜ M abdc ( x ) (32)8s the tensor ˜ M abcd has the particular form (30), it already presents the symmetry ˜ M abcd = ˜ M cdab andtherefore ˜ m and ˜ ε must be even functions, ˜ m ( − x ) = ˜ m ( x ) and ˜ ε ( − x ) = ˜ ε ( x ) , (33)The action (31) includes the non-dispersive case which corresponds to ˜ M abcd ( x − y ) = M abcd δ ( x − y ) ,where M abcd is a constant tensor.The Lagrangian density for the non-local action is L = 14 F ab ( x ) (cid:90) d y ˜ M abed ( y ) F ed ( x − y ) = 14 F ab (cid:16) ˜ M abed ∗ F ed (cid:17) (34)which is obviously non-local, because L ( x ) depends on the field values A b ( x ) and A b ; a ( x ) and, due tothe convolution, it also depends on the values F ed ( y ) at any other point.A way of dealing with a non-local Lagrangian consists in transforming it into an infinite order La-grangian, by replacing F ed ( x − y ) in the integral (34) with its Taylor expansion around y = 0 F ed ( x − y ) = ∞ (cid:88) k =0 y c . . . y c k k ! ( − k F ed ; c ...c k (35)where F ed ; c ...c k = F ed ; c ...c k ( x ) is understood. Using this, the Lagrangian (34) becomes L = 14 F ab ∞ (cid:88) k =0 M ab ed | c ...c k F ed ; c ...c k (36)where the coefficients M ab ed | c ...c k = ( − k k ! (cid:90) d y M ab ed ( y ) y c . . . y c k = 1 k ! (cid:90) d y M ed ab ( y ) y c . . . y c k (37)have been introduced.The symmetry property (32) implies the skewsymmetry in each pair ab and ed , and also M ab ed | c ...c k = ( − k M ed ab | c ...c k (38)In terms of the vector field A a and its derivatives the Lagrangian (36) reads L = ∞ (cid:88) k =0 M abed | c ...c k A a ; b A e ; dc ...c k (39)Notice that, as A e ; dc ...c k is completely symmetric with respect to the indices dc . . . c k , only the fullsymmetrization of M abed | c ...c k with respect to these indices, which we shall denote by M abed | c ...c k ,will be relevant. This is an infinite order Lagrangian from which we shall derive the equations of mo-tion and Noether theorem following the methods outlined in Appendices A1 and A2 for an n -th orderLagrangian for the vector field A a and then replacing n with ∞ . Although it is true that this substitution has only a heuristic value, because F does not need to be full real analytic, themethod has proved really efficient in other instances: [7], [8] and [9] to quote a few .1 The field equations Notice that, while the term k = 0 is quadratic in A a ; b , the other terms, k > , depend linearly onthe higher order derivatives A m ; nc ...c k . To derive the field equations and conserved currents for theLagrangian (39), the canonical momenta (73) Π a | b = 2 M abcd | A c ; d + ∞ (cid:88) l =1 ( − l (cid:104) M cdab | c ...c l + M cdab | c ...c l (cid:105) A c ; dc ...c l (40)and Π a | bd ...d k = ∞ (cid:88) l =0 ( − l M feab | d ...d k c ...c l A f ; ec ...c l , k ≥ (41)will be a useful intermediate step (the symmetry relation (38) has been included). With the help of therelations (35) and (37), these two formal series for the canonical momenta are summed in Appendix Band converted into the integrals (95) and (97), which we can write as the unified expression for k ≥ a | bd ...d k = − (cid:90) d y ˜ M aecd ( y ) ∂∂y e (cid:90) d λ (1 − λ ) k k ! y b y d . . . y d k F cd ( x − λy ) − δ k ˜ M abcd ∗ F cd (42)As the Lagrangian (39) does not depend explicitly on A b , the field equation (75) reduces to − ∂ b Π a | b = 0 (43)with Π a | b defined by the integral (42) with k = 0 . Using this and the definition (30), the field equationbecomes − ∂ b H ab − (cid:90) d y ˜ M aecd ( y ) ∂∂y e (cid:90) d λ y b F cd ; b ( x − λy ) = 0 . (44)Now we can use that y b F cd ; b ( x − λy ) = − dd λ F cd ( x − λy ) to obtain (cid:90) d λ y b F cd ; b ( x − λy ) = − F cd ( x − y ) + F cd ( x ) which permits to transform the second term of equation (44) into − (cid:90) d y ˜ M aecd ( y ) F cd ; e ( x − y ) = − ∂ e (cid:104) ˜ M aecd ∗ F cd (cid:105) and, using the properties of the convolution, the field equation can be finally written as ∂ b H ab = 0 , where H ab = ˜ M abcd ∗ F cd (45)(the symmetry relation (32) is included). 