In physics, the Euler-Heisenberg Lagrangian describes the non-linear dynamics of electromagnetic fields in vacuum. It takes into account vacuum polarization to one loop, and it is valid for electromagnetic fields that change slowly compared to the inverse electron mass. It was first obtained by Werner Heisenberg

Werner Heisenberg

Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

 and Hans Heinrich Euler

Hans Heinrich Euler

Hans Heinrich Euler was a German physicist. He received his PhD in 1935 at the University of Leipzig under Werner Heisenberg with a thesis Über die Streuung von Licht an Licht nach der Diracschen Theorie .Euler was the first physicist who was able to show that Paul Dirac's...

, and can be expressed as:


Here m is the electron mass, e the electron charge,
,

and


In the weak field limit, this becomes: