content_image/344905.jpg content_image/344906.jpg content_image/344907.jpg content_image/344908.jpg content_image/344913.jpg content_image/344914.jpg
is sufficient to have at one’s disposal only the effective Lagrangian L ( F , G ) , from where the dependance on the time- and space-derivatives of the field F$_{µν}$ is disregarded [35]. In the limit of vanishing momenta the invariant coefficients Λ$_{1}$$_{,}$$_{2}$$_{,}$$_{3}$$_{,}$$_{4}$ are quadratic functions of k$_{µ}$ expressed in terms of the (momentum-independent) derivatives L$_{F}$ = ∂ L /∂ F , L$_{FF}$ = ∂ $^{2}$L /∂ F $^{2}$, L$_{GG}$ = ∂ $^{2}$L /∂ G $^{2}$, L$_{FG}$ = ∂ $^{2}$L /∂ F ∂ G as follows 

<img src='content_image/344905.jpg'>

<img src='content_image/344906.jpg'>

<img src='content_image/344907.jpg'>

<img src='content_image/344908.jpg'>

We have written these formulae referring to the special frame. The equivalence relations (4) allow to immediately restore their invariant form valid in any frame. Eqs. (17) – (20) are obtained using the definition of the polarization operator components as the second derivatives with respect to vector-potentials components (see, e .g. [3]). Insofar as one is interested in the quantities Λ$_{i}$ | k$_{µ}$ → 0 up to one-loop accuracy one should either take the Heisenberg-Euler expression for L here or pass to the infrared limit in the expressions for Λ$_{i}$ calculated within one-loop approximation in [4]. The two-loop approximation for L is also available [10]. 

From eqs. (17) – (20) the vanishing of the eigenvalues in the zero-momentum point 

<img src='content_image/344913.jpg'>

follows. This property is, in the end, a consequence of the gauge invariance that requires that the effective Lagrangian should depend only on the field strengthes, and not potentials. 

## III. DISPERSION CURVES 

In the special frame dispersion equations (14) can be represented in the form 

<img src='content_image/344914.jpg'>