10 .2 The canonical energy-momentum and angular momentum tensors As the Lagrangian (34) is invariant under spacetime translations, Noether theorem implies the localconservation of the canonical energy-momentum tensor —Appendix A2, eq. (87) replacing n with ∞ — T ab = ∞ (cid:88) k =0 Π d | c ...c k a A d ; c ...c k b − L δ ab (46)and, using the techniques displayed in Appendix B, the series can be summed to obtain T ab = 12 H ae A e ; b − δ ab F ef H ef − (cid:90) d y ˜ M efcd ( y ) ∂∂y f (cid:90) d λ y a F cd ( X − y ) A e ; b ( X ) (47)where X = x + y (1 − λ ) . Notice that, besides being non-symmetric, T ba is gauge dependent. Thepotential A a can be eliminated in the above expression by using the inverse of the relation F ed = A d ; e − A e ; d , indeee, by the Poincaré Lemma [16] we have that A b ( x ) = (cid:90) d τ τ x c F cb ( τ x ) + ∂ b f ( x ) (48)where f ( x ) is an arbitrary function that is related with gauge transformations. Substituting this in (53)we can split Θ ba in one part that only depends on F cd and is gauge independent, and another one thatdepends linearly on ∂ e f , i. e. a gauge dependent contribution. However it can be easily proved that bothparts are separately conserved an therefore we can take as the definition of Θ ba the gauge independentpart only.At the same time the angular momentum current is —Appendix A2, eq. (88)— J abc = 2 x [ c T ab ] + S abc , where the first term in the rhs is the orbital contribution and S bca is the intrinsic angular momentum(spin) current (89). The latter can be written as S bca = P [ bc ] a + Q [ bc ] a (49)where P bca = 2 ∞ (cid:88) k =0 Π c | c ...c k a A b ; c ...c k and Q bca = 2 ∞ (cid:88) k =0 k Π d | cc ...c k − a A bd ; c ...c k − which can be added using the techniques displayed in Appendix B to obtain P bca = − A b H ca − (cid:90) d y ˜ M cfed ( y ) ∂∂y f (cid:90) d λ y a F ed ( X − y ) A b ( X ) (50)and Q bca = − (cid:90) d y ˜ M lfed ( y ) ∂∂y f (cid:90) d λ (1 − λ ) y a y c F ed ( X − y ) A bl ; ( X ) (51)As the dielectric tensor (30), ˜ M abcd , privileges the timelike vector u b , the Lagrangian (34) that weare considering is translation invariant but is not Lorentz invariant. Therefore the conservation of thecanonical energy-momentum tensor follows from from Noether theorem but the conservation of the an-gular momentum tensor does not. As a result, the Belinfante-Rosenfeld tensor obtained by the techniquepresented in Appendix A3 may be non-symmetric.11 .3 The Belinfante-Rosenfeld tensor By applying the “symmetrization” techniques presented in Appendix A3 to the canonical tensor (47) andthe spin tensor (49) we obtain the Belinfante-Rosenfeld tensor Θ ba = T ba + ∂ c W cab , (52)where —see eq. (91)— W cab = 12 (cid:16) S cab + S cba − S abc (cid:17) or, using (49), W cab = 12 ∆ cabklm (cid:16) P klm + Q klm (cid:17) , with ∆ cabklm = δ c [ k δ al ] δ bm + δ c [ k δ bl ] δ am − δ a [ k δ bl ] δ cm which, including (50) and (51), leads to W cab = −
12 ∆ cabklm (cid:26) A k H lm − (cid:90) d y ˜ M hfed ; f ( y ) (cid:90) d λ F ed ( X − y ) y m (cid:104) δ lh A k ( X ) + (1 − λ ) y l A ; kh ( X ) (cid:105)(cid:27) Using the field equations (45) and some integration by parts, we obtain after some algebra that ∂ c W cab = − H ac A b ; c + 12 F ed (cid:20) ˜ M edh [ b ∗ F a ] h + ˜ M edh ( b ; a ) ∗ A h + 14 (cid:16) y b ˜ M edhf (cid:17) ∗ F ; ahf (cid:21) + 12 (cid:90) d y ˜ M hfed ; f ( y ) (cid:90) d λ (cid:26) y b F ed ( X − y ) A ; ah ( X ) − y ( b (cid:104) F ed ( X − y ) A a ) ( X ) (cid:105) ; h + y ( a δ b ) h (cid:104) F ed ( X − y ) A k ( X ) (cid:105) ; k + (1 − λ ) y a y b (cid:104) F ed ( X − y ) A ; kh ( X ) (cid:105) ; k (cid:27) , and combining this with (52) for the Belinfante-Rosenfeld tensor, we arrive at Θ ba = 12 H ca F bc − η ab F ed H ed + 12 F ed (cid:20) ˜ M edh [ b ∗ F a ] h + ˜ M edh ( b ; a ) ∗ A h + 12 (cid:16) y b ˜ M edhf (cid:17) ∗ F ; ahf (cid:21) − (cid:90) d y ˜ M hfed ( y ) ∂∂y f (cid:90) d λ y ( a (cid:110) F ed ( X − y ) (cid:104) A ; b ) h ( X ) + F b ) h ( X ) (cid:105) − F ed ; h ( X − y ) A b ) ( X ) + δ b ) h (cid:104) F ed ( X − y ) A k ( X ) (cid:105) ; k + (1 − λ ) y b ) (cid:104) F ed ( X − y ) A ; kh ( X ) (cid:105) ; k (cid:27) (53)where X = x + (1 − λ ) y and the potentials are to be eliminated by the expression (48). The non-dispersive case
