Wave propagation in electromagnetic systems with a linear response

Transcription

1 Wave propagation in electromagnetic systems with a linear response Yakov Itin Institute of Mathematics, Hebrew University of Jerusalem and Jerusalem College of Technology GIF4, JCT, Jerusalem October, 2014 Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 1 / 22

2 Introduction Premetric electrodynamics - why we need it? Lorentz invariance violation models It is well known that interior and exterior symmetries play a central role in classical and quantum field theory. Violation of these symmetries is certainly a key point of possible extension of the standard field theories. Moreover the dynamical violation of the interior symmetry is often incorporated in field theoretic construction as a necessary element (Higgs scalar field). Violation of exterior symmetries, such as CPT and Lorentz symmetry, if it really exists, must yield crucial modifications of the very basis of the field theory. Modern field theories usually predict modifications of the light cone structure expressed by an anisotropic dispersion relation, a birefringent vacuum and a violation of local Lorentz and CPT invariance. Such theoretical phenomena emerge in Loop quantum gravity String theory Very Special Relativity models Since these models are still very far from their complete form and observational predictions, it is very important to have a phenomenological model that predicts the indicated optics phenomena. Moreover it is rather possible that even when we will already have a good quantum gravity we will need a phenomenology models in order to give observational predictions. It is due to the huge gap between the energies that are acceptable now and the energies for which quantum gravity effects come to be relevant. Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 2 / 22

3 Introduction Premetric electrodynamics - why we need it? Standard Model Extension (SME) construction due to Kostelecky and al. Modification of the electrodynamics action by an 1-parametric axion term is an old idea. It was applied by Ni in 1977 for studying possible violation of the equivalent principle. In Carroll-Field-Jackiw electrodynamics, modified energy-momentum tensor and birefringent dispersion relation for axion model was derived and the astronomical data was applied for deducing the upper magnitude of the axion. In Kostelecky s model, an extension of the axion model was proposed in the form of the modified Lagrangian L = 1 4 ηik η jl F ij F kl 1 4 κijkl F ij F kl. (1) Here the fixed coefficients κ ijkl represent small violation from the standard electrodynamics model. The symmetries of κ ijkl are readily read off from Eq.(1): κ ijkl = κ jikl = κ ijlk = κ klij. (2) Moreover the leading term is usually absorbed by field redefinition, thus 20 independent parameters of κ ijkl are left over. An extension of the construction (1) to the standard model, called Standard Model Extension (SME) was worked out. Theoretical work in this model has spanned a number of areas that include string theory, gravity theory, quantum field theory and cosmology. Several exceptionally accurate Lorentz tests have recently been conducted, and other experiments are currently under way. Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 3 / 22

4 Introduction Premetric electrodynamics - why we need it? Problem with SME To my opinion, a central problem in theses approaches is an absence of a clear physical meaning of the the individual terms in the modified Lagrangian, apart from their covariance property. Premetric Electrodynamics Premetric Electrodynamics as an alternative Lorentz violation model. Premetric electrodynamics is mainly inspired by ideas from solid state physics. Consequently the individual modified terms of such Lorentz violation model can be expressed in well known physics notations such as dielectric permittivity and magnetic permeability. Premetric Electrodynamics as an extension of electromagnetic sector of SME. It includes 36 independent parameters instead of 20 (or 21) parameters of Kostelecky s model. Premetric Electrodynamics as a bridge between macrophysics and (very) microscopic physics. Premetric electrodynamics as an extension of macroscopic physics Metamaterials,... Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 4 / 22

5 Premetric electrodynamics - a brief account In terms of differential forms, the integral Maxwell equations are given in a compact covariant form F = 0, H = J. (3) M 2 N 2 N 3 Here F is the even (untwisted) 2-form of field strength, while H is the odd (twisted) 2-form of excitation, J is the odd 3-form of electric current. The integration domain M 2 is a smooth closed 2-dimensional surface, while N 3 is a smooth 3-dimensional oriented hyper-surface with a smooth boundary N 2 = N 3. Eqs.(3) have a clear physical interpretation. The first equation represents the conservation law of the magnetic flux. The second equation of is a consequence of the electric current conservation law. For the details of such an interpretation, see the Hehl-Obukhov book. In the differential form, the Maxwell equations read df = 0, dh = J. (4) Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 5 / 22

