content_image/352657.jpg content_image/352658.jpg
vector c − ν does not transform through any of then, because it is Lorentz-boost-invariant. We conclude that c − ν and c + ν can no longer mix together to form eigenvectors, but should be eigenvectors separately. The mixture becomes impossible if and only if Λ$_{3}$ disappears from (10), (12). Then, according to (17), ♭ (2) ν becomes ∼ c + ν , hence the degeneracy is expressed as the relation 

accompanied by the relations 

<img src='content_image/352657.jpg'>

<img src='content_image/352658.jpg'>

Now we use the fact that the electric fields in eigenmodes are defined up to a common factor to renomalize them all according to ˜ e = e ( k 2 0 − k 2 $_{3}$) 1 / $^{2}$. Then, under the special kinematic condition under consideration k 2 0 − k 2 3 = 0, the electric fields in modes 1 and 2 (see eqs. (32), (35)) are finite and equal in length $^{|}$˜ e $^{(1)}$| = $^{|}$˜ e $^{+}$| = k$_{⊥}$k$_{3}$ , while that in mode 3 (36) disappears, ˜ e − = 0. Consequently this degree of freedom is impossible. As pointed in Section III, in the exceptional point k 2 0 − k 2 3 = 0, the dispersion law k 2 = 0 may correspond to an actual mode-1 photon. In view of (49) it must be accompanied by a mode-2 partner. For k 2 = 0 the electric fields in modes 1 and 2, besides the fact that they are equal in size, $^{become polarized in transverse directions, (}$˜ e $^{+}$˜ e $^{(1)}$) = | k$_{⊥}$ | k$_{3}$$_{k$_{0}$}$√$_{k}$ 2 = 0. Therefore, we have again two photon degrees of freedom. 

The symmetry relations (47)–(50) are confirmed in the infrared limit by eqs. (17) – (20) and – for any value k$_{µ}$ of the momentum – by the one-loop calculations in QED of [4]. In a theory with the dual invariance, which is not QED, another degeneracy is possible in the one-invariant case G = 0 [3] that equates the eigenvalues κ 2 and κ $_{3}$, since under the continuous duality transformation the vector ♭ (2) µ ∣ G =0 = ( ˜ F k )$_{µ}$ and the pseudovector ♭ (3) µ ∣ G =0 = ( F k )$_{µ}$ transform through each other. 

## VI. CONCLUSIONS 

In this paper we studied on the most general basis properties of small perturbations of the vacuum, filled with a constant and homogeneous background electromagnetic field with its both invariants different from zero. To this end the eigenvector decomposition of 