The Lagrangian for a non-dispersive medium is a particular case of (34), with ˜ M abcd ( y ) = M abcd δ ( y ) and M abcd = constant . With this choice the expressions (30), (34), (47), (49) and (53) respectivelyyield H ab = M abcd F cd , L = 14 F ab M abcd F cd , T ab = A e ; b M aecd F cd − F ab M abcd F cd bca = − A [ b H c ] a and W cab = H ca A b , and therefore Θ ba = H ca F bc − η ab F ed H ed (54)that is, the energy-momentum tensor of Minkowski electrodynamics. We now consider plane wave solutions, namely F cd = f cd e ik b x b with f cd − f dc that, substituted intoequation (45), yields M abcd ( k ) f cd k b = 0 (55)where M abcd ( k ) is the Fourier transform of ˜ M abcd ( k ) . Now in order that F cd is derivable from anelectromagnetic potential, it must fulfill the first pair of Maxwell equations which for plane waves reads k b f cd + k c f db + k d f bc = 0 , whose general solution is f cd = f c k d − f d k c (56)where f c is the wave polarization vector and admits the addition of any multiple of k c . Substituting thisinto equation (55), we arrive at M abcd ( k ) k b k d f c = 0 (57)This is a linear homogeneous system and admits non-trivial solutions for the polarization vector if, andonly if, det (cid:2) M abcd ( k ) k b k d (cid:3) = 0 .We shall assume that the dielectric tensor ˜ M abcd has the form (30), hence its Fourier transform is M abcd ( k ) = m ( k ) ˆ η a [ c ˆ η d ] b + 2 ε ( k ) u [ a ˆ η b ][ c u d ] (58)Due to the isotropy, ε ( k ) and m ( k ) = µ − ( k ) depend on the wave vector k b through the scalars ω = − k b u b and q := k b k b + ω , where we have taken k a = ωu a + q ˆ q a , with ˆ q a ˆ q a = 1 , ˆ q a u a = 0 Furthermore, from the symmetry property (32) and the fact that ˜ M caed ( y ) is real it follows that M caed ( k ) = M edca ( − k ) = M edca ∗ ( k ) and therefore m ( k ) = m ( − k ) = m ∗ ( k ) , ε ( k ) = ε ( − k ) = ε ∗ ( k ) (59)This is rather a limitation of the Lagrangian model adopted here because physical refractive indices havean imaginary part that accounts for the absortive properties of the medium.In case that ω (cid:54) = 0 , as f a is determined except for a multiple of k a , it can be chosen so that f b u b = 0 .Thus equation (57) implies that f b k b = 0 and (cid:0) q − ω n (cid:1) f ⊥ a = 0 (60)where n = √ εµ is the refractive index.If on the contrary ω = 0 , the projection of equation (57) leads to q f b u b = 0 that, substituted inequation (57), implies that f a is proportional to k a and therefore equivalent to f a = 0 , that is the trivialsolution.Maxwell equations thus imply that: (a) the waves are transverse to the plane spanned by k b and u b ,with an arbitray polarization and (b) the phase velocity satisfies the dispersion relation q = ω n .13 Real dispersive media: absorption and causality
The action integral (31) implies that the dielectric and magnetic functions present the symmetry (33), i. e.are even functions. Hence their Fourier transforms ε and µ are real valued and so it must be the refractiveindex n as well. Now this is too restrictive because it excludes absorptive media, for the damping ofenergy by absorption is connected with I m( n ) . Moreover, in the optical regime, ε and µ only depend onthe angular frequency ω and causality implies that the real and imaginary parts of each of these functionsare related by the Kramers-Krönig relations [17]. As a consequence of these, if ε ( ω ) and µ ( ω ) are realvaluated, then they must be constant and the medium is trivially non-dispersive.However, the definitions for energy-momentum tensor presented here can still be useful for electro-magnetic field in a real linear medium, in which the Maxwell equations (45) ∂ b H ab = 0 , where H ab = M abcd ∗ F cd are fulfilled but the parity condition (33) is not. By a simple calculation that uses the Maxwell equationsand that y b ∂ b [ F cd ( x − λy ) A e ; b ( X )] = − dd λ [ F cd ( x − λy ) A e ; b ( X )] , we obtain that the 4-divergencesof both energy-momentum tensors, the canonical one (47) and the Belinfante-Rosenfeld tensor (53) is ∂ a T ab = ∂ a Θ ab = − F cd ˜ M cdef − ∗ F ef ; b where ˜ M cdef − = (2 π ) − (cid:104) ˜ m − ˆ η a [ c ˆ η d ] b + 2˜ ε − u [ a ˆ η b ][ c u d ] (cid:105) with ˜ m − ( y ) = ˜ m ( y ) − ˜ m ( − y )2 and similarly for ˜ ε − ( x ) .Now, using (16) to separate the electric and magnetic parts, we arrive at ∂ a T ab = (2 π ) − (cid:20) E d (cid:16) ˜ ε − ∗ E d (cid:17) ; b − B d (cid:16) ˜ m − ∗ B d (cid:17) ; b (cid:21) and, as the Fourier transforms of ˜ m − ( x ) and ˜ ε − ( x ) are connected to the absorptive parts of the magneticand dielectric functions, we have that the failure of local conservation of energy-momentum in a realmedium is due to absorption. Aiming to compare the energy-momentum tensor obtained here with other proposals that can be foundin the literature, e. g. [18], [19] and [20], we particularize the expression (53) for the Belinfante tensorfor a wave packet as F cd ( x ) = R e (cid:16) F cd ( x ) e ik a x a (cid:17) = | F cd ( x ) | cos ( k a x a + ϕ cd ) , k a = q a + ωu a (61)where F cd ( x ) is a “slowly” varying amplitude (if compared with the rapidly oscillating carrier e ik c x c ). As T ba and Θ ba are related by (52) it does not matter which one is used ε and µ only depend on the fre-quence ω = − u a k a , and we shall take ˜ M abcd ( y ) = (2 π ) − / δ ( y ) ˜ m abcd ( τ ) with ˜ m abcd = ˜ m ( τ ) ˆ η a [ c ˆ η d ] b + 2˜ ε ( τ ) u [ a ˆ η b ][ c u d ] (62)and y a = y a − τ u a is the spatial part of y a .Thus the displacement tensor (30) is H ca ( x ) = R e (cid:16) H ca ( x ) e ik b x b (cid:17) with H ca ( x ) ≈ m caed F ed ( x ) (63)where ≈ means that the “slow variation” approximation has been included to evaluate the integral (2 π ) − / (cid:90) d τ ˜ m caed ( τ ) F ed ( x b − τ u b ) e ik b x b + iωτ ≈ ˜ m caed ( τ ) F ed ( x ) e ik b x b Using (63), the Maxwell equations lead to D c H ca ≈ D b F cd + D c F db + D d F bc = 0 where D b = ∂ b + ik b . For slowly varying amplitudes they reduce to the Maxwell equations for a planewave and we can write F cd ≈ − ω ( E c k d − E d k c ) and m caed k c E e k d ≈ (64)where E c = F cd u d is the electric field. Similarly as in Section 3.4, the second equation implies that E c k c = 0 and q = ω ε ( ω ) µ ( ω ) (65)Moreover, from (61) and (64) it follows that the electromagnetic potential is A b ( x ) = R e (cid:16) A b ( x ) e ik c x c (cid:17) with A c ≈ − iω E c + α k c , (66)where α ( x ) is a gauge arbitrary function.If we now substitute the wave packet (61) in the expression (53) for the Belinfante tensor, we findthat: • The evaluation of the convolution products in the first line yields ˜ M edh [ b ∗ F a ] h ≈ R e (cid:16) m edh [ b ( ω ) F a ] h e ik c x c (cid:17) , ˜ M edh ( b ; a ) ∗ A h ≈ R e (cid:16) m edh ( b ( ω ) D a ) A h e ik c x c (cid:17) and (cid:16) y b ˜ M edhf (cid:17) ∗ F ; ahf ≈ R e (cid:16) − i D a F hf m (cid:48) edhf ( ω ) (cid:17) where a “prime” means derivative with respect to ω .15 Each term in the first line of (53) has the form
Φ Ψ = R e (cid:16) Φ e ik c x c (cid:17) R e (cid:16) Ψ e ik c x c (cid:17) = 12 R e (cid:16) Φ Ψ e ik c x c + Φ Ψ ∗ (cid:17) (67)and consists of a slowly varying part plus a rapidly oscillating one. Taking the average over aperiod of the carrier we obtain (cid:104) Φ Ψ (cid:105) = 12 R e (Φ Ψ ∗ ) • To evaluate the integral in the second and third lines in (53) we realize that each term containsa group of the kind of (67) and a Fourier integral of ˜ m hfed ( τ ) . We shall use the slow variationapproximation and take the mean over a carrier period.A tedious calculation leads to (cid:104) Θ ba (cid:105) ≈ R e (cid:20) H ca F ∗ bc − H c [ a F ∗ b ] c − H cd F ∗ cd η ab + F ∗ ed m edh [ b F a ] h + i m edh ( a k b ) ( F ∗ ed A h − F ed A ∗ h ) − u a k b F ed m (cid:48) edhf F ∗ hf which, using the relations (64-66), can be simplified to (cid:104) Θ ba (cid:105) ≈ R e (cid:20) H ca F ∗ bc − H cd F ∗ cd η ab − u a k b F ed m (cid:48) edhf F ∗ hf (cid:21) (68)From the latter we easily obtain the energy density in the medium rest frame ( u a = δ a ) and it yields U = (cid:104) Θ ≈ R e (cid:2) ( ε + ωε (cid:48) ) E · E ∗ + ( m − ωm (cid:48) ) B · B ∗ (cid:3) , where E · E ∗ = E a E ∗ a , or U ≈ R