6 Premetric electrodynamics - brief account In order to derive the wave-front conditions, we apply the covariant jump conditions, that are a consequence of the integral Maxwell equations (3). Consider an arbitrary smooth non-degenerate hypersurface Σ in space-time. Let its implicit description be given by the equation ϕ(x i ) = 0. Thus, at an arbitrary point of Σ, the covector q i = ϕ/ x i is assumed to be well-defined and non-zero. Here and in the sequel, the Roman indices take values in the range {0, 1, 2, 3}. We apply the following covariant description Definition 1: A generalized electromagnetic wave is a set of the solutions, F and H, of Maxwell s system (3) that are non-zero on one side of some hypresurface ϕ(x i ) = 0 and zero on its other side. With this definition, the generic jump conditions, take the form of Wavefront conditions F dϕ = 0, H dϕ = 0. (5) In tensorial representation with the permutation pseudotensor ɛ ijkl, they read ɛ ijkl F jk q l = 0, H ij q j = 0. (6) The hypersurface ϕ(x i ) = 0 appearing in Definition 1 is referred to as a wavefront. Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 6 / 22

7 Constitutive pseudotensor Eqs.(3) present only the formal structure of electromagnetism. They are endowed with a physical content only when a constitutive relation between the fields H and F is postulated. We consider Linear, local constitutive relation: H ij = 1 2 χijkl F kl. (7) Due to its definition, the constitutive pseudotensor χ ijkl possess the symmetries χ ijkl = χ jikl = χ ijlk. (8) Hence, in 4-dimensional space, χ ijkl has 36 independent components. It is easy to compare the model (4) with (7) to Kostelecky s Lagrangian (1). When a Lagrangian of the standard form is considered and the constitutive relation (7) is substituted one arrives to that is equivalent to (1) providing L = 1 2 F ij H ij = 1 4 χijkl F ij F kl (9) ( ) χ ijkl = η ik η jl η il η jk + κ ijkl. (10) Thus two models are rather similar. However, they are not completely equivalent. Indeed, SME starts from a Lagrangian, i.e. κ ijkl = κ klij. Thus it has 21 indep. param. The premetric electrodynamics starts from the conservation laws, it has the full set of 36 indep. param. SME Lagrangian involves explicitly the metric tensor η ik. In a contrast, premetric electrodynamics does not use any metric tensor nor a connection. Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 7 / 22

8 Constitutive pseudotensor irreducible decomposition A useful way to deal with a multi-component tensor, such as χ ijkl, is to decompose it into the sum of simpler independent sub-tensors with fewer components. A common way to derive such a decomposition is to apply the Young diagram technique. The Littlewood-Richardson rule restricts the number of the relevant Young s diagrams. In our case, there are only 3 different diagrams = (11) with the dimensions 36 = It is a decomposition of the tensor space into the direct sum of invariant subspaces of smaller dimensions. Thus the canonical decomposition of the constitutive pseudotensor into three pieces is unique and irreducible. Following the Hehl-Obukhov book, we denote the decomposition depicted in (11) as χ ijkl = (1) χ ijkl + (2) χ ijkl + (3) χ ijkl. (12) Here the axion part (3) χ ijkl is depicted by the third diagram of (11). It has only 1 comp. (3) χ ijkl = χ [ijkl] = αε ijkl. (13) The skewon part (2) χ ijkl of 15 independent comp. corresponds to the middle diagram of (11). (2) χ ijkl = 1 ( ) χ ijkl χ klij. (14) 2 The principal part of 20 independent comp. corresponds to the first diagram (1) χ ijkl = 1 ( ) 2χ ijkl + 2χ klij χ iklj χ ljik χ iljk χ jkil. (15) 6 Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 8 / 22

