^{3}.

By using the gauge-invariant, but path-dependent, variables formalism, we study both massive Euler-Heisenberg-like and Euler-Heisenberg-like electrodynamics in the approximation of the strong-field limit. It is shown that massive Euler-Heisenberg-type electrodynamics displays the vacuum birefringence phenomenon. Subsequently, we calculate the lowest-order modifications to the interaction energy for both classes of electrodynamics. As a result, for the case of massive Euler-Heisenbeg-like electrodynamics (Wichmann-Kroll), unexpected features are found. We obtain a new long-range (

The phenomenon of vacuum polarization in quantum electrodynamics (QED), arising from the polarization of virtual electron-positron pairs and leading to nonlinear interactions between electromagnetic fields, remains as exciting as in the early days of QED [

Interestingly, it should be recalled here that the physical effect of vacuum polarization appears as a modification in the interaction energy between heavy charged particles. In fact, this physical effect changes both the strength and the structural form of the interaction energy. This clearly requires the addition of correction terms in the Maxwell Lagrangian to incorporate the contributions from vacuum polarization process. Two important examples of such a class of contributions are the Uehling and Serber correction and the Wichmann-Kroll correction, which can be derived from the Euler-Heisenberg Lagrangian. Incidentally, as explained in [

In this perspective, we also point out that extensions of the Standard Model (SM) such as Lorentz invariance violating scenarios and fundamental length have become the focus of intense research activity [

Inspired by these observations, the purpose of this paper is to extend our previous studies [

The organization of the paper is as follows: In Section

In our conventions, the signature of the metric is

As already expressed, we now reexamine the interaction energy for Maxwell theory with an additional term corresponding to the Uehling correction (Uehling electrodynamics). This would not only provide the setup theoretical for our subsequent work, but also fix the notation. To do that we will calculate the expectation value of the energy operator

Before going on, two remarks are pertinent at this point. First, the modification of Coulomb’s law in (

Second, it should be noted that the theory described by (

Now, we move on to compute the canonical Hamiltonian. For this end we perform a Hamiltonian constraint analysis. The canonical momenta are found to be

It must be clear from this discussion that the presence of the new arbitrary function,

With the foregoing information, we can now proceed to obtain the interaction energy. As already mentioned, in order to accomplish this purpose, we will calculate the expectation value of the energy operator

Making use of the above Hamiltonian structure [

We note that term (

Since the second and third term on the right-hand side of (

Before we proceed further, we wish to show that this result can be written alternatively in a more explicit form. Making use of [

Before concluding this subsection, we discuss an alternative way of stating our previous result (

Proceeding in the same way as we did in the foregoing section, we will now consider the interaction energy for Euler-Heisenberg-like electrodynamics. Nevertheless, in order to put our discussion into context, it is useful to describe very briefly the model under consideration. In such a case, the Lagrangian density reads

Having made these observations, we can write immediately the field equations for

To illustrate this important feature, we introduce the vectors

In accordance with our previous procedure [

As a consequence, we have two different situations. First, if

We now pass to the calculation of the interaction energy between static point-like sources for a massive Wichmann-Kroll-like model; our analysis follows closely that of [

Before we proceed to work out explicitly the interaction energy, we will first restore the gauge invariance in (

We are now ready to compute the interaction energy. In this case, the canonical momenta are

In the same way as was done in the previous subsection, the expectation value of the energy operator

In such a case, by employing (

Finally, with the aid of expressions (

Before we proceed further, we should comment on our result. In the case of QED (Euler-Heisenberg Lagrangian density), the parameter ^{9} m). In other words, we see that detectable corrections induced by vacuum polarization with a mass term would be present at low energy scales.

From (

Inserting these expressions in (

We now want to extend what we have done to Euler-Heisenberg-like electrodynamics at strong fields. As already mentioned, such theories show a power behavior that is typical for critical phenomena [

In the same way as was done in the previous section, one can introduce an auxiliary field,

A similar procedure can be used to manipulate the quadratic term in (

It is once again straightforward to apply the gauge-invariant formalism discussed in the foregoing section. The canonical momenta read

Requiring the primary constraint

As before, requiring the primary constraint

Following the same steps that led to (

Finally, within the gauge-invariant but path-dependent variables formalism, we have considered the confinement versus screening issue for both massive Euler-Heisenberg-like and Euler-Heisenberg electrodynamics in the approximation of the strong-field limit. Once again, a correct identification of physical degrees of freedom has been fundamental for understanding the physics hidden in gauge theories. Interestingly enough, their noncommutative version displays an ultraviolet finite static potential. The analysis above reveals the key role played by the new quantum of length in our analysis. In a general perspective, the benefit of considering the present approach is to provide a unification scenario among different models as well as exploiting the equivalence in explicit calculations, as we have illustrated in the course of this work.

The author declares that there is no conflict of interests regarding the publication of this paper.

It is a pleasure for the author to thank J. A. Helayël-Neto for helpful comments on the paper. This work was partially supported by Fondecyt (Chile) Grant 1130426 and DGIP (UTFSM) internal project USM 111458.