  We explain a conjecture which states that the proper-time series expansion of
the one-loop effective Lagrangian of quantum electrodynamics can be partially
summed in all terms containing the field-strength invariants $\mathcal{F} =
\frac{1}{4} F_{\mu\nu}F^{\mu\nu} (x)$, $\mathcal{G}= \frac{1}{4} \tilde
F_{\mu\nu}F^{\mu\nu}(x)$. This summation is encapsulated in a factor with the
same form as the (spacetime-dependent) Heisenberg-Euler Lagrangian density. We
also discuss some implications and a possible extension in presence of gravity.
We will focus on the scalar field case.