9 Characteristic system Substituting F kl = (1/2)(a k q l a l q k ) into the system Eqs.(6) we obtain the characteristic system where M ik a k = 0, (16) M ik = χ ijkl q j q l. (17) Eq.(16) is a linear system of 4 covariant equations for the 4 components of the covector a k. However, the relations M ik q k = 0 and M ik q i = 0 (18) hold identically. Consequently the system (16) is singular. This fact is an algebraic expression of the gauge invariance. In order to have a non-trivial solution a k of the characteristic system (16), the matrix M ik must satisfy certain consistent relation. This dispersion relation is a polynomial algebraic equation for q i with the coefficients depending on the media parameters. Usually an equation of the form (16) is consistent (has non-trivial solutions) when the relation det(m) = 0 holds. Due to gauge invariance, the electromagnetic system is a singular, so its determinant vanishes identically. Hence, the system (16) accepts a physical meaning, only if it has at least two linearly independent solutions one for gauge freedom and one for physics. We recall a known fact from linear algebra: A linear system has two (or more) linearly independent solutions if and only if the rank of its matrix M ij is of 2 (or less). It means that the adjoint of the matrix M ij must be equal to zero. Recall that the adjoint matrix is constructed from the cofactors of M ij. Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 9 / 22

10 Dispersion relation Thus, in electromagnetism, as well as in an arbitrary U(1)-gauge invariant system, existence of physically meaningful solutions of a system M ik (q)a k = 0 requires A ij = adj(m) = 0. (19) This condition is somewhat unusual because it is given in a matrix form. However, we have here only one independent condition. Indeed, for gauge invariant system we can observe the following algebraic fact: The adjoint matrix A ij = adj(m) is proportional to the tensor product of the covectors q i, A ij = λ(q)q i q j, (20) where λ(q) is a polynomial of q. Since q i is non-zero, the dispersion relation takes the ordinary scalar form λ(q) = 0. (21) It follows that the scalar function λ(q) is a 4-th order homogeneous polynomial of the wave covector q i. It means that, for an arbitrary linear response medium, there are at most two independent quadratic wave cones at every points of the space. Observe that in a widely used non-covariant treatment of the characteristic system (16) one obtains instead of (21) the equation of the form ω 2 λ(q) = 0. In an addition to two light cones mentioned above, this equation has a degenerate solution q i = (0, k 1, k 2, k 3 ), called zero frequency electromagnetic wave. In our covariant description, this unpleasant solutions are absent. Thus the problem how to interpret or how to remove the zero frequency waves is not emerges in our approach at all. The explicit form of the coefficients of the quartic form λ(q) called Tamm-Rubilar tensor. To our opinion, it is more convenient to work with the generic form adj(m) = 0. Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 10 / 22

11 Skewon contribution to dispersion relation Since skewon field provides a central difference between SME and Premetric Electrodynamics, we provide some recent results about skewon influence on the wave propagation. The optic matrix is irreducibly decomposed into the sum of two terms M ik = P ik + Q ik with P ik = (1) χ ijkl q j q l, Q ik = (2) χ ijkl q j q l. (22) (1) Symmetry: The principle optic tensor P ik is symmetric while the skewon optic tensor Q ik is antisymmetric: P ik = P ki, Q ik = Q ki. (23) (2) Linear relations: Since the partial pseudo-tensors (1) χ ijkl and (2) χ ijkl preserve the symmetries of the original pseudo-tensor χ ijkl, the linear relations of the type (18) hold also for the matrices P ik and Q ik, i.e., P ik q k = 0, and Q ik q k = 0. (24) (3) Determinants: Due to the linear relations between the columns of the matrices, we have, det(p) 0, and det(q) 0. (25) It is in addition to the relation det(m) 0. (4) Adjoint of the skewon optics tensor: In order to have a non-trivial (non-zero) expression for the adjoint matrix, the rank of the original 4 4-matrix must be equal to 3. But the rank of an arbitrary antisymmetric matrix is even. Thus our skewon matrix satisfies adj(q) 0. (26) Thus the skewon part alone does not provide a non-trivial dispersion relation. Thus skewon can serve only as a supplement to the principle part. Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 11 / 22

12 Skewon covector Since the skewon part of the constitutive tensor contributes to the wave propagation only via the antisymmetric tensor Q ij, it has a simpler representation, which we now derive We start with the relation Q ij q j = 0. (27) A solution of this equation reads Q ij = 1 2 ɛijrs (Y r q s Y sq r ). (28) Using the skew-symmetry, this expression can be rewritten finally as Q ij = ɛ ijkl q k Y l. (29) with an arbitrary covector Y i. We will refer to Y i as the skewon optic covector. Let us list its basic properties: (1) Since Q ij is quadratic in q i, the components of Y i are the first order homogeneous functions of q i ; (2) Y i is a covector density, because Q ij is a tensor density; (3) Due to (29), Y i is defined only up to an arbitrary addition of the wave covector q i. So we have here a type of a gauge symmetry for Y which is similar to the ordinary gauge symmetry of the Maxwell system; (4) An additional gauge fixing condition on Y i can be applied. Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 12 / 22

