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

TL;DR

This paper compares the effects of Born-Infeld and Euler-Heisenberg-like models on static potentials in a magnetic field, revealing non-commutativity of quantized actions and showing long-range corrections in one model but not the other.

Contribution

It demonstrates the non-commutativity between quantized truncated actions and truncated quantized actions in these models, with explicit calculations of potential corrections.

Findings

Born-Infeld model exhibits a long-range $1/r^5$ correction to Coulomb potential.

Euler-Heisenberg-like model's potential remains Coulombian.

Quantized truncated actions and truncated quantized actions do not commute.

Abstract

We calculate the lowest-order corrections to the static potential for both the generalized Born-Infeld Electrodynamics and an Euler-Heisenberg-like model, in the presence of a constant external magnetic field. Our analysis is carried out within the framework of the gauge-invariant but path-dependent variables formalism. 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) to the Coulomb potential for the generalized Born-Infeld Electrodynamics. Interestingly enough, in the Euler-Heisenberg-like model, the static potential remains Coulombian. Therefore, contrary to popular belief, the quantized truncated action and the truncated quantized action do not commute at all.