e (cid:20) d( εω )d ω E · E ∗ + µ ∗ µ d( µω )d ω H · H ∗ (cid:21) (69)which reproduces previous results in the literature —see [18], [19] and [20]-The momentum density in the medium rest frame is G i = (cid:104) Θ i (cid:105) and, in an obvious vector notation,we obtain from (68) that G ≈ R e (cid:20) ε E × B ∗ + 12 (cid:0) ε (cid:48) E · E ∗ − m (cid:48) B · B ∗ (cid:1) q (cid:21) Now, from the Maxwell equations (64) it follows that E × B ∗ = E · E ∗ ω q and B · B ∗ = εµ E · E ∗ This is the physically meaningful quantity as far as the carrier period lasts much less than an actual measurement G ≈ R e (cid:20) ωµ d( εµω )d ω E × B ∗ (cid:21) (70)The Poynting vector is S i = (cid:104) Θ i (cid:105) ≈ R e [ E ∗ × H ∗ ] and the Maxwell stress tensor is T ij = −(cid:104) Θ ij (cid:105) ≈ R e (cid:20) E ∗ i D j + H i B ∗ j −
12 ( D · E ∗ + H · B ∗ ) δ ij (cid:21) We have tackled the derivation of an energy-momentum tensor for electromagnetic field in a linear,isotropic, homogeneous dispersive medium. Our set up is based on a quadratic Lagrangian for the elec-tromagnetic field. Due to dispersivity this Lagrangian must be non-local or, equivalently, of infinite order,i. e. it must depend on derivatives of the field at any order. In the non-dispersive limit, the Lagrangianbecomes local and first order, and Minkowski theory[2] is recovered.Homogeneity implies that the Lagrangian is invariant by spacetime translations. Hence the conser-vation of some energy-momentum current must follow from an eventual Noether theorem for non-localLagrangians. As we are aware that this subject is not currently found in textbooks, we have devotedAppendix A to outline the derivation of both the field Euler-Lagrange equations and Noether theoremfor an infinite order Lagrangian.As a result we have obtained an explicit expression for the canonical energy-momentum tensor T ab which depends quadratically and non-locally on the Faraday tensor and its first order derivatives. In thenon-dispersive limit this tensor does not coincide with the Minkowski energy-momentum tensor; the dif-ference is the 4-divergence of an antisymmetric tensor of order three. We have derived this correction byapplying the Belinfante-Rosenfeld technique[6] and obtained an energy-momentum tensor Θ ab whichin the non-dispersive limit does reduce to Minkowski tensor. In general the tensor Θ ab is not symmetric,as Minkowski tensor is not either. This is due to the fact that the angular momentum current is not con-served because the Lagrangian is not Lorentz invariant, as it should be expected since the rest referencesystem of the medium has a privileged position.It must be said that our model has the disadvantage that its scope is restricted to non-absorptivemedia. Indeed, the action (31) implies the symmetry conditions (32) and (59), whence it follows that ε ( ω, k ) is real for real ω and k , and it must be recalled that the absorptive behavior of a medium isconnected with the imaginary part of its dielectric function ε . Moreover, if this imaginary part vanishes,it follows from Kramers-Krönig relations that ε and µ must be constant.To avoid this drawback we keep the definition (53) for the canonical energy-momentum tensor Θ ba and evaluate its 4-divergence provided that the field equations (45) are fulfilled. We than find that itis not locally conserved and that local conservation fails due to I m( ε ) and I m( µ ) , i. e. the absorptivecomponents.We have then specialized our Belinfante-Rosenfeld energy-momentum tensor for an electromagneticfield of slowly varying amplitude over a rapidly oscillating carrier wave for a medium in the optical17pproximation —that is ε and µ only depend on the frequency ω . Taking the average over one periodof the carrier and using the slow motion approximation we have evaluated the energy and momentumdensities, the Poynting vector and the Maxwell stress tensor in the rest reference frame. We have foundthen compared the energy density with the approximated formulae given in some textbooks [18], [19],[20]. Acknowledgment
Funding for this work was partially provided by the Spanish MINCIU and ERDF (project ref. RTI2018-098117-B-C22).
