E151.4ELECTROMAGNETICDUALITYUnder certain conditions, for instance for very low field strengths, we mayassumethatthe response ofa substanceis linear sothatD =(t,x)E(1.45)H=μ-I(t,x)B(1.46)i.e., that the derived fields are linearly proportional to the primary fields andthattheelectricdisplacement(magnetisingfield)isonlydependentontheelec-tric (magnetic) field.The field equations expressed in terms of the derived field quantities D andHareV·D=p(t,x)(1.47a)aBVxE+=0(1.47b)atV.B=0(1.47c)aDVXH-= j(t,x)(1.47d)atand are called Maxwell's macroscopic equations.1.4ElectromagneticDualityIfwelookmorecloselyatthemicroscopicMaxwell equations(1.48),weseethat they exhibita certain,albeitnota complete,symmetry.Letusfor explicit-ness denote the electric charge density p =p(t,x) by Pe and the electric currentdensity j = j(t,x) by je-We further make the ad hoc assumption that thereexistmagneticmonopolesrepresentedbyamagneticchargedensity,denotedPm =Pm(t,x), and a magnetic current density, denoted jm =jm(t,x). With thesenewquantities included inthetheory,theMaxwellequations canbewrittenV.E-Pe(1.48a)80aB(1.48b)VxE+-μojmat·B=μopm(1.48c)OE(1.48d)×B+80H0t=HojeWe shall call these equations the Dirac-Maxwell equations or the electromag-netodynamicequationsDraft versiored13thNovber2000at22:01.Downloadedfromhttp://ww-pSe/CED/Boo!④由由
“main” 2000/11/13 page 15 1.4 ELECTROMAGNETIC DUALITY 15 Under certain conditions, for instance for very low field strengths, we may assume that the response of a substance is linear so that D = ε(t,x)E (1.45) H = µ −1 (t,x)B (1.46) i.e., that the derived fields are linearly proportional to the primary fields and that the electric displacement (magnetising field) is only dependent on the electric (magnetic) field. The field equations expressed in terms of the derived field quantities D and H are ∇·D = ρ(t,x) (1.47a) ∇×E+ ∂B ∂t = 0 (1.47b) ∇·B = 0 (1.47c) ∇×H− ∂D ∂t = j(t,x) (1.47d) and are called Maxwell’s macroscopic equations. 1.4 Electromagnetic Duality If we look more closely at the microscopic Maxwell equations (1.48), we see that they exhibit a certain, albeit not a complete, symmetry. Let us for explicitness denote the electric charge density ρ = ρ(t,x) by ρe and the electric current density j = j(t,x) by je. We further make the ad hoc assumption that there exist magnetic monopoles represented by a magnetic charge density, denoted ρm = ρm(t,x), and a magnetic current density, denoted jm = jm(t,x). With these new quantities included in the theory, the Maxwell equations can be written ∇·E = ρe ε0 (1.48a) ∇×E+ ∂B ∂t = −µ0jm (1.48b) ∇·B = µ0ρm (1.48c) ∇×B+ε0µ0 ∂E ∂t = µ0je (1.48d) We shall call these equations the Dirac-Maxwell equations or the electromagnetodynamic equations Draft version released 13th November 2000 at 22:01. Downloaded from http://www.plasma.uu.se/CED/Book
E16CLASSICALELECTRODYNAMICSTakingthedivergence of(1.48b),wefindthat0V-(V×E)=(V·B)-μoV·jm=0(1.49)atwherewe usedthefactthat,accordingto formula (M.82)on page175,thedivergence ofa curlalways vanishes.Using(1.48c)to rewritethisrelation,weobtain the equation of continuityfor magnetic monopolesapm +V-jm=0(1.50)atwhich has the same form as that for the electric monopoles (electric charges)and currents, Equation (1.21) on page 9.We notice that the new Equations (1.48) on the preceding page exhibit thefollowing symmetry (recall that εoμo = 1/c2):E→cB(1.51a)cB→-E(1.51b)(1.51c)cpe→Pm(1.51d)Pm--cpe(1.51e)cje→ jm(1.51f)jm→-cjewhich is a particular case ( = /2) of the general duality transformation (de-picted by the Hodge star operator)*E-Ecoso+cBsing(1.52a)(1.52b)c*B=-Esina+cBcose(1.52c)c*pe=cpecos+Pmsina*pm=-cpesing+Pmcoso(1.52d)c*je= cjecos+jmsing(1.52e)(1.52f)*jm=-cjesin+jmcosowhich leaves the Dirac-Maxwell equations, and hence the physics they de-scribe(oftenreferredtoaselectromagnetodynamics),invariant.SinceEandjeare (true or polar) vectors, B a pseudovector (axial vector), pe a (true) scalarthen Pm and , which behaves as a mixing angle in a two-dimensional “chargespace," must be pseudoscalars and jm a pseudovector.DownloaSe/CED/BookDraft version released 13th November 2000 at 22:01.p://④由由
“main” 2000/11/13 page 16 16 CLASSICAL ELECTRODYNAMICS Taking the divergence of (1.48b), we find that ∇·(∇×E) = − ∂ ∂t (∇·B)−µ0∇·jm ≡ 0 (1.49) where we used the fact that, according to formula (M.82) on page 175, the divergence of a curl always vanishes. Using (1.48c) to rewrite this relation, we obtain the equation of continuity for magnetic monopoles ∂ρm ∂t +∇·jm = 0 (1.50) which has the same form as that for the electric monopoles (electric charges) and currents, Equation (1.21) on page 9. We notice that the new Equations (1.48) on the preceding page exhibit the following symmetry (recall that ε0µ0 = 1/c 2 ): E → cB (1.51a) cB → −E (1.51b) cρe → ρm (1.51c) ρm → −cρe (1.51d) cje → jm (1.51e) jm → −cje (1.51f) which is a particular case (θ = π/2) of the general duality transformation (depicted by the Hodge star operator) ?E = Ecosθ+cBsinθ (1.52a) c ?B = −Esinθ+cBcosθ (1.52b) c ? ρe = cρe cosθ+ρm sinθ (1.52c) ? ρm = −cρe sinθ+ρm cosθ (1.52d) c ? je = cje cosθ+jm sinθ (1.52e) ? jm = −cje sinθ+jm cosθ (1.52f) which leaves the Dirac-Maxwell equations, and hence the physics they describe (often referred to as electromagnetodynamics), invariant. Since E and je are (true or polar) vectors, B a pseudovector (axial vector), ρe a (true) scalar, then ρm and θ, which behaves as a mixing angle in a two-dimensional “charge space,” must be pseudoscalars and jm a pseudovector. Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01
E171.4ELECTROMAGNETICDUALITYDUALITYOFTHEELECTROMAGNETODYNAMICEQUATIONSShowthatthesymmetric,electromagnetodynamicformof Maxwell's equations(theDirac-Maxwell equations), Equations (1.48) on page 15 are invariant under the dualityEXAMPLEtransformation(1.52)1.1ExplicitapplicationofthetransformationyieldsPecos&+cμoPm singV.*E=V·(Ecos+cBsin)=E0(1.53)111*PePecos+=PmsingE0E0a*BaV×*E+=V×(Ecos+cBsin)+Esin+BcosatatOB10Ecoso-μojmcososing+cμojesing(1.54)atcat1EaBsing+coso=-μojmcoso+cpojesingcatdr-μo(-cjesin+jmcos)=-μo*jmQEDand analogouslyfortheothertwoDirac-MaxwellequationsENDOFEXAMPLE11>MAXWELLFROMDIRAC-MAXWELLEQUATIONSFORAFIXEDMIXINGANGLEShowthatfor afixed mixing anglesuch thatEXAMPLEPm=cpetang(1.55a)1.2(1.55b)jm=cjetangthe Dirac-Maxwell equations reduce to the usual Maxwell equationsExplicitapplication ofthefixed mixing angle conditions on the duality transformation(1.52) on the facing page yields=Pming=Pecoso+-cpetangsing*Pe=Pecose+-C(1.56a)11cos6(Pecos-0+Pesin?0)=cosgPe(1.56b)*pm=-cpesing+cpetangcos=-cpesing+cpesing=0*je=jecoso+jetangsin=(jecos?0+jesin?0)=(1.56c)cosocosA(1.56d)*jm=-cjesing+cjetangcos=-cjesing+cjesing=0Hence, a fixed mixing angle, or, equivalently,a fixed ratio between the electric andmagnetic charges/currents,“hides"the magnetic monopole influence (pmand jm)onthe dynamic equationsWe notice that the inverse of the transformation given by Equation (1.52) on page 16Draft version releaed13thNoveber2000at22:01.Downloadedfromhttp://www.plaSe/CED/Book①由④
