POYNTING'S THEOREM AND LUMINAL TOTAL ENERGY PHYSICAL REVIEW E 64 046610 nite and infinite volume energies, respectively. We also Let va be an eigenvector of E for eigenvalue AEP. The showed how a total dynamical energy density demonstrates eigenvalue can be expressed in terms of the eigenvector and that a causal media responds to a virtual, Instantaneous field the real and imaginary parts of E via the formula spectrum weighted by system resonances, that this density gives the heat energy eventually dissipated to the medium, but that this density is not the same as the one derived from va Relelva va lmlElv kinematic arguments vv,“+ port velocity and the global concept of total energy center- The kinetic symmetry (b)of e implies this symmetry of its of-mass velocity was also established, and the importance of real and imaginary parts, which, since they are each real, avoiding subsets of a total energy (i.e, forms that are not shows that they are each trivially Hermitian. Taking the Her- conserved)in establishing the global notion of luminality mitian conjugate, then, of our formula, and using the Hermit- emphasized. For the Lorentz model(a specific example ian properties (ust established)of the real and imaginary of the general media considered at the onset), a certain subset parts of E shows that the imaginary part of A can be ex of the total energy was shown to have the potential for dem- pressed as onstrating what appears(in the center of mass picture)to be an arbitrarily fast global energy transport mechanism. It was emphasized, however, that this appearance of high velocity A≈A- a+ viLle]vx does not constitute energy transport from one detector to another, but simply indicates that other energy already down- At this point it is already obvious that Im[x]>0 since Im[e] positive definite, but we make the proof even more ex- illustrates the general principle that classical"superluminal plicit since we will use a certain notation that these details effects are intimately linked with an incomplete energy ac- provide in the main text(Sec. Il B)where we derive the tota counting. energy of the system; since e assume (c), i.e. since we APPENDIX: KINETIC SYMMETRY AND PASSIVITY assume that Im[E] is a positive definite tensor, and since, as IMPLY DISSIPATION we have just shown, Im[ e] is trivially Hermitian, this tensor can be factored and expressed We prove(c)and(b)→c’fore. That the converse(c’) and(b)=c) is false(without invoking more structure)is Im[E]=B'B verified by specific counter examples. However, as is dis- so that the imaginary part of A is then expressed cussed in the main text, if we invoke the structure that the eigenvectors of the tensors can be taken as real, it is then tru BB、Bv2 that(c) and (b)=(c). Physically this extra structure means Im[A] that the eigenvectors of the tensors can correspond to vAVA vAl2 bonafide directions in real 3 space, the principal axes, say, of a crystal. Otherwise these eigenvectors correspond to"direc- regardless of the relationship of the eigenvector va of E to the tions"only in complex 3 space, but whose real and imagi- tensor Im[E]. However, since equality is achieved only if va nary parts in real 3 space can be made to correspond to happens to be a nullvector of B and, hence, of Im[e], and directions together with rotation angles since we assume that Im[e] has no null vectors, the result is In order to avoid needless repetition, we state here onc that in the following paragraph all statements about the per- lOvall mittivity tensor e are valid when it is evaluated at real posi- ImA]= v/2-0 tive frequencies @, i.e., when the restrictions of assumption (c)are enforced which is(c’) II D. Mugnai, A Ranfagni, and R Ruggeri, Phys. Rev. Lett. 8 4830(2000) [7E. L. Bolda, J C. Garrison, and R. Y, Chiao, Phys. Rev. A 49, []L J. Wang, A. Kuzmich, and A. Dogariu, Nature(Lond 2938(1994) 406,277-279(2000 B3JJ. Peatross, S. Glasgow, and M. Ware(unpublished) [9]S. A Glasgow and J B Peatross(unpublished/982) 8]S Chu and S. Wong, Phys. Rev. Lett. 48, 738(13 44L. Brillouin, Wave Propagation and Group Velocity(Aca- [10JJ. D. Jackson, Classical Electrodynamics, 2nd.