Given the electromagnetic Lagrangian density $$ \mathcal{L}~=~-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}~=~\frac{1}{2}(E^2-B^2) $$ is a Lorentz invariant, how many other electromagnetic invariants exists that can be incorporated into the electromagnetic Lagrangian?
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As mentioned in the comments, to find all possible terms we normally only consider local, gauge invariant, Lorentz invariant interactions. There are in fact an infinite number of these. This is easiest understood using the Lagrangian. The gauge invariant field stength tensor is given by
\begin{equation}
F _{ \mu \nu } = \partial _\mu A _\nu - \partial _\nu A _\mu
\end{equation}
The only other tensors with Lorentz indices are
\begin{equation}
\epsilon _{ \alpha \beta \gamma ... } \quad , \quad g _{ \mu \nu }
\end{equation}
To lowest order in $ F $ the only non-zero invariants are:
\begin{equation}
F _{ \mu \nu } F ^{ \mu \nu} \quad , \quad \epsilon _{ \alpha \beta \gamma \delta } F ^{ \gamma \delta } F ^{ \alpha \beta }
\end{equation}
If we restrict ourselves to terms with mass dimension of $4$ or lower these are the only options (these terms are called renormalizable terms). However, one can also write down other invariants which have higher mass dimensions. One such example is the mass dimension six term,
\begin{equation}
\partial ^\mu F _{ \mu \nu } \partial ^\alpha F _\alpha ^{ \,\, \nu }
\end{equation}
Such terms are small at low energies and are often ignored. In general there are an infinite number of allowed (non-renormalizable) terms in the Lagrangian. Though it may not be trivial, such terms could be written in terms of the electric and magnetic fields to find the different combinations of $\bf E$ and $\bf B$ that form Lorentz invariants.
JeffDror
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