“main” 2000/11/13 page 17 1.4 ELECTROMAGNETIC DUALITY 17 ✑ DUALITY OF THE ELECTROMAGNETODYNAMIC EQUATIONS EXAMPLE 1.1 Show that the symmetric, electromagnetodynamic form of Maxwell’s equations (the Dirac-Maxwell equations), Equations (1.48) on page 15 are invariant under the duality transformation (1.52). Explicit application of the transformation yields ∇· ?E = ∇·(Ecosθ+cBsinθ) = ρe ε0 cosθ+cµ0ρm sinθ = 1 ε0 ✒ ρe cosθ+ 1 c ρm sinθ✓ = ?ρe ε0 (1.53) ∇× ?E+ ∂ ?B ∂t = ∇×(Ecosθ+cBsinθ)+ ∂ ∂t ✒ − 1 c Esinθ+Bcosθ✓ = − ∂B ∂t cosθ−µ0jm cosθ+ 1 c ∂E ∂t sinθ+cµ0je sinθ − 1 c ∂E ∂t sinθ+ ∂B ∂t cosθ = −µ0jm cosθ+cµ0je sinθ = −µ0(−cje sinθ+jm cosθ) = −µ0 ? jm (1.54) and analogously for the other two Dirac-Maxwell equations. QED ✔ END OF EXAMPLE 1.1 ✕ ✑ MAXWELL FROM DIRAC-MAXWELL EQUATIONS FOR A FIXED MIXING ANGLE EXAMPLE 1.2 Show that for a fixed mixing angle θ such that ρm = cρe tanθ (1.55a) jm = cje tanθ (1.55b) the Dirac-Maxwell equations reduce to the usual Maxwell equations. Explicit application of the fixed mixing angle conditions on the duality transformation (1.52) on the facing page yields ? ρe = ρe cosθ+ 1 c ρm sinθ = ρe cosθ+ 1 c cρe tanθ sinθ = 1 cosθ (ρe cos2 θ+ρe sin2 θ) = 1 cosθ ρe (1.56a) ? ρm = −cρe sinθ+cρe tanθ cosθ = −cρe sinθ+cρe sinθ = 0 (1.56b) ? je = je cosθ+je tanθ sinθ = 1 cosθ (je cos2 θ+je sin2 θ) = 1 cosθ je (1.56c) ? jm = −cje sinθ+cje tanθ cosθ = −cje sinθ+cje sinθ = 0 (1.56d) Hence, a fixed mixing angle, or, equivalently, a fixed ratio between the electric and magnetic charges/currents, “hides” the magnetic monopole influence (ρm and jm) on the dynamic equations. We notice that the inverse of the transformation given by Equation (1.52) on page 16 Draft version released 13th November 2000 at 22:01. Downloaded from http://www.plasma.uu.se/CED/Book
E18CLASSICALELECTRODYNAMICSyields(1.57)E=*Ecos-c*BsingThis means thatV.E=V.*Ecoso-cV.*Bsino(1.58)Furthermore,from the expressionsfor thetransformed charges andcurrents above, wefindthat*pe1peV.*E=(1.59)COSOEOE0andV.*B= μo*Pm= 0(1.60)so thatp.coso-0=peV.E=(1.61)COSOOE0QEDand so on for the other equations.ENDOFEXAMPLE1.2<The invarianceof theDirac-Maxwell equations under the similaritytrans-formation means that the amount of magnetic monopole density Pm is irrel-evant for the physics as long as the ratio pm/pe = tan is kept constant. Sowhetherweassumethattheparticles areonlyelectricallychargedorhavealsoa magnetic charge with a given, fixed ratio between the two types of chargesis a matter of convention, as long as we assume that this fraction is the samefor all particles.Such particles are referred to as dyons.By varying the mix-ing angle we can change the fraction ofmagnetic monopoles at will withoutchangingthelaws ofelectrodynamics.For=Owerecovertheusual Maxwellelectrodynamicsasweknowit.THE COMPLEX FIELD SIX-VECTORThecomplexfield six-vectorEXAMPLE(1.62)F(t,x) = E(t,x)+icB(t,x)1.3where EBe R and henceFeC,has a number ofinteresting properites:1. The inner product of Fwith itselfF·F =(E+icB)·(E +icB)= E?-c?B? +2icE·B(1.63)is conserved. ILe.,Downloadedfromhttp://wr.se/CED/BookDraft version released 13th November 2000 at 22:01.tnl:u①由④
