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We calculate the lowest-order corrections to the static potential for boththe generalized Born-Infeld Electrodynamics and an Euler-Heisenberg-like model,in the presence of a constant external magnetic field. Our analysis is carriedout within the framework of the gauge-invariant but path-dependent variablesformalism. The calculation reveals a long-range correction (${\raise0.7ex\hbox{$1$} \mathord{\left/ {\vphantom {1{r^5}}}\right.\kern-\nulldelimiterspace} \lower0.7ex\hbox{${r^5}$}}$-type) tothe Coulomb potential for the generalized Born-Infeld Electrodynamics.Interestingly enough, in the Euler-Heisenberg-like model, the static potentialremains Coulombian. Therefore, contrary to popular belief, the quantizedtruncated action and the truncated quantized action do not commute at all.

international journal of modern physics a

Born-Infeld Electrodynamics and Euler-Heisenberg-like Model: outstanding  examples of the lack of commutativity among quantized truncated actions and  truncated quantized actions