Noether s theorem applied to classical electrodynamics

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1 Noether s theorem applie to classical electroynamics Thomas B. Mieling Faculty of Physics, University of ienna Boltzmanngasse 5, 090 ienna, Austria (Date: November 8, 207) The consequences of gauge invariance an Poincaré invariance of the action of the electromagnetic fiel in the special theory of relativity are investigate. The conservation of electric charge an the laws governing the tensors of energy-momentum an angular momentum of the electromagnetic fiel in the presence of charges an currents are euce from Noether s theorem. CONTENTS I. Introuction A. Conventions II. Conservation of charge III. Non-ynamical fiels as sources 2 A. Conservation of energy-momentum 2 B. Conservation of angular momentum 3 I. Interpretation 4 A. Electric Charge 4 B. Energy an Momentum 4 C. Angular momentum an mass moment 4. Conclusion 5 I. References 5 A. Appenix 5. Derivatives of the Lagrangian 5 2. ariations of the fiels 6 a. Gauge transformations 6 b. Translations 6 c. Lorentz transformations 6 References 6 I. INTRODUCTION Noether s theorem states that every ifferentiable symmetry of the action correspons to a conserve quantity. Consiering classical electroynamics, two symmetries immeiately come to min: Poincaré invariance (Lorentz transformations an translations) an gauge invariance. In the absence of sources, Poincaré invariance implies the conservation of energy-momentum an angular momentum of the fiel, but in the presence of electric charges an currents, these quantities are no longer conserve. In this article, we will erive the corresponing equations using Noether s theorem. A. Conventions This paper uses the Einstein summation convention where greek inices take on values between 0 an 3 an latin inices take on values between an 3. The metric signature is (+,,, ). We consier the action from classical Electroynamics in the Gaussian unit system S = c with the Lagrangian A µ j µ t 6π F µν F µν t () L = c A µj µ 6π F µν F µν, (2) where the electromagnetic fiel tensor F is efine as F µν = µ A ν ν A µ. We will make use of the various erivatives of the Lagrangian (equations A2, A3 an A5) which are calculate in appenix A. II. CONSERATION OF CHARGE It is well known that the continuity equation for the electric charges follows irectly from Maxwell s equations. Here, we erive the continuity equation from the gauge invariance of the action. We want to show that µ j µ = 0 everywhere. We o this by showing that this equation hols on all compact subsets of R 4. Let Ω R 4 be a compact set, an χ : R 4 R an arbitrary smooth function with supp χ Ω (supp enotes the support of a function). Consier the variation of the 4-potential ue to a gauge transformation δa µ = µ χ. (3) A formal escription of such variations is iscusse in appenix A 2 a. The variation of the Lagrangian is then δl = c ( µχ)j µ = c µ(χj µ ) + c χ µj µ. (4) thomas.mieling@unvie.ac.at This implies the following expression for the variation

2 2 of the action cδs = δl 4 x R 4 = lim [ µ (χj µ ) + χ µ j µ ] 4 x r c = lim r c = c + lim r c B(r) B(r) B(r) R 4 χ µ j µ 4 x χj µ n µ S χ µ j µ 4 x (5) where the surface integral over the bounary of B(r), a sphere of raius r with normal n, vanishes for sufficiently large r ue to supp χ Ω, which is compact an thus boune (by the Heine-Borel theorem). At the same time we have δl = δa ν + µ δa ν A ν = ( ) x µ δa ν [ + ( ) ] A ν x µ δa ν = ( ) x µ δa ν (6) ue to the Euler-Lagrange equations for fiels. Computing the variation of the action from this expression, one obtains ( ) cδs = lim r x µ δa ν 4 x = lim r B(r) B(r) ν χn µ S = 0 (7) since χ is smooth an has compact support. Comparing with equation (5), we have χ µ j µ 4 x = 0 (8) Ω for all smooth functions χ with compact support in Ω. By the Du Bois-Reymon lemma (the funamental lemma of calculus of variations), this implies that µ j µ = 0 on Ω. Since the compact set Ω R 4 was arbitrary, this implies µ j µ = 0 everywhere. III. NON-DYNAMICAL FIELDS AS SOURCES For the further iscussion, we consier a certain form of Noether s theorem which allows for nonynamical fiels (here: j µ ). This means that our moel contains only the equations of motion for the electromagnetic fiel an not those of