k$_{⊥}$ = 0 . The symmetry of the polarization operator should manifest itself as a degeneracy that implies that two out of its three eigenvalues should coincide, and their corresponding eigenvectors should transform through one another under the symmetry transformations, while remaining eigenvectors. By inspecting (31) we see that ♭ (1) µ and c − µ do possess this mutual property, when k$_{⊥}$ = 0 , but c + µ does not (recall that | k $^{2}$| = k 2 0 − k 2 $_{3}$, once k$_{⊥}$ = 0 ). Hence the latter cannot be admixed to c − µ in (10), and we conclude that, the same as in the one-invariant external field, Λ$_{3}$ = 0 , and κ 1 = κ $_{3}$, when k 2 ⊥ = 0 . (47) (The condition k 2 ⊥ = 0 in the general frame should be replaced by k $^{2}$E2 − kF $^{2}$k = 0). Then eqs. (10), (12) imply that Λ$_{1}$ = − Λ$_{2}$ , when k 2 ⊥ = 0 . (48) Now from (32)–(37) we see that, when k 2 ⊥ = 0 , modes 1 and 3 carry mutually perpendicular and equal in magnitude transverse electric fields e ( 1 ) and e ( 3 ) = e − polarized in the plane orthogonal to their propagation direction and to the external fields. The same is true for the magnetic fields in these modes. Therefore, in this special case the mode 1 does correspond to a real photon, as explained in Section III. Simultaneously, mode-2 becomes a purely longitudinal wave that may exist only as long as k 2 3 − k 2 0 = 0, since (35) otherwise disappears, i.e. it may only be massive. The conclusion is that the number of massless modes remains two, as expected. Another degeneracy of polarization operator is provided by the kinematical situation k 2 0 − k 2 3 = 0. This condition is invariant under Lorentz boosts along the common direction of the fields in the special frame. (Its invariant equivalent is k $^{2}$B2 + kF $^{2}$k = 0). On the other hand, the external field in the special frame is also invariant under this transformation: it does not lead out of this frame, since the constant electric and magnetic fields are not transformed by it. Hence, the polarization operator should be also invariant, which implies that some two of its eigenvalues must coincide, while the corresponding eigenvectors are transformed through one another by the Lorentz rotation in the ( 3,0 ) hyperplane. This is the case for the vectors ♭ (1) ν and c + ν in (31), since in the limit under consideration the 2-vector in the upper row of the former is negligible as compared to the other two components, and | k$_{⊥}$ | = √ | k $^{2}$| , so that ♭ (1) ν matches c + ν as a Lorentz boost partner. On the contrary, the 23