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ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC PHYSICAL REVIEW E 73. 026606(2006) B=V×A e(1)=at)+-|do0(a)(t) which solves the two homogeneous Maxwell equations v×E+aB=0 (10) d(o) TJo 02-(o'+iy? exp(io't) v.B=0 (19) The Euler-Lagrange equations for the potentials are the two It follows that the Fourier transform of the dielectric function inhomogeneous" Maxwell equations where is given by because in the present context there are no free charges and (a) currents so that these equations are in fact homogeneous as e(o)=1+= do-m2-(o+iy)2 v.D=0 (12) 1+ ,(a2 v×H-aD=0 1+ (20) which can be demonstrated with textbook manipulations a d,G(w) [12,13] where it has been used that d(o)=G(o). Using that The Euler-Lagrange equation for the material field F is the equation of a driven harmonic oscillator =P一+ima), (21) 2F+ (14) The inhomogeneous solution of this equation is (with depen. where the capital"P"indicates the principal value, it follows dence on o and t explicit) G(o' Gw) id(o) (22) where g(o, t)is a Greens function of the harmonic oscillator equation. The homogeneous solution is not present in this classical theory. However, in the quantum theory it must be By construction, this function satisfies the Kramers-Kronig taken into account. There it describes a noise polarization, a relations as well as the symmetry relation E(o)=E(-o).As quantity which can even be interpreted as the basic ingredi- a consequence, the dielectric function in the time domain is ent of the quantum theory on which all other fields depend real [e(t=a(t)'1 and causal [e()=0 if tsoJ. This proves [14-17]. In the classical theory it turns out that the solution that the constitutive relation for media with arbitrary disper- is causal provided the Green's function is chosen to be the sion and absorption is properly described by the present retarded Greens function model. As a consequence, the equations of motion for the electromagnetic field in such media are formally equivalent G(a,t)=6(r) sin(ot) (16) to the Euler-Lagrange equations for the proposed Lagrangian dens where A(r) is the step function [a(r=l if t>0, a(0)=1/2 if IIL CONSERVATION LAWS 1=0, 0(0)=0 if t<o]. The retarded Greens function has Fourier representation The transport and dissipation of electromagnetic energy is 2Tw'-('+iy)2 p(-io'D,(17 described by the energy balance equation a I M+VS=-w (23) where y is a positive infinitesimal quantity. The resulting where the electromagnetic field energy density u M, energy expression for the material field F leads to a dielectric dis- flux density S( Poynting vector), and the density of the rate of work on the material subsystem W are defined by D)=Eodr’e(t-t)E() n=E2+-B2, with the dielectric function S=E×H 026606-3B = A, 9 which solves the two homogeneous Maxwell equations E + t B = 0, 10 · B = 0. 11 The Euler-Lagrange equations for the potentials are the two “inhomogeneous” Maxwell equations where we use quotes because in the present context there are no free charges and currents so that these equations are in fact homogeneous as well · D = 0, 12 H − t D = 0, 13 which can be demonstrated with textbook manipulations 12,13. The Euler-Lagrange equation for the material field F is the equation of a driven harmonic oscillator t 2 F + 2 F = E. 14 The inhomogeneous solution of this equation is with dependence on and t explicit F,t = − dtG,t − tEt, 15 where G,t is a Green’s function of the harmonic oscillator equation. The homogeneous solution is not present in this classical theory. However, in the quantum theory it must be taken into account. There it describes a noise polarization, a quantity which can even be interpreted as the basic ingredient of the quantum theory on which all other fields depend 14–17. In the classical theory it turns out that the solution is causal provided the Green’s function is chosen to be the retarded Green’s function G,t = t sint , 16 where t is the step function t= 1 if t0, t= 1/ 2 if t=0, t= 0 if t0. The retarded Green’s function has a Fourier representation G,t = − d 2 1 2 − + i 2 exp− it, 17 where is a positive infinitesimal quantity. The resulting expression for the material field F leads to a dielectric displacement Dt = 0 − dtt − tEt, 18 with the dielectric function t = t + 2 0 dˆt sint = − d 2 1 + 2 0 d ˆ 2 − + i 2 exp− it. 19 It follows that the Fourier transform of the dielectric function is given by ˆ =1+ 2 0 d ˆ 2 − + i 2 =1+ 1 0 d ˆ 1 − − i − 1 + + i =1+ 1 − d ˆ 1 − − i , 20 where it has been used that ˆ=ˆ−. Using that 1 − i = P 1 + i , 21 where the capital “P” indicates the principal value, it follows that ˆ =1+ 1 P − d ˆ − + iˆ =1+ 2 P 0 d ˆ 2 − 2 + iˆ . 22 By construction, this function satisfies the Kramers-Kronig relations as well as the symmetry relation ˆ=ˆ− * . As a consequence, the dielectric function in the time domain is real t=t * and causal t= 0 if t0. This proves that the constitutive relation for media with arbitrary dispersion and absorption is properly described by the present model. As a consequence, the equations of motion for the electromagnetic field in such media are formally equivalent to the Euler-Lagrange equations for the proposed Lagrangian density. III. CONSERVATION LAWS A. Energy The transport and dissipation of electromagnetic energy is described by the energy balance equation t uEM + · S = − W, 23 where the electromagnetic field energy density uEM, energy flux density S Poynting vector, and the density of the rate of work on the material subsystem W are defined by uEM = 0 2 E2 + 1 2 0 B2, 24 S = E H, 25 ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC... PHYSICAL REVIEW E 73, 026606 2006 026606-3