“main” 2000/11/13 page 18 18 CLASSICAL ELECTRODYNAMICS yields E = ?Ecosθ−c ?Bsinθ (1.57) This means that ∇ ·E = ∇· ?Ecosθ−c∇· ?Bsinθ (1.58) Furthermore, from the expressions for the transformed charges and currents above, we find that ∇ · ?E = ?ρe ε0 = 1 cosθ ρe ε0 (1.59) and ∇ · ?B = µ0 ? ρm = 0 (1.60) so that ∇ ·E = 1 cosθ ρe ε0 cosθ−0 = ρe ε0 (1.61) and so on for the other equations. QED ✔ END OF EXAMPLE 1.2 ✕ The invariance of the Dirac-Maxwell equations under the similarity transformation means that the amount of magnetic monopole density ρm is irrelevant for the physics as long as the ratio ρm/ρe = tanθ is kept constant. So whether we assume that the particles are only electrically charged or have also a magnetic charge with a given, fixed ratio between the two types of charges is a matter of convention, as long as we assume that this fraction is the same for all particles. Such particles are referred to as dyons. By varying the mixing angle θ we can change the fraction of magnetic monopoles at will without changing the laws of electrodynamics. For θ = 0 we recover the usual Maxwell electrodynamics as we know it. ✑ THE COMPLEX FIELD SIX-VECTOR EXAMPLE 1.3 The complex field six-vector F(t,x) = E(t,x)+icB(t,x) (1.62) where E,B ∈ ✖3 and hence F ∈ ✗3 , has a number of interesting properites: 1. The inner product of F with itself F·F = (E+icB)·(E+icB) = E 2 −c 2B 2 +2icE·B (1.63) is conserved. I.e., Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01
E191.4ELECTROMAGNETICDUALITYE?-B?= Const(1.64a)E-B=Const(1.64b)as we shall see later2.The inner product ofF with the complex conjugate of itselfF·F* = (E + icB)-(E icB) = E +B2(1.65)is proportional to the electromagnetic field energy3.Aswithanyvector,thecrossproductofFitselfvanishes:FxF=(E+icB)x(E+icB)=E×E-B×B+ic(E×B)+ic(B×E)(1.66)=0+0+ic(E×B)-ic(E×B)=04.ThecrossproductofFwiththecomplexconjugateof itselfF×F*= (E+icB)×(E-icB)(1.67)=E×E+cB×B-ic(E×B)+ic(B×E)=0+0-ic(E×B)-ic(E×B)=-2ic(E×B)is proportional to the electromagnetic powerfluxENDOFEXAMPLE1.3△DUALITY EXPRESSEDIN THE COMPLEX FIELD SIX-VECTORExpressed in the complex field vector, introduced in Example 1.3 on the facing page,EXAMPLEthedualitytransformationEquations(1.52)onpage16become1.4*F=*E+ic*B=Ecos+cBsin-iEsin+icBcoso(1.68)=E(cos@-isin)+ icB(cos-isin) = e-i(E + icB)= e-igFfrom which it is easy to see that*F.*F"=|*F|2=e-"F.eF"=|FP(1.69)while*F.*F=e2ifF.F(1.70)Furthermore,assuming that =(t,x),we see that the spatial and temporal differenti-Draft version releaSed13thNovenmber2000at 22:01.Downloadedfromhttp://www.plasruu.se/CED/Book①由由
“main” 2000/11/13 page 19 1.4 ELECTROMAGNETIC DUALITY 19 E 2 −c 2B 2 = Const (1.64a) E·B = Const (1.64b) as we shall see later. 2. The inner product of F with the complex conjugate of itself F·F ∗ = (E+icB)·(E−icB) = E 2 +c 2B 2 (1.65) is proportional to the electromagnetic field energy. 3. As with any vector, the cross product of F itself vanishes: F×F = (E+icB)×(E+icB) = E×E−c 2B×B+ic(E×B)+ic(B×E) = 0+0+ic(E×B)−ic(E×B) = 0 (1.66) 4. The cross product of F with the complex conjugate of itself F×F ∗ = (E+icB)×(E−icB) = E×E+c 2B×B−ic(E×B)+ic(B×E) = 0+0−ic(E×B)−ic(E×B) = −2ic(E×B) (1.67) is proportional to the electromagnetic power flux. END OF EXAMPLE 1.3 ✕ ✑ DUALITY EXPRESSED IN THE COMPLEX FIELD SIX-VECTOR EXAMPLE 1.4 Expressed in the complex field vector, introduced in Example 1.3 on the facing page, the duality transformation Equations (1.52) on page 16 become ?F = ?E+ic ?B = Ecosθ+cBsinθ−iEsinθ+icBcosθ = E(cosθ−isinθ)+icB(cosθ−isinθ) = e −iθ (E+icB) = e −iθF (1.68) from which it is easy to see that ?F· ?F ∗ = ✘ ✘ ?F ✘ ✘ 2 = e −iθF· e iθF ∗ = |F| 2 (1.69) while ?F· ?F = e 2iθF·F (1.70) Furthermore, assuming that θ = θ(t,x), we see that the spatial and temporal differentiDraft version released 13th November 2000 at 22:01. Downloaded from http://www.plasma.uu.se/CED/Book