the source fiels j µ. We thus obtain the tensors for energy-momentum an angular-momentum of the fiel only an expect these quantities not to be conserve (only the total quantities are expecte to be conserve in general). We consier a symmetry transformation of the fiels A µ an j µ, which changes the Lagrangian by a ivergence δl = µ D µ. At the same time we have δl = j µ δjµ + δa µ + A µ = j µ δjµ + x µ ( δa ν ν δa µ ) (9) by the same argument as before. (We have only use the equations of motion for the electromagnetic fiel, not those of the source fiels.) Comparing the two expressions for δl, we fin ( ) x µ δa ν D µ + j µ δjµ = 0. (0) We see that the non-ynamical 4-current ensity acts as a source of the quantity in the parenthesis. In the next two sections, we will examine two instances of this equation where the 4-current ensity acts as a source of the electromagnetic energymomentum tensor an raiates angular momentum via the electromagnetic fiel. A. Conservation of energy-momentum Consier a translation of both the source fiels an the electromagnetic fiel in the irection of an arbitrary 4-vector n, couple with a gauge transformation by a function χ. The variations of the fiels are then given by δj µ = ρ j µ n ρ δa µ = ρ A µ n ρ + µ χ () A erivation of the variations of the fiels ue to translations is given in appenix A 2 b. We choose χ = A ρ n ρ to make the variation of the 4-potential gauge invariant, since we then have δa µ = ρ A µ n ρ + µ χ = ( ρ A µ µ A ρ )n ρ = F ρµ n ρ. This gives us the following variations (2) δj µ = ρ j µ n ρ δa µ = F ρµ n ρ (3) Using the continuity equation µ j µ = 0, we fin the variation of the Lagrangian ensity to be a ivergence δl = µ L n µ c ( µχ)j µ = µ D µ, (4) Whether the gauge transformation is performe prior or after the translation oes not change the variation, which epens only on terms of first orer in the variation parameter.

3 3 where the first term stems from the translation (ue to the chain rule) an the secon one originates from the gauge transformation. W we have set D µ = (L δ µρ c ) A ρj µ n ρ. (5) Plugging everything into equation (0), we obtain ( x µ 4π F νµ F ρν + ) 6π F αβ F αβ δ ρ µ n ρ + ( x µ c A νj ν δ ρ µ ) c A ρj µ n ρ (6) c A µ ρ j µ n ρ = 0 The terms in parentheses in the first line efine the energy-momentum tensor of the electromagnetic fiel T µ ν = 4π F µρ F ρν + 6π F ρσ F ρσ δ µ ν. (7) Since the vector n is arbitrary, we can euce (T µρ x µ + c A νj ν δ µρ c ) A ρj µ c A µ ρ j µ = 0 (8) The terms apart from the energy-momentum tensor can be simplifie using the continuity equation: µ (A ν j ν δ µ ρ A ρ j µ ) A ν ρ j ν = ρ A ν j ν µ A ρ j µ A ρ µ j µ = F ρν j ν (9) Finally, we arrive at the result µ T µν + f ν = 0. (20) where the 4-force ensity is f µ = c F µν j ν. Consequently, the ivergence of the energy-momentum tensor of the electromagnetic fiel vanishes in the absence of charges an currents. B. Conservation of angular momentum We now consier the variations of the fiels ue to a Lorentz transformation (as erive in appenix A 2 c) couple with a gauge transformation by a function χ. δj µ = ρ j µ a ρ σx σ a µ νj ν δa µ = ρ A µ a ρ σx σ + a ν µa ν + µ χ (2) Choosing the gauge function χ = A ρ a ρ σx σ, we have µ χ = µ A ρ a ρ σx σ A ρ a ρ µ an the variation of A µ becomes gauge-inepenent δj µ = ρ j µ a ρ σx σ a µ νj ν δa µ = F ρµ a ρ σx σ (22) The variation of the Lagrangian is δl = µ L a µ σx σ c µχj µ (23) The two terms arise, again, from the chain rule an the gauge transformation, respectively. Using the continuity equation µ j µ = 0 an the vanishing trace of the coefficients a µ µ = a µ σδ σ µ = a µ σ µ x σ = 0, this can be rewritten as a ivergence δl = µ L a µ σx σ + L a µ σ µ x σ c µχj µ c χ µj µ ( = µ L a µ σx σ ) c χjµ We can rea off the vector D µ as D µ = L δ µ ρ a ρ σx σ c χjµ = 6π F αβ F αβ δ µ ρ a ρ σx σ c A αj α δ µ ρ a ρ σx σ (24) + c A ρa ρ σx σ j µ (25) Plugging everything into equation (0), we fin [ µ 4π F νµ F ρν x σ + ] 6π F αβ F αβ δ ρ µ x σ a ρ σ [ + µ c A αj α δ ρ µ a ρ σx σ ] c A ρa ρ σx σ j µ (26) c A µ ρ j µ a ρ σx σ + c A ρa ρ σj σ = 0 We can recognise the first term as the ivergence of T µ ρa ρ σx σ. The secon an thir line form one single coefficient of a ρ σ, which we can simplify (here without the factor /c) using the continuity