13 Skewon part of the dispersion relation How the skewon part contributes to the dispersion relation? Calculate the adjoint adj(m) of the optic tensor M ij = P ij + Q ij we have A ij = 1 3! ɛ ( ii 1 i 2 i 3 ɛ jj1 j 2 j 3 P i 1 j 1 P i 2j 2 P i 3j 3 + 3P i 1j 1 Q i 2j 2 Q i 3j 3 + 3P i 1j 1 P i 2j 2 Q i 3j 3 + Q i 1j 1 Q i 2j 2 Q i ) 3j 3 (30). The left hand side of this equation is a symmetric matrix, thus the antisymmetric matrices in its right hand side must vanish. Indeed, we can check straightforward that the identities ɛ ii1 i 2 i 3 ɛ jj1 j 2 j 3 Q i 1j 1 Q i 2j 2 Q i 3j 3 0, ɛ ii1 i 2 i 3 ɛ jj1 j 2 j 3 P i 1j 1 P i 2j 2 Q i 3j 3 0 (31) hold for an arbitrary symmetric matrix P and antisymmetric matrix Q. Note that Eq.(31) represents the mentioned fact that the adjoint of an antisymmetric matrix Q vanishes identically. Thus we remain in (30) with the relation λ(p, Q)q i q j = 1 3! ɛ ii 1 i 2 i 3 ɛ jj1 j 2 j 3 ( P i 1 j 1 P i 2j 2 P i 3j 3 + 3P i 1j 1 Q i 2j 2 Q i 3j 3 ) = 0. (32) The first term in the right hand side is the adjoint of the symmetric matrix P ij. Consequently, the dispersion relation takes the form λ(p)q i q k + 1 2! ɛ ii 1 i 2 i 3 ɛ jj1 j 2 j 3 P i 1j 1 Q i 2j 2 Q i 3j 3 = 0, (33) where λ(p) is the quartic form evaluated on the principle part only. Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 13 / 22

14 Skewon part of the dispersion relation We calculate the second term of (33). The result is surprisingly simple: Proposition 1: For a most generic linear constitutive pseudo-tensor χ ijkl, the dispersion relation reads Observe some immediate consequences of this formula: λ(p) + P ij Y i Y j = 0. (34) (1) The skewon modification of the light cone is provided by a quadratic form P ij Y i Y j. (2) Due to the identity P ij q i = 0, Eq.(34) is invariant under a gauge transformation of the skewon covector Y i Y i + αq i. (3) If there is a solution q i of Eq.(34) for which Y i (q) q i, then λ(p) = 0, i.e., the wave vector q i lies on the non-modified light cone. (4) If Y i (q) q i for all solutions of Eq.(34), then the corresponding skewon does not modify the light cone structure. (5) If λ(p) = 0 for all solutions of Eq.(34), then simultaneously P ij Y i (q)y j (q) = 0 holds. In this case we remain with a subset of the original (non-skewon) light cone. (6) For a non-trivial solution of the dispersion relation, two scalar terms λ(p) and P ij Y i (q)y j (q) must be of opposite signs. In order to analyze the contributions of the skewon part to wave propagation, we restrict in the next section two media with a simplest principal part. Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 14 / 22

15 Skewon contributions to the (pseudo) Riemannian vacuum Let the principal part be constructed from a generic metric tensor of Euclidean or Minkowski signature. The skewon part, however, let be of the most general form. How, in this case, the skewon modifies the standard light cone structure? An extensive analysis of this problem was given by Obukhov and Hehl in their PRD-paper. Let the principal part of the constitutive pseudo-tensor be presented in the metric-type form (1) χ ijkl = ( ) g g ik g jl g il g jk. (35) From (35), the principal optic tensor reads P ik = g ( g ik q 2 q i q k ). (36) Here the indices are raised by the metric g ik, i.e., q i = g ij q j, q 2 = g ij q i q j. (37) The principle part contribution to the dispersion function reads λ(p) = sgn(g) g q 4. (38) The contribution of the skewon part follows straightforward from (35) P ij Y i Y j = ( g Y 2 q 2 (Y, q) 2). (39) where the scalar product is defined by the use of the metric, (q, Y ) = g ij Y i q j. Proposition 2: The dispersion relation for the most generic skewon media in the (pseudo) Riemannian vacuum is represented as sgn(g)q 4 + Y 2 q 2 (Y, q) 2 = 0. (40) Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 15 / 22