Appendix A. Vector field ruled by a higher order Lagrangian
A1: The field equations and the action principle
Consider a Lagrangian for a vector field in Minkowski spacetime, A a ( x ) , a = 1 . . . , that depends onthe field derivatives up to the n -th order; the action is S = (cid:90) V L ( A a , A a ; b . . . A a ; b ...b n ) d x (the subindices after a semicolon mean partial derivatives) and the field equations follow from δS = 0 for field variations δA a such that δA a ; c ...c k = 0 , ≤ k < n on the boundary of V , (71)The vanishing of the action variation reads (cid:90) V n (cid:88) k =0 ∂ L ∂A a ; c ...c k δA a ; c ...c k d x which, after several integrations by parts, can be written as (cid:90) V (cid:40) ∂ b (cid:34) n − (cid:88) k =0 Π a | c ...c k b δA a ; c ...c k (cid:35) + Π a δA a (cid:41) d x = 0 , (72)where Π a | c ...c k , k = 0 , . . . n , are the canonical momenta (the stroke | is meant to separate theindices corresponding to the field component from those corresponding to partial derivatives) and aredefined by the descending algorithm Π a | c ...c k = − ∂ b Π a | c ...c k b + ∂ L ∂A a ; c ...c k , n ≥ k ≥ , starting from Π a | c ...c n +1 = 0 , that is Π a | c ...c k = n − k (cid:88) l =0 ( − l ∂ b ...b l (cid:18) ∂ L ∂A a ; b ...b l c ...c k (cid:19) , ≤ k ≤ n (73)18he first part of the integrand in (72) is a divergence that, by Gauss theorem, can be reduced to an integralon the boundary of V , which vanishes due to the variation condition (71) on the boundary. Therefore(72) reduces to (cid:90) V Π a δA a d x (74)for all variations fulfilling (71), whence the field equation follows Π a = 0 , with Π a ≡ n (cid:88) k =0 ( − k ∂ b ...b k (cid:18) ∂ L ∂A a ; b ...b k (cid:19) (75) A2: Noether theorem
Now assume that the Lagrangian is invariant by Poincaré transformations L ( A a , A a ; b . . . A a ; b ...b n ) = L (cid:0) A (cid:48) a , A (cid:48) a ; b . . . A (cid:48) a ; b ...b n (cid:1) (76)Infinitesimal Poincaré transformations act on coordinates as x (cid:48) a = x a + δx a , δx a = ε a + ω ab x b , ω ab + ω ba = 0 (77)( ε a and ω ab are constants) while the vector field transforms as A (cid:48) a ( x (cid:48) ) = A a ( x ) − ω ba A b ( x ) . (78)To derive the transformation law for the field derivatives we must recall that x c = x (cid:48) c − δx c , apart fromsecond order terms, and the Jacobian matrix is ∂x c ∂x (cid:48) b = δ cb − δx c ; b (79)Now defining δA a ( x ) = A (cid:48) a ( x ) − A a ( x ) and including (77) and (78) we have that δA a = − ω ba A b − A a ; c δx c , δA a ; c = − ω ba A b ; c − (cid:16) A a ; b δx b (cid:17) ; c (80)and therefore δA a ; c ...c k = − ω bd (cid:0) η da A b ; c ...c k + η dc A a ; b...c k + . . . + η dc k A a ; c ...c k − b (cid:1) − δx b A a ; c ...c k b (81)By a Poincaré transformation the integration volume V transforms into V (cid:48) and, as the Lagrangian isassumed invariant, S = (cid:90) V L ( A a ( x ) , A a ; b ( x ) . . . A a ; b ...b n ( x )) d x = (cid:90) V (cid:48) L (cid:0) A (cid:48) a ( x (cid:48) ) , A (cid:48) a ; b ( x (cid:48) ) . . . A (cid:48) a ; b ...b n ( x (cid:48) ) (cid:1) d x (cid:48) (82)By substracting both expressions and writing x instead of x (cid:48) in the segond integral (the integration vari-able is a dummy one), we arrive at (cid:90) V L ( A a ( x ) , A a ; b ( x ) . . . A a ; b ...b n ( x )) d x − (cid:90) V (cid:48) L (cid:0) A (cid:48) a ( x ) , A (cid:48) a ; b ( x ) . . . A (cid:48) a ; b ...b n ( x ) (cid:1) d x = 0 As depicted in the figure, the volumes V and V (cid:48) differ very little: they share a large common part V and differ in an infinitesimal part near the boundary ∂ V For our immediate interest we shall restrict to the case of Poincaré invariance d x = dΣ a δx a , where dΣ a is the hyper-surface element on the boundary. Hence the variation of the action can be written as (cid:90) V (cid:2) L (cid:0) A (cid:48) a ( x ) , A (cid:48) a ; b ( x ) . . . A (cid:48) a ; b ...b n ( x ) (cid:1) − L ( A a ( x ) , A a ; b ( x ) . . . A a ; b ...b n ( x )) (cid:3) d x + (cid:90) ∂ V L δx a dΣ a = 0 (83)Following a similar scheme as in the derivation of the field equations and recalling eqs. (80) and (81),we obtain that the latter equation amounts to (cid:90) V (cid:104) ∂ b J b + Π a δA a (cid:105) d x = 0 (84)with − J b = Π a | b δA a + Π a | cb δA a ; c + . . . + Π a | c ...c n − b δA a ; c ...c n − + L δx b (85)Now, provided that A a is a solution of field equations, we have that Π a = 0 and (84) becomes (cid:82) V ∂ b J b d x = 0 , for any spacetime region V , whence the local conservation of the current J b follows ∂ b J b = 0 Substituting (81) in (85) and using (77) we have that the conserved current can be written as J a = ε b T ab + 12 ω bc J abc (86)where the factors of ε b and ω bc are respectively the canonical energy-momentum tensor T ab = n − (cid:88) k =0 Π d | c ...c k a A d ; c ...c k b − L δ ab (87)and the angular momentum current J abc = 2 x [ c T ab ] + S abc , (88)where the first term in the rhs is the orbital contribution and S abc = ∞ (cid:88) k =0 Π d | c ...c k a (cid:0) η [ cd A b ]; c ...c k + η [ cc A d ; b ] ...c k + . . . + η [ cc n A d ; c ...c k − b ] (cid:1)
20s the intrinsic angular momentum (spin) contribution which, using the symmetry of Π d | c ...c k with re-spect to the indices on the right of the ‘stroke”, can be written as S abc = 2 ∞ (cid:88) k =0 (cid:104) Π | c ...c k a [ c A b ]; c ...c k + k Π d | c ...c k − a [ c A d ; b ] c ...c k − (cid:105) (89)The local conservation of the current (86) for all values of ε b and ω bc implies that both parts of J a are separately conserved. For the energy-momentum tensor we have that ∂ a T ab = 0 and, using this, the angular momentum conservation implies that T [ bc ] + 12 S abc = 0 Therfore, if the spin current does not vanish, the canonical energy-momentum tensor is not symmetric.