equation µ [A α j α x σ δ µ ρ A ρ j µ x σ ] A µ ρ j µ x σ + A ρ j σ = ρ A α j α x σ + A α j α δ σ ρ µ A ρ j µ x σ A ρ µ j µ x σ = ρ A α j α x σ α A ρ j α x σ + A α j α δ σ ρ = F ρα j α x σ + A α j α δρ σ (27) The last term gives zero when contracte with a ρ σ, ue to its vanishing trace. Consequently, we have [ µ (T µρ x σ ) + c F ραj ] α x σ a ρσ = 0 (28) Since this equation has to be satisfie for all antisymmetric a ρσ, the antisymmetric part of the bracket must vanish. Thus, we finally arrive at µ (x σ T µρ x ρ T µσ ) + x σ f ρ x ρ f σ = 0. (29) The term in the parenthesis efines the (orbital) angular momentum ensity M µνρ = x µ T ρν x ν T ρµ (30) an the above equation now reas ρ M µνρ + x µ f ν x ν f µ = 0. (3) Thus, the ivergence of the angular momentum tensor vanishes in the absence of charges an currents.

4 4 I. INTERPRETATION Consiering a fixe inertial system, the iscusse coorinate transformations have a irect physical interpretation: they represent temporal an spatial translations, rotations as well as Lorentz boosts an are each associate with a separate ynamical quantity. Their mathematical treatment is completely analogous to the one for the electric charge, which we will iscuss first. A. Electric Charge The continuity equation µ j µ = 0 motivates the efinition of the electric charge Q insie a volume Q = j 0. (32) c where ϱ = j 0 /c is the charge ensity. The time erivative of the charge in is then given by t Q = 0 j 0 = ( µ j µ k j k ) (33) = j k A k ue to the ivergence theorem. The charge containe in can thus only change by means of an electric current through the bounary, which is given by the surface integral of the current ensity j µ over the surface B. Energy an Momentum Similar to the electric charge, we can introuce four quantities associate with a volume P µ = T 0µ (34) c which are relate to translations along the four coorinate axes. Their time erivative can be expresse in the following way t P µ = 0 T 0µ = = f µ ( ν T νµ k T kµ ) T kµ A k (35) a. Energy The quantity associate with translations in time is energy. The electromagnetic energy containe in a volume is E = cp 0 = W, (36) where W = T 00 is the energy ensity of the electromagnetic fiel. The equation for the rate of change of the fiel energy in a given volume is t E = cf 0 S k A k (37) The electromagnetic energy content of a region thus changes ue to work one on the charges insie the region an the energy flux through the surface area via the energy flux ensity or Poynting vector S with the components S k = ct k0. b. Momentum The quantity associate with spatial translations is the momentum P with the components P i. The quantities T 0i /c form the momentum ensity of the electromagnetic fiel an ue to the symmetry of the energy-momentum tensor, the energy flux ensity S is equal to the momentum ensity multiplie by c 2. The equation for the change of electromagnetic momentum in the region is t P i = f i σ ki A k (38) The electromagnetic momentum in a given volume thus changes ue to the Lorentz force applie to charges an ue to raiation through the bounary surface. The ensity of the momentum raiation is given by the Maxwell stress tensor σ with components σ ik = T ik. Note that some authors efine σ with the opposite sign. Written in matrix form, the components of the electromagnetic energy-momentum tensor are given by W S /c S 2 /c S 3 /c (T µν S ) = /c σ σ 2 σ 3 S 2 /c σ 2 σ 22 σ (39) 23 S 3 /c σ 3 σ 32 σ 33 where we lower inices of spatial tensors with the spatial metric coefficients δ ij which are the negative of the spatial components of the four-imensional metric η ij. C. Angular momentum an mass moment In complete analogy to the preceing iscussion, introuce the oubly inexe quantity M µν = c M µν0, (40) which is (as we saw) relate to Lorentztransformations. The time erivative thereof is given by t M µν = (x µ f ν x ν f µ ) M µνk A k (4) This antisymmetric tensor can be ecompose into a polar vector N with components M 0i /c an an antisymmetric spatial tensor with components M ij. This spatial antisymmetric tensor is ual to an axial vector M. Symbolically, we write this as (M µν ) = (cn, M). This is completely analogous to the ecomposition of the electromagnetic fiel tensor into the electric an magnetic fiels, written as (F µν ) = (E, B).

