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SJOERD STALLINGA PHYSICAL REVIEW E 73. 026606(2006) satisfying a transport equation of the form a,p+Vv=0 (with and the dependence of F on(angular) eigenfrequency a obvious generalization to conserved quantities with a vecto- position r, and time t is suppressed, except when this com- rial character). Balance equations for two subsystems, for pact notation can give rise to ambiguity. Vector notation is example the"field!"and"matter"subsystems, have the form used if convenient and the tensor notation in all other cases p1+V.v1=-Q, (1) ne partial derivative with respect to time is denoted by an the partial derivative with respect to the spatial coordinates by da where a=x,y, z, and the Einstein summation conven- dp ,+v v2=Q (2) tion is used. Partial derivatives only apply to the quantity here p=p1+p2 and V=V+V2, and where Q represents the directly following the derivative unless brackets indicate oth dissipation of field energy, momentum or angular momentum erwise. The tensor Sap is the Kronecker tensor(SB=l if a from subsystem 1 to 2. It turns out that the exchange of =B and O otherwise), and the tensor eaBy is the Levi-Civita and material parts is such that the dissipation integrated over odd permutations, and O otherwise) the duration of the interaction is always positive. This irre- versibility is related to the coupling of the electromagnetic IL. EQUATIONS OF MOTION degrees of freedom to a continuum of harmonic oscillators. rather than to a finite number of degrees of freedom The action is the integral over time and space of the La- The split of the conservation laws into balance equations grangian density for the field and material subsystems is to some extent arbi trary, and various definitions will do. As a consequence the 1= darl dissipation of energy, momentum, and angular momentum of the field to matter are also ambiguous. a key point of inter- where the Lagrangian density is the sum of an electromag pretation is thus how to relate these quantities to the ab- sorbed heat, force, and torque on the medium that are actu- etc contribution, a contribution from the material field, and lly observed in experiment. It may therefore be the case that an interaction contribution different experimental circumstances require the application of different descriptions of momenta and forces. The answer C=E2 LB2+ do(o)[(F)2-2F2+2F·E] to the abraham- Minkowksi debate in this view is not a defi- 2 nition of "the" momentum of light in dielectric media but tum. An attempt is made in this paper to find out for which The function G(o)is positive for all nonzero o and defined physical situation the total field-plus-matter momentum of for negative angular frequencies by d(a)=G(o). The ab- he auxiliary field model is a useful quantity ence of free charges and currents implies that G(o)-0 if The main shortcoming of the auxiliary field model is that 0-0. It may be defined for complex o by analytical con- it does not take into account deformation or displacement of tinuation and is assumed to have no poles in the upper half the material medium. It is assumed that the position of each complex plane (in view of causality). The electromagnetic material point is kept fixed throughout the interaction with part of the Lagrangian density is just the vacuum electromag the electromagnetic field. This implies that the distinction netic Lagrangian density, the material part describes a con- between the space-fixed coordinate frame and the coordinate tinuous set of harmonic oscillators, and the interaction term frame fixed to the material points is lost so that a clear iden- describes the interaction of the electric field with a continu- tification as to which quantity is momentum and which quan- ous set of electric dipoles. The polarization P is thus entirely tity is pseudomomentum cannot be made. This indistinguis defined in terms of the material field F ability of uniformity of space and homogeneity of matter has the consequence that only one meaningful momentumlike dodo) conserved quantity exists within the model. This total system momentum corresponds to what is called pseudomomentum by Gordon [2], wave momentum(the sum of momentum and The dielectric displacement D and magnetic field H are then pseudomomentum)by Nelson [3] and canonical momentum defined by by Garrison and Chiao [4] The paper is organized as follows. In Sec. Il the equations of motion are derived and shown to be equivalent to Max E+2E0 wells equations in general linear dielectrics. The conserva tion laws are treated in Sec. ll and sec. lv focuses on the energy and momentum of a one-dimensional wave packet H B The paper is concluded in Sec. V with a discussion of the obtained results and an outlook on possibilities for future The scalar potential and vector potential A are introduced explorations Concerning the notation. it is mentioned that in the fol- lowing the dependence of E and B on position r and time t E=-Vd-aA 0266062satisfying a transport equation of the form t +·v= 0 with obvious