A3: The Belinfante-Rosenfeld tensor
In case that the energy-momentum tensor is not symmetric, there is a technique —see e.g. [6] and [14] toquote a few—that permits to construct the Belinfante-Rosenfeld energy-momentum tensor Θ ca which issymmetric and is in some sense “equivalent” to T ca because: (a) the total energy-momentum containedin a hyperplane t = constant is the same for both tensors (cid:90) d x Θ b ( x , t ) = (cid:90) d x T b ( x , t ) (b) the 4-divergences are equal too, ∂ a Θ ab = ∂ a T ab = 0 and the current Θ ac is also conserved, and(c) the new spin current Σ bca = J bca − x [ a Θ bc ] and the new orbital angular momentum current x [ a Θ bc ] are separately preserved. This is achieved by defining Θ ca = T ca + ∂ b W bac where W bac = −W abc (90)and W cab = 12 (cid:16) S cab + S cba − S abc (cid:17) , (91)Recalling the expression (89), the latter tensor can be written as W bca = P ( ab ) c − P ( ac ) b + P [ cb ] a + 2 Q [ cb ] a (92)with P bca = n (cid:88) k =0 Π c | ad ...d k A b ; d ...d k , Q bca = n (cid:88) k =0 k Π d | acd ...d k − A bd ; d ...d k − (93)21 ppendix B: The sum of some useful series The series in the momenta
To sum the series in Π a | b we rewrite (40) as Π a | b = − M abcd | F cd − ∞ (cid:88) l =1 ( − l F cd ; c ...c l (cid:20) l + 2 l + 1 M cdab | c ...c l + ll + 1 M cdac | bc ...c l (cid:21) (94)Now, including (35), (37) and (38), the first part of the series in the rhs is ∞ (cid:88) l =1 ( − l F dc ; c ...c l l + 2 l + 1 (cid:90) d y ˜ M abcd ( y ) y c . . . y c l l ! == (cid:90) d y ˜ M abcd ( y ) ∞ (cid:88) l =1 F dc ; c ...c l ( − l y c . . . y c l l ! (cid:18) l + 1 (cid:19) = (cid:90) d y ˜ M abcd ( y ) (cid:34) ∞ (cid:88) l =1 F dc ; c ...c l ( − l y c . . . y c l l ! + ∞ (cid:88) l =1 F dc ; c ...c l ( − l y c . . . y c l ( l + 1)! (cid:35) = (cid:90) d y ˜ M abcd ( y ) (cid:20) F dc ( x − y ) + (cid:90) d λ F dc ( x − λy ) − F dc ( x ) (cid:21) where the equality l + 1)! = 1 l ! (cid:90) d λ λ l , has been used. Similarly, for the second part in the series inthe rhs of (94), we have ∞ (cid:88) l =1 ( − l l ( l + 1)! F dc ; c ...c l (cid:90) d y ˜ M ac cd ( y ) y b y c . . . y c l == (cid:90) d y ˜ M ac cd ( y ) y b ∞ (cid:88) k =0 F dc ; c d ...d k (cid:90) d λ λ k +1 ( − k +1 y d . . . y d k k != − (cid:90) d y ˜ M aecd ( y ) (cid:90) d λ λ y b F dc ; e ( x − λy ) where the equality l ( l + 1)! = 1( l − (cid:90) d λ λ l and k = l − , has been considered. Substituting theseintermediate results in (94) we arrive at Π a | b ( x ) = −
12 ˜ M abcd ∗ F cd − (cid:90) d y ˜ M aecd ( y ) ∂∂y e (cid:90) d λ y b F cd ( x − λy ) (95)To sum the series for Π a | bd ...d k we write (41) as Π a | bd ...d k = − ∞ (cid:88) l =0 ( − l M ceab | d ...d k c ...c l F ce ; c ...c l (96)22ince after the “stroke” | everything is symmetric, we have that M ceab | d ...d k c ...c l = 1 l + k + 1 (cid:104) M ceab | d ...d k c ...c l + k M cead | bd ...d k c ...c l + l M ceac | d ...d k bc ...c l (cid:105) (symmetrization over the underlined indices is assumed). Each one of the three terms contributes to theseries (96). Using (37) and (38), we find that the contribution of the first term is − ∞ (cid:88) l =0 ( − k ( l + k + 1)! (cid:90) d y ˜ M ceab ( y ) y d ...d k y c ...c l F ce ; c ...c l which can be easily