5 5 a. Angular Momentum By our erivation, the spatial components M ik are relate to spatial rotations. The components of the antisymmetric tensor M ik can be combine to form the pseuovector of angular momentum M with components M i = 2 ε ijkm jk such that M ij = ε ijk M k, (42) where ε ijk is the permutation symbol. The rate of change of the angular momentum evaluates to t M i = ε ijk x j f j ε ijk x j σ kl A l (43) So the angular momentum of the electromagnetic fiel in a given volume changes ue to the torque applie to charges insie the volume an the raiation of momentum through the surface of the volume. b. Mass Moment The components M 0i form the vector of ynamical mass moment N by N i = c M 0i = (tt 0i /c x i W/c 2 ) (44) = tp i (E /c 2 )R i where R is the center of energy given by R i = x i W. (45) E The rate of change of the mass moment is given by t N i = (tf i x i f 0 /c 2 ) (46) (tσ ik x i S k /c 2 ) A k We see that mass moment of the electromagnetic fiel is transferre to charges via the Lorentz force an is raiate through the surface area. The raiation term consists of two parts: the raiation of momentum (escribe by the stress tensor σ) an the energy flux (escribe by the Poynting vector S ivie by c 2 ). To summarise, we write the components of the angular momentum tensor in matrix form. 0 cn cn 2 cn 3 (M µν cn ) = 0 M 3 M 2 cn 2 M 3 0 M (47) cn 3 M 2 M 0. CONCLUSION We have erive the conservation of charge an the laws governing the energy-momentum tensor an the angular momentum tensor of the electromagnetic fiel in the presence of charges an currents from the gauge symmetry an Poincaré symmetry of the action. Gauge transformations, which were chosen such that the variations became gauge inepenent, irectly lea us to the symmetric an gauge inepenent energy-momentum tensor, instea of the canonical energy-momentum which is neither gauge invariant nor symmetric. The presente erivation suggests the interpretation of the foun quantities, for example by relating energy to translations in time an momentum to translations in space. This lea us to a consistent escription of all ynamical quantities foun. I. REFERENCES Lanau an Lifshitz provie in [] an excellent overview of electroynamics an iscusses all ynamical quantities introuce in this paper. In [2] Pauli iscusses both the historical introuction of the energy-momentum tensor of the fiel ue to Minkowski an a variational approach similar to ours. In [3] Rinler introuces the energy-momentum tensor of the fiel by consiering the forces acting on charge istributions. The metho of obtaining a gauge invariant energy-momentum tensor from Noether s theorem by consiering transformations together with a couple gauge transformation was iscusse before e.g. in [4]. Appenix A: Appenix. Derivatives of the Lagrangian We compute the erivatives of the Lagrangian L = c A µj µ 6π F µν F µν, (A) with respect to the fiels. The fiel tensor is F µν = µ A ν ν A µ = A ν,µ A µ,ν. We can immeiately rea off j µ = c A µ A µ = c jµ (A2) (A3) For the erivative with respect to A µ,ν we first compute F ρσ = A σ,ρ A ρ,σ = δ µ σδ ν ρ δ ρ µδ σ ν (A4) But since F µν F µν is a quaratic form in the components of F, we have (F ρσ F ρσ ) = 2F ρσ F ρσ = 2F ρσ (δ µ σδ ν ρ δ ρ µδ σ ν ) = 2F νµ 2F µν = 4F µν