16 Lorenz-type gauge The dispersion relation (40) may be given even in a more simpler form. Since the covector Y i is defined only up to an arbitrary addition of the covector q i, we can apply an arbitrary scalar gauge condition. On a space endowed with a metric it can be used in a form similar to the Lorenz gauge condition (Y, q) = g ij Y i q j = 0. (41) Note that this condition is applicable for an arbitrary signature of the metric tensor, even it is usually used for the Lorentz metric. With this expression at hand, the dispersion relation takes a form of a system sgn(g)q 4 + q 2 Y 2 = 0 and (q, Y ) = 0. (42) Euclidean signature: For a positive signature of the metric, the dispersion relation reads q 4 + Y 2 q 2 = 0. (43) Both terms in (43) are nonnegative. Consequently q 2 = 0. It yields Proposition 3: For a Euclidean signature metric space endowed with an arbitrary skewon, the dispersion relation has a unique trivial solution, q i = 0. Lorentz signature: Also for a negative signature of the metric, it is more convenient to use the dispersion relation in the Lorenz gauge. Thus we are dealing with the system Proposition 4: q 4 = Y 2 q 2, and (Y, q) = 0. (44) In the Lorentz signature metric space, the solution of the skewon modified dispersion relation can be spacelike or null, q 2 0. Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 16 / 22

17 Photon propagator Photon propagator and its decomposition A solution of the linear system can be written formally as M im a m = j i. (45) a i = D ik j k. (46) The tensor D ik is called the photon propagator. It is the momentum representation of the Green function of the system. Usually in quantum electrodynamics, one considers the symmetric propagators. We will see that the antisymmetric part of the propagator will exhibit the skewon modification of the standard electrodynamics model. For a generic 2-nd order tensor, a decomposition into a symmetric and antisymmetric parts is admissible. Moreover, using the metric tensor we can extract the traceless piece from the symmetric part. Consequently we come to an irreducible decomposition of the photon propagator Here the trace part is The symmetric traceless part is denoted by Ďij. Evidently, D ij = 1 4 Dg ij + Ďij + D ij. (47) D = g ij D ij. (48) Ď ij = D (ij) 1 4 Dg ij, (49) with the properties g ij Ď ij = 0, Ď ij = Ďji, Dij = D ji, (50) Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 17 / 22

18 Photon propagator Gauge transformations of the photon propagator Observe that due to the electric charge conservation, q k j k = 0, the propagator D ik is defined only up to an additional term proportional to q k. Also a term proportional to q i can be freely added to D ik. It is due to the gauge invariance, a i a i + Cq i. Consequently the gauge transformation of the propagator can be written in the form D ik D ik + q i φ k + q k ψ j. (51) Here φ i and ψ i are arbitrary. When they are chosen as covectors one gets Lorentz invariant gauge transformation. Alternatively, in some problems it is useful to use a non-invariant Coulomb-type gauge. In this case, some components of φ i and ψ i are fixed. For the ordinary symmetric propagator, the transformation (51) is used with only one arbitrary function. In a correspondence with the usual practice, we call the simplest photon propagator as the Feynman propagator and denote it as FĎ ik for the symmetric and F D ik for the antisymmetric case. Recall that Feynman propagator is obtained by removing all longitudinal components. The Landau-gauge photon propagator LĎ ik is defined by the conditions L Ď ik q k = 0, L Dik q k = 0. (52) The relations between these two covariant propagators are derived by the gauge transformations ( ) L D ik = F D ik F q D im q k + F m D mk q i q 2 + q 4 q i q k. (53) Here we define a scalar = F D mnq m q n. (54) Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 18 / 22