converted into − (cid:90) d y ˜ M abce ( y ) y d ...d k ( − l ∞ (cid:88) l =0 y c ...c l ( l + k + 1)! F ce ; c ...c l Including now that l + k + 1)! = 1 l ! k ! (cid:90) d λ λ l (1 − λ ) k , we can sum the series to obtain − (cid:90) d y (cid:90) d λ ˜ M abce ( y ) y d ...d k (1 − λ ) k k ! F ce ( x − λy ) We can similarly find the sum of the other two contributions and, adding all three terms together, weeasily arrive at ( k > ) Π a | bd ...d k = − (cid:90) d y ˜ M aecd ( y ) (cid:90) d λ (1 − λ ) k k ! ∂∂y e (cid:104) y b y d . . . y d k F cd ( x − λy ) (cid:105) (97) Some sums in the canonic energy-momentum and Belinfante tensors (87-92)
For the canonical energy-momentum tensor we have to calculate ∞ (cid:88) k =0 Π e | ad ...d k A e ; bd ...d k that using (97)yields − (cid:90) d y M efmn ( y ) (cid:26) δ af F mn ( x − y ) A e ; b ( x ) + (cid:90) d λ ∂∂y f [ y a F mn ( x − λy ) A e ; b ( x − λy + y )] (cid:27) Now, combining this with (78) i (87), we arrive at T ab ( x ) = − (cid:2) δ af A e ; b ( x ) + δ ab F ef ( x ) (cid:3) ˜ M efmn ∗ F mn − (cid:90) d y ˜ M efmn ( y ) (cid:90) d λ ∂∂y f [ y a F mn ( x − λy ) A e ; b ( x − λy + y )] (98)For the tensors P abc and Q cba in (93) we similarly obtain P abc ( x ) = − (cid:90) d y ˜ M cfmn ( y ) (cid:90) d λ ∂∂y f (cid:104) y a A b ( x − λy + y ) F mn ( x − λy ) (cid:105) (99)23 cba ( x ) = − (cid:90) d y ˜ M dfmn ( y ) (cid:90) d λ (1 − λ ) ∂∂y f (cid:104) y a y c A bd ; ( x − λy + y ) F mn ( x − λy ) (cid:105) (100)which, substituted in (92), yield W cab ( x ) = − (cid:90) d y ˜ M dfmn ( y ) ∂∂y f (cid:26)(cid:90) d λ F mn ( x − λy ) (cid:104) y b δ [ ad A c ] ( x − λy + y )+ δ bd y [ a A c ] ( x − λy + y ) + δ [ ad y c ] A b ( x − λy + y ) + 2(1 − λ ) y b y [ a A c ] d ; ( x − λy + y ) (cid:105)(cid:111) (101) References [1] Jackson J D,
Classical Electrodynamics , 3rd Edition, John Wiley (1999)[2] Minkowski H,
Göt Nachr Math-phys Klasse (1908) 53[3] Vladimirov V S,
Equations of Mathematical Physics , URSS (1996)[4] Jackson J D,
Op cit , equations (4.34) and (5.81)[5] Jackson J D,
Op cit , Section 6.7[6] Dixon W G,
Special Relativity , Cambridge University Press (1982)[7] Marnelius R,
Phys Rev
D10 (1974) 2535[8] Jaen X, Jauregui R, Llosa J and Molina A,
Phys Rev
D36 (1987) 2385[9] Gomis J, Kamimura K and Llosa J,
Phys Rev
D63 (2001) 045003[10] Noether E,
Nachr Ges Wiss Goettingen; Math-phys Klasse , 235-237 (1918); English translation:Travel M A,
Transp Theory Stat Phys , (1971) 183H[11] Brevik I, Phys Rep (1979) 133; Antoci S and Mihich L, EPJ direct (2000) 1: 1.https://doi.org/10.1007/s101059800d001[12] Ramos T, Rubilar G F and Obhukov Y N,
J Opt (2015) 025611[13] Alfonso A, “Electromagnetic energy-momentum tensor. Abraham vs. Minkowski”, TFG in PhysicsUB (2015)[14] Landau L D and Lifshitz E M, The Classical Theory of Fields , Pergamon Press (1985)[15] Belinfante F J,
Physica (1940) 449; Rosenfeld L, Acad R Belgique Classe Sci mém. no 6 1(1940)[16] Spivak M, Calculus on Manifolds , th 4-11, p 94, W A Benjamin (New York, 1965)[17] Jackson Kramers-Krönig 2418] Landau L D, Lifshitz E M and Pitaevskiï L P,
Electrodynamics of Continuous Media , Section §61 ,2nd Edition, Pergamon Press (1985)[19] Jackson J D,
Op cit , Section §6.8[20] Schwinger J, De Raad L L, Milton K A, and Tsai W,