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?
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5$\begingroup$ I am not sure if this is exactly what you're asking, could this answer by Nikos M. and other users be related? $\endgroup$– VangiCommented Jun 27, 2019 at 20:49
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$\begingroup$ One Natural non-linear generalization of EM is DBI (Dirac-Born-Infield)action. $\endgroup$– HkwCommented Jun 28, 2019 at 3:53
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$\begingroup$ 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. $\endgroup$– user105620Commented Jun 28, 2019 at 13:31
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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.
Explicit construction of algebraic invariants of field strength tensor in arbitrary dimensions and the discussion of algebraically special cases could be found here:
- Sokolowski, L. M., Occhionero, F., Litterio, M., & Amendola, L. (1993). Classical electromagnetic radiation in multidimensional space-times. Annals of Physics, 225(1), 1-47, doi:10.1006/aphy.1993.1050.