6 which gives us the result = 4π F µν (A5) ourselves to scalar fiels, which transform in the following way. f( x) = f(x) = f( x + ɛn) = f( x) + ɛ ρ f( x)n ρ + O(ɛ 2 ) 6 (A) 2. ariations of the fiels We efine the variation of a function with respect to some variational parameter ɛ in the following way. Let f(ɛ, x, y,...) be any analytic expression in ɛ an an arbitrary number of other arguments. We then set δf(x, y,...) = [ ] f(ɛ, x, y,...) ɛ ɛ=0 (A6) δf(x, y,...) is thus the secon coefficient of the Taylor expansion of f(ɛ, x, y,...) aroun ɛ = 0 f(ɛ, x, y,...) = f(x, y,...) + ɛδf(x, y,...) + O(ɛ 2 ) (A7) Note that this variation obeys a chain rule. For brevity, we will not write the argument ɛ explicitly in our formulas. a. Gauge transformations Given any smooth function χ, we can construct a one-parameter family of gauge transformations by which trivially satisfies Ā µ (x) = A µ (x) + ɛ µ χ(x) δa µ = µ χ(x). b. Translations (A8) (A9) We compute the variations of arbitrarily value tensor fiels ue to translations. Let n be an arbitrary vector an efine the one-parameter family of coorinate transformations x µ = x µ + ɛn µ x µ = x µ ɛn µ (A0) Since the coorinates are not rotate, the components of tensor fiels transform trivially an we can restrict So the variation of any fiel f is simply δf(x) = ρ f(x)n ρ. c. Lorentz transformations (A2) We erive the general formula for the variations of scalar, vector an covector value fiels ue to Lorentz transformations. Let (a µ ν) be an element of the Lie algebra of the Lorentz group an consier the one-parameter family of Lorentz transformations obtaine from this element by exponentiation. x µ = L µ ν x ν = x µ + ɛa µ ν x ν + O(ɛ 2 ) x µ = (L ) µ ν xν = x µ ɛa µ νx ν + O(ɛ 2 ) (A3) As is well known, we have to require that a µν = a νµ for this to efine a Lorentz transformation. (The generators of the Lorentz group are antisymmetric matrices). For a scalar fiel f we then have f( x) = f(x) = f( x + ɛa x + O(ɛ 2 )) = f( x) + ɛ µ f( x)a µ σ x σ + O(ɛ 2 ). A vector fiel transforms as µ ( x) = xµ x ν ν (x) = (δ µ ν ɛa µ ν + O(ɛ 2 )) ( ν ( x) + ɛ ρ ν ( x)a ρ σ x σ + O(ɛ 2 )) = µ ( x) (A4) + ɛ[ ρ µ ( x)a ρ σx σ a µ ν ν ( x)] + O(ɛ 2 ), (A5) while for a covectorfiel α we have ᾱ µ ( x) = xν x µ α ν(x) = (δ ν µ + ɛa ν µ + O(ɛ 2 )) (α ν ( x) + ɛ ρ α ν ( x)a ρ σ x σ + O(ɛ 2 )) = α µ ( x) + ɛ[ ρ α µ ( x) + a ν µα ν ( x)] + O(ɛ 2 ) (A6) So the variations of scalar, vector an covector value fiels are of the following form. δf(x) = ρ f(x)a ρ σx σ δ µ (x) = ρ µ (x)a ρ σx σ a µ ν ν (x) δα µ (x) = ρ α µ (x)a ρ σx σ + a ν µα ν (x) (A7) [] L. D. Lanau an E. M. Lifshitz, Course of Theoretical Physics 2: The Classical Theory of Fiels. Pergamon Press, 3. e., 97.

7 7 [2] W. Pauli, Relativitätstheorie. Springer Berlin Heielberg, [3] W. Rinler, Relativitätstheorie. Wiley CH, 206. [4] M. Bañaos an I. Reyes, A short review on Noether s theorems, gauge symmetries an bounary terms, International Journal of Moern Physics D, vol. 25, p , sep 206.