(Wiley, New demic Press, New York, 1960) Yok,1975),pp.236and307 5]J. Peatross, S.A. Glasgow, and M. Ware, Phys. Rev. Lett. 84, [11]M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A 2370(2000 Tip, Phys. Rev. Lett. 66, 3132(1991 6]C. G. B. Garrett and D. E. McCumber, Phys. Rev. A 1, 305 [12]G L J. A Rikken and B. A van Tiggelen, Phys. Rev. Lett. 78 046610-13nite and infinite volume energies, respectively. We also showed how a total dynamical energy density demonstrates that a causal media responds to a virtual, instantaneous field spectrum weighted by system resonances, that this density gives the heat energy eventually dissipated to the medium, but that this density is not the same as the one derived from kinematic arguments. The connection between the local concept of energy trans￾port velocity and the global concept of total energy center￾of-mass velocity was also established, and the importance of avoiding subsets of a total energy ~i.e., forms that are not conserved! in establishing the global notion of luminality was emphasized. For the Lorentz model ~a specific example of the general media considered at the onset!, a certain subset of the total energy was shown to have the potential for dem￾onstrating what appears ~in the center of mass picture! to be an arbitrarily fast global energy transport mechanism. It was emphasized, however, that this appearance of high velocity does not constitute energy transport from one detector to another, but simply indicates that other energy already down￾stream has been converted to the type being observed. This illustrates the general principle that classical ‘‘superluminal’’ effects are intimately linked with an incomplete energy ac￾counting. APPENDIX: KINETIC SYMMETRY AND PASSIVITY IMPLY DISSIPATION We prove ~c! and (b)⇒c8 for eˆ. That the converse (c8) and (b)⇒c) is false ~without invoking more structure! is verified by specific counter examples. However, as is dis￾cussed in the main text, if we invoke the structure that the eigenvectors of the tensors can be taken as real, it is then true that (c8) and (b)⇒(c). Physically this extra structure means that the eigenvectors of the tensors can correspond to bonafide directions in real 3 space, the principal axes, say, of a crystal. Otherwise these eigenvectors correspond to ‘‘direc￾tions’’ only in complex 3 space, but whose real and imagi￾nary parts in real 3 space can be made to correspond to directions together with rotation angles. In order to avoid needless repetition, we state here once that in the following paragraph all statements about the per￾mittivity tensor eˆ are valid when it is evaluated at real posi￾tive frequencies v, i.e., when the restrictions of assumption ~c! are enforced. Let vl be an eigenvector of eˆ for eigenvalue lPr. The eigenvalue can be expressed in terms of the eigenvector and the real and imaginary parts of eˆ via the formula l5 vl † Re@eˆ#vl vl † vl 1i vl † Im@eˆ#vl vl † vl . The kinetic symmetry ~b! of eˆ implies this symmetry of its real and imaginary parts, which, since they are each real, shows that they are each trivially Hermitian. Taking the Her￾mitian conjugate, then, of our formula, and using the Hermit￾ian properties ~just established! of the real and imaginary parts of eˆ shows that the imaginary part of l can be ex￾pressed as Im@l#5 l2l† 2i 5vl † Im@eˆ#vl vl † vl . At this point it is already obvious that Im@l#.0 since Im@eˆ# is positive definite , but we make the proof even more ex￾plicit since we will use a certain notation that these details provide in the main text ~Sec. II B! where we derive the total energy of the system: since we assume ~c!, i.e. since we assume that Im@eˆ# is a positive definite tensor, and since, as we have just shown, Im@eˆ# is trivially Hermitian, this tensor can be factored and expressed as Im@eˆ#5bˆ †bˆ so that the imaginary part of l is then expressed as Im@l#5 vl †bˆ †bˆ vl vl † vl 5 ibˆ vli 2 ivli 2 >0, regardless of the relationship of the eigenvector vl of eˆ to the tensor Im@eˆ#. However, since equality is achieved only if vl happens to be a nullvector of bˆ and, hence, of Im@eˆ#, and since we assume that Im@eˆ# has no null vectors, the result is Im@l#5 ibvli 2 ivli 2 .0, which is (c8). @1# D. Mugnai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84, 4830 ~2000!. @2# L. J. Wang, A. Kuzmich, and A. Dogariu, Nature ~London! 406, 277-279 ~2000!. @3# J. Peatross, S. Glasgow, and M. Ware ~unpublished!. @4# L. Brillouin, Wave Propagation and Group Velocity ~Aca￾demic Press, New York, 1960!. @5# J. Peatross, S. A. Glasgow, and M. Ware, Phys. Rev. Lett. 84, 2370 ~2000!. @6# C. G. B. Garrett and D. E. McCumber, Phys. Rev. A 1, 305 ~1970!. @7# E. L. Bolda, J. C. Garrison, and R. Y. Chiao, Phys. Rev. A 49, 2938 ~1994!. @8# S. Chu and S. Wong, Phys. Rev. Lett. 48, 738 ~1982!. @9# S. A. Glasgow and J. B. Peatross ~unpublished!. @10# J. D. Jackson, Classical Electrodynamics, 2nd. ~Wiley, New York, 1975!, pp. 236 and 307. @11# M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, Phys. Rev. Lett. 66, 3132 ~1991!. @12# G. L. J. A. Rikken and B. A. van Tiggelen, Phys. Rev. Lett. 78, POYNTING’S THEOREM AND LUMINAL TOTAL ENERGY . . . PHYSICAL REVIEW E 64 046610 046610-13