Lagrangian density of a free electromagnetic field

Question

How do you derive the result for the lagrangian density of a free electromagnetic field

Answer 1

We have following arguments:

• Lagrangian cannot contain potential because they are not defined uniqely due to gauge invariance
• The action should be scalar

These arguments give structure $$S=a\int dtd^3x\,F_{\mu\nu}F^{\mu\nu},$$ where $F_{\mu\nu}F^{\mu\nu}=2({\bf H}^2-{\bf E}^2)$. Electric field contains term $\partial{\bf A}/\partial t$. The term $(\partial{\bf A}/\partial t)^2$ should appears in $S$ with sign "+". If not, for enough fast variation of ${\bf A}$ action $S$ becomes infinite and has not minimum. So, the constant $a$ should be negative. Finally, $$S=-a\int dtd^3x\,F_{\mu\nu}F^{\mu\nu}.$$