Comment on: "Observation of Locally Negative Velocity of Electromagnetic Field in Free Space''
aa r X i v : . [ qu a n t - ph ] M a r Comment on: “Observation of Locally NegativeVelocity of Electromagnetic Field in Free Space”
In a very interesting Letter [1], and in related publica-tions [2, 3], the near- and intermediate- as well as far-fieldcausal properties of classical electro-magnetic fields havebeen discussed in great detail by making use of a repre-sentation of a Greens function where the various spatialdependencies of the components of the electric field areexplicit, i.e., E k ( x , t ) = − πǫ Z D d x ′ ( ∂ k ∂ ′ n | x − x ′ | ) Z t R t dt ′ J n ( x ′ , t ′ ) − πǫ c Z D d x ′ | x − x ′ | ( ∂ k ∂ ′ n | x − x ′ | ) J n ( x ′ , t R )+ 14 πǫ c Z D d x ′ | x − x ′ | ( θ k θ n − δ kn ) ∂ t J n ( x ′ , t R ) . (1)Here t R = t − | x − x ′ | /c is a retarded time-variableand t a suitably chosen initial time such that electriccharge density ρ ( x ′ , t ′ ) vanish for t ′ = t . Furthermore, θ n = ( x n − x ′ n ) / | x − x ′ | is a component of a unit vec-tor in terms of the observation point x and a sourcepoint x ′ ∈ D in the domain D of the current source J n ( x ′ , t ′ ). The partial derivatives ∂ k and ∂ ′ n are act-ing on the x or the x ′ dependence, respectively. Byperforming the partial derivatives ∂ k ∂ ′ n in Eq.(1) it isseen that Eq.(1) has exactly the same form as used inRef.[1]. The non-local time dependence in the first termof Eq.(1) is due to the elimination of a dependence ofthe charge density ρ ( x ′ , t ′ ) in the electric charge conser-vation law ∂ t ρ ( x ′ , t ′ ) + ∂ k J k ( x ′ , t ′ ) = 0. It was noticed inRef.[1] that the representation Eq.(1) can lead to locallynegative velocities and apparent superluminal features ofelectro-magnetic fields also demonstrated experimentally[1, 3]. These observations do not challenge our under-standing of causality since they describe phenomena thatoccur behind the light front of electro-magnetic signals(see, e.g., Refs.[3, 4] and references cited therein). It was,however, also remarked in Ref.[1] that the derivation ofEq.(1) ” appears if potentials are linked by the relativis-tic Lorentz gauge, which is something to think about ”.We find the content of this quotation misleading. Theobservable electric field as in Eq.(1) should, of course, be gauge invariant. By inspection of the actual mathemati-cal derivation as referred to (in particular Chapter 26 ofRef.[5]), one indeed observes that the Lorentz gauge isused. Now Eq.(1) appears to be different from the stan-dard and gauge-invariant expression in terms of retardedcharge- and current-densities (see, e.g., Chapter 6.5 inRef.[6]), i.e., E k ( x , t ) = − ∂∂t (cid:18) πǫ c Z d x ′ J k ( x ′ , t R ) | x − x ′ | (cid:19) − πǫ Z d x ′ ∂ ′ k ρ (cid:0) x ′ , t R (cid:1) | x − x ′ | , (2)where ∂ ′ k ρ ( x ′ , t ′ ) in Eq.(2) has to be evaluated for a fixedvalue of t ′ = t R . By making use of current conserva-tion and elementary partial integrations one finds, aftersome remarkable and non-trivial cancellation of terms,that Eqs.(1) and (2) are, actually equivalent apart frompossible boundary terms that vanishes if, e.g., J n ( x ′ , t ′ )goes to zero sufficiently fast at the boundary of D . Fur-thermore, the gauge-independence of Eq.(2) has beendiscussed in various contexts (see, e.g., Ref.[7] and ref-erences cited in Ref.[8]). With regard to another issueas raised by Budko [1] on the role of Eq.(1) in quantumoptics, we notice at least the following fact. Interpretedas an expectation value of second-quantized physical de-grees of the electro-magnetic field in the presence of arbi-trary but classical charge- and current-densities sources,Eq.(2) is obtained by exactly solving for the correspond-ing quantum-mechanical equations of motion (see Ref.[8]and references cited therein). As long as characteristicdimensions of the sources are small compared to any typ-ical wave-length of photons, this should be an exact re-sult according to the analysis of the infrared problem inquantum field theory. In view of these remarks, we donot find that the near- and intermediate-field terms asexhibited explicitly in Eq.(1) are by ”far less understoodin quantum optics” but, nevertheless they deserves moreattention in a quantum-mechanical context.B-S. K. SkagerstamDepartment of PhysicsNTNUN-7491 TrondheimNorwayPACS numbers: 03.50.De, 41.20.Jb [1] N. V. Budko, Phys. Rev. Lett. , 020401 (2009).[2] N. V. Budko, Phys. Rev. A 80 , 053817 (2009).[3] N. V. Budko, “
Superluminal, subluminal, and negativevelocities in free-space electromagnetic propagation” , in“
Advances in Imaging and Electron Physics” , pp. 47-72,Vol , Ed. P. W. Hawkes (Academic Press, 2011).[4] A. D. Jackson, A. Lande, and B. Lautrup, Phys. Rev.
A64 , 044101 (2001).[5] A. T. Hoop, “
Handbook of Radiation and Scattering of Waves” (Academic Press, New York, 1995).[6] J. D. Jackson, “
Classical Electrodynamics” , Third Edition(Wiley, New York, 1999).[7] J. D. Jackson, Am. J. Phys. , 917 (2002), where also aprinting sign error in B.-S. K. Skagerstam, Am. J. Phys. , 1148 (1983), and Eq.(9), was observed.[8] B.-S. K. Skagerstam, K. E. Eriksson, and P. K. Rekdal,Classical and Quantum Gravity36