Antisymmetric matrices in effective field theory

I'm trying to construct a nonlinear $$d$$-dimensional version E&M as an effective field theory. Let $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$$ be the field strength. The most general action I can write down will be a generic function of the following? Treating $$F_\mu^\nu$$ as a $$d$$-dimensional antisymmetric matrix, I can create Lorentz invariant combinations by taking $$\text{tr}(F^2),~~\text{tr}(F^4),~~\text{tr}(F^6),\dots,\text{tr}(F^{2n}).$$ My question is: What is the largest value of $$n$$ such that the above traces are all independent of one another?

• I am not sure if this is exactly what you're asking, could this answer by Nikos M. and other users be related? Jun 27, 2019 at 20:49
• One Natural non-linear generalization of EM is DBI (Dirac-Born-Infield)action.
– Hkw
Jun 28, 2019 at 3:53
• By looking at the characteristic polynomial in 4D, i found that the only non-trivial combinations are $E^2-B^2$ and $(E\cdot B)^2$, which agrees with the above comments. Im mostly interested in 4D, so this essentially answers my question, but just for the sake of curiosity, I would like to know how this generalizes to higher dimension. Jun 28, 2019 at 13:31

The number of independent invariants of electromagnetic field in $$d$$ dimensions is $$\lfloor \frac d 2 \rfloor$$, where $$\lfloor \,. \rfloor$$ is the floor function. The number could be guessed because for a general (non-null) field there would be a Lorentz transform of the field strength tensor (see e.g. here) which would have at most $$\lfloor \frac d 2 \rfloor$$ independent real parameters: $$F'_{\mu\nu} = \Lambda \Lambda F = \begin{bmatrix} \begin{matrix}0 & B_1 \\ -B_1 & 0\end{matrix} & 0 & \cdots & 0 \\ 0 & \begin{matrix}0 & B_2 \\ -B_2 & 0\end{matrix} & & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & \begin{matrix}0 & B_r\\ -B_r & 0\end{matrix} \\ & & & & \begin{matrix}0 \\ & \ddots \\ & & 0 \end{matrix} \end{bmatrix}$$
So, for $$\text{tr}(F^n)$$ with $$n>d$$ there should be recurrence relation expressing the trace through traces of lower powers.