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ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC PHYSICAL REVIEW E 73. 026606(2006) fa= dod w)( Fp) TaB=TEn+TMT -,BB+ED+HB saB Eo dod(o)(Fp0 ,FB-2 FBdFp) doGo)[(O,F)2-0'F2+FE]sag.(50) The total system momentum proposed here corresponds to o, =0 dod()(Fp0.d FB-d Fgd Fg .(42) the pseudomomentum of Gordon [2]. the wave momentum of Nelson [3] and the canonical momentum of Garrison and A third step is rewriting this expression using the following Chiao [4] According to nelson the wave momentum is the sum of momentum and pseudomomentum. The momentum contri bution from the material subsystem in the present theory corresponds to Nelsons pseudomomentum contribution to the wave momentum. a difference with Nelson is in the gen- Eo dwd(o)[awl eral form of the momentum density and stress tensor. These quantities are not unique in the sense that terms can be shifted from the density to the fux density and vice versa. In +FB0 FB+a FBd FBl particular, any multiple of the identity Eq. (43)can be added or subtracted from the total momentum conservation law Eq =o- dod(o)(FB0ad, FB+0. FB0FB) (48). An example of such a redefinition of the momentum density and stress tensor using the identity Eq.(43)is 1时 ub,、2e0doao)lFBF This identity follows from the equation of motion of the material field Eq (14). This gives that TaB=-EODB-HBB+ED+HB SaB f=al doG(o)F80,, F Eo dod(o)[(F)2-0 F+F E]8og. (52) dodo[(aF)2-0F+F These forms correspond quite closely to the density and flux The Lorentz force density can now be expressed as (44) density of wave momentum of Nelson [3].Apparently, an independent requirement is needed to justify the form of these quantities. The point of view taken here is motivated fo=gm+aRTo by an an of wave packets, and will be discussed in the next section. It here the material momentum density and momentum flux turns out that the present choice, Eqs. (49)and(50), results nsity are given by in transport of energy and momentum with the same velocity, s opposed to the alternative choice, Eqs. (51)and(52) saT=EapxPBBy ed dod o)Fpoa FB.(46)which leads to transport of energy and momentum at differ mentum travel at the same speed, which implies that Eqs Ta=-EPp-Eo dd( o)[(O, F)2-w0F210ng 49)and(50) are the correct forms of the density and flux density of the total momentum. Similar to the energy case the total momentum can be (47) divided into nondispersive and dispersive parts, with densi- Conservation of momentum of the total system is expressed tes B aga+dBlaB=0 where the momentum density and stress tensor of the total dodo)FBdoo, F system are and flux densities gaM+ga=∈DpB, dod(o)Fga. aF 026606-5f = 0 0 dˆFE − EF = 0 0 dˆFt 2 F − t 2 FF = t 0 0 dˆFt F − t FF . 42 A third step is rewriting this expression using the following identity: 0 = 0 0 dˆt 2 F + 2 F − EF = 0 0 dˆ2 FF − EF + Ft 2 F + Ft 2 F = t 0 0 dˆFt F + t FF − 0 0 dˆt F 2 − 2 F2 + F · E . 43 This identity follows from the equation of motion of the material field Eq. 14. This gives that f = t 20 0 dˆFt F − 0 0 dˆt F 2 − 2 F2 + F · E . 44 The Lorentz force density can now be expressed as f = t g MT + T MT, 45 where the material momentum density and momentum flux density are given by g MT = PB + 20 0 dˆFt F, 46 T MT = − EP − 0 0 dˆt F 2 − 2 F2 . 47 Conservation of momentum of the total system is expressed by t g + T = 0, 48 where the momentum density and stress tensor of the total system are g = g EM + g MT = DB + 20 0 dˆFt F, 49 T = T EM + T MT = − ED − HB + 1 2 E · D + 1 2 H · B − 0 0 dˆt F 2 − 2 F2 + F · E. 50 The total system momentum proposed here corresponds to the pseudomomentum of Gordon 2, the wave momentum of Nelson 3, and the canonical momentum of Garrison and Chiao 4. According to Nelson, the wave momentum is the sum of momentum and pseudomomentum. The momentum contribution from the material subsystem in the present theory corresponds to Nelson’s pseudomomentum contribution to the wave momentum. A difference with Nelson is in the general form of the momentum density and stress tensor. These quantities are not unique in the sense that terms can be shifted from the density to the flux density and vice versa. In particular, any multiple of the identity Eq. 43 can be added or subtracted from the total momentum conservation law Eq. 48. An example of such a redefinition of the momentum density and stress tensor using the identity Eq. 43 is g = DB − 20 0 dˆt FF, 51 T = − ED − HB + 1 2 E · D + 1 2 H · B + 0 0 dˆt F 2 − 2 F2 + F · E. 52 These forms correspond quite closely to the density and flux density of wave momentum of Nelson 3. Apparently, an independent requirement is needed to justify the form of these quantities. The point of view taken here is motivated by an analysis of the relation between energy and momentum of wave packets, and will be discussed in the next section. It turns out that the present choice, Eqs. 49 and 50, results in transport of energy and momentum with the same velocity, as opposed to the alternative choice, Eqs. 51 and 52, which leads to transport of energy and momentum at different velocities 19. It seems natural to have energy and momentum travel at the same speed, which implies that Eqs. 49 and 50 are the correct forms of the density and flux density of the total momentum. Similar to the energy case the total momentum can be divided into nondispersive and dispersive parts, with densities g ND = DB , 53 g DS = 20 0 dˆFt F, 54 and flux densities T ND = − ED − HB + 1 2 E · D + 1 2 H · B , 55 ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC... PHYSICAL REVIEW E 73, 026606 2006 026606-5