generalization to conserved quantities with a vectorial character. Balance equations for two subsystems, for example the “field” and “matter” subsystems, have the form t 1 + · v1 = − Q, 1 t 2 + · v2 = Q, 2 where =1+2 and v=v1+v2, and where Q represents the dissipation of field energy, momentum or angular momentum from subsystem 1 to 2. It turns out that the exchange of energy, momentum, and angular momentum between field and material parts is such that the dissipation integrated over the duration of the interaction is always positive. This irreversibility is related to the coupling of the electromagnetic degrees of freedom to a continuum of harmonic oscillators, rather than to a finite number of degrees of freedom. The split of the conservation laws into balance equations for the field and material subsystems is to some extent arbitrary, and various definitions will do. As a consequence the dissipation of energy, momentum, and angular momentum of the field to matter are also ambiguous. A key point of interpretation is thus how to relate these quantities to the absorbed heat, force, and torque on the medium that are actually observed in experiment. It may therefore be the case that different experimental circumstances require the application of different descriptions of momenta and forces. The answer to the Abraham-Minkowksi debate in this view is not a defi- nition of “the” momentum of light in dielectric media but rather a prescription of when to use which type of momentum. An attempt is made in this paper to find out for which physical situation the total field-plus-matter momentum of the auxiliary field model is a useful quantity. The main shortcoming of the auxiliary field model is that it does not take into account deformation or displacement of the material medium. It is assumed that the position of each material point is kept fixed throughout the interaction with the electromagnetic field. This implies that the distinction between the space-fixed coordinate frame and the coordinate frame fixed to the material points is lost so that a clear identification as to which quantity is momentum and which quantity is pseudomomentum cannot be made. This indistinguishability of uniformity of space and homogeneity of matter has the consequence that only one meaningful momentumlikeconserved quantity exists within the model. This total system momentum corresponds to what is called pseudomomentum by Gordon 2, wave momentum the sum of momentum and pseudomomentum by Nelson 3 and canonical momentum by Garrison and Chiao 4. The paper is organized as follows. In Sec. II the equations of motion are derived and shown to be equivalent to Maxwell’s equations in general linear dielectrics. The conservation laws are treated in Sec. III, and Sec. IV focuses on the energy and momentum of a one-dimensional wave packet. The paper is concluded in Sec. V with a discussion of the obtained results and an outlook on possibilities for future explorations. Concerning the notation, it is mentioned that in the following the dependence of E and B on position r and time t and the dependence of F on angular eigenfrequency , position r, and time t is suppressed, except when this compact notation can give rise to ambiguity. Vector notation is used if convenient and the tensor notation in all other cases. The partial derivative with respect to time is denoted by t , the partial derivative with respect to the spatial coordinates by , where =x, y ,z, and the Einstein summation convention is used. Partial derivatives only apply to the quantity directly following the derivative unless brackets indicate otherwise. The tensor is the Kronecker tensor = 1 if = and 0 otherwise, and the tensor is the Levi-Civita tensor = 1 for even permutations of xyz, −1 for odd permutations, and 0 otherwise. II. EQUATIONS OF MOTION The action is the integral over time and space of the Lagrangian density I = dt d3 rL, 3 where the Lagrangian density is the sum of an electromagnetic contribution, a contribution from the material field, and an interaction contribution L = 0 2 E2 − 1 2 0 B2 + 0 0 dˆt F 2 − 2 F2 + 2F · E. 4 The function ˆ is positive for all nonzero and defined for negative angular frequencies by ˆ=ˆ−. The absence of free charges and currents implies that ˆ→0 if →0. It may be defined for complex by analytical continuation and is assumed to have no poles in the upper half complex plane in view of causality. The electromagnetic part of the Lagrangian density is just the vacuum electromagnetic Lagrangian density, the material part describes a continuous set of harmonic oscillators, and the interaction term describes the interaction of the electric field with a continuous set of electric dipoles. The polarization P is thus entirely defined in terms of the material field F P = 20 0 dˆF. 5 The dielectric displacement D and magnetic field H are then defined by D = L E = 0E + 20 0 dˆF, 6 H = − L B = 1 0 B. 7 The scalar potential and vector potential A are introduced via E = − − t A, 8 SJOERD STALLINGA PHYSICAL REVIEW E 73, 026606 2006 026606-2