19 Photon propagator Photon propagator for a generic linear response system Our problem is to derive the solutions a m of the singular system M im a m = j i. (55) It is instructive to recall how one is dealing with a similar regular system in the basic linear algebra. One multiplies both sides of Eq.(55) by the adjoint matrix A ki = adj(m). The classical Laplace expansion of the determinant is written in tenor notations as Thus one remains with the equation M ij A kj = ( detm) δ i k, (56) (det M) a i = A ij j i For invertible matrix, the solution is accepted now by a division of both sides of the equation by the scalar factor det M, i.e. by forming the inverse matrix. It is well known that, for a non-singular matrix M ij, the solution of Eq.(45) is unique. In our case, the procedure described above does not work because the matrix is not invertible, det M = 0. For a singular system, we use a similar procedure but with the second adjoint B ijkl instead of the first adjoint A ij. The second adjoint is defined by removing two rows and two columns from the original matrix. Formally it can be written as The Laplace identity for the second adjoint reads B ijkl = 1 2! ɛ ijabɛ klcd M ac M bd. (57) M ij B krjs = δ i k Ars δi r A ks. (58) Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 19 / 22

20 Photon propagator Photon propagator for a generic linear response system We multiply two sides of the equation by the tensor B irks and obtain M ij a j = j i. (59) B irks M ij a j = B irks j i. (60) Using the second order Laplace expansion (58), we rewrite this equation as Substituting here the expression of the adjoint we obtain A rsa k A rk a s = B irks j i. (61) λq r (q sa k q k a s) = B irks j i. (62) The problem now is to extract the wave covector from this equation. We multiply both sides of Eq.(62) by the metric tensor g rs and use the Lorenz gauge condition a sq s = 0. Consequently it takes the form a k = 1 λq 2 grs B irks j i. (63) From Eq.(63), the propagator tensor reads of as F D ij = 1 λq 2 Bm ijm. (64) This expression is derived by removing all longitudinal terms, thus it must be treated as the Feynman photon propagator. Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 20 / 22

21 Photon propagator Photon propagator for skewon media we calculate the propagator for the Minkowski vacuum modified by a generic skewon field. D ij = 1 ( ( ) λq 2 q 2 g ij + 2q i q j q 2 + ɛ ikmj q k Y m q 2 + ) q 2 Y i Y j + Y 2 q i q j. (65) Due to the gauge invariance (51), we can remove from this expression the terms proportional to the components q i and q j. Consequently, the q 2 factor in the denominator is canceled and we remain with the following expression for the Feynman propagator F D ij = 1 ) (g ij q 2 + ɛ ikmj q k Y m + Y i Y j, (66) λ or, explicitly, F D ij = g ij q 2 + ɛ ikmj q k Y m + Y i Y j q 4 q 2 Y 2 + (q, Y ) 2. (67) We can simplify this expression by using the skewon covector that satisfies the Lorenz-type gauge condition (q, Y ) = 0. Consequently, F D ij = g ij q 2 + ɛ ikmj q k Y m + Y i Y j q 4 q 2 Y 2. (68) In this gauge of the skewon, the Landau propagator also takes a simple form L D ij = ( gij q 2 q i q j ) + ɛikmj q k Y m + Y i Y j q 4 q 2 Y 2. (69) Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 21 / 22

22 Photon propagator Photon propagator for skewon media Observe some basic properties of this propagator: (1) For the vanishing skewon field, the propagators return into their standard vacuum form F D ij = g ij q 2, L D ij = g ij q 2 q i q j q 4. (70) The same is true also for the non-zero but physically trivial skewon covector Y i q i. (2) Skewon propagator has nontrivial symmetric and antisymmetric parts. In the case of the Feynman gauge, they are expressed as F D (ij) = g ij q 2 + Y i Y j q 4 q 2 Y 2, F D [ij] = ɛ ikmj q k Y m q 4 q 2 Y 2. (71) (3) The propagator is singular in at most four roots of the denominator. As a result the are at most two light cones at every space-time point birefringence. (4) If the skewon covector Y i is regular in q then in the first order approximation of a small skewon field F D (ij) = g ij q 2, F D [ij] = ɛ ikmj q k Y m q 4. (72) Thus the small skewon field affects only the antisymmetric part of the propqagator. Yakov Itin (Institute of Mathematics) Wave propagation in electromagnetic systems with a linear response 22 / 22