Expansion of Dirac-Born-Infeld (DBI) action

I want to compute Dirac-Born-Infeld (DBI) Lagrangian \begin{align} L = 1-\sqrt{-\det(\eta_{\mu\nu}+ F_{\mu\nu})} = 1 - \sqrt{1 - (E^2 - B^2) - (E \cdot B)^2} \end{align} without actual computation of determinant, $i.e$, without computing as follows

For maxwell field tensor $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ \begin{align} -|\delta_{kl}+f_{kl}|& = - \begin{vmatrix} -1 & f_{12} & f_{13} & f_{14} \\ f_{21} & -1 & f_{23} & f_{24} \\ f_{31} & f_{32} & -1 & f_{34} \\ f_{41} & f_{42} & f_{43} & 1 \\ \end{vmatrix} \\ & = 1+(f_{23}^{2}+f_{31}^{2}+f_{12}^{2}-f_{14}^{2}-f_{24}^{2}-f_{34}^{2})-(f_{23}f_{14}+f_{31}f_{24}+f_{12}f_{34})^{2} \\ & = 1+(B^{2}-E^{2})-(E\cdot B)^{2} \end{align}

Cf what i want to do is defining $\tilde{F}^{\mu\nu} = \frac{1}{2} \epsilon^{\mu\nu\rho\sigma} F_{\rho\sigma}$ \begin{align} - \det(\delta + F) & =- \det(\delta - F) = -\exp \left[ tr(\log(\delta - F)) \right] \\ &= \left[ 1 + \frac{1}{2}F^2 - \frac{1}{8} (F \tilde{F})^2 \right] \\ & = 1+(B^{2}-E^{2})-(E\cdot B)^{2} \end{align}

I have in trouble of computing the second line, usual expansion of $\log(1-x) = -\sum_{n=1}^{\infty} \frac{1}{n} x^n$ and $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$, i try to derive the second step, but having some problems.

1. The above expansion are infinite but the results are finite

2. I couldn't see how $\tilde{F}$ comes out.

• Hint: In n dimensions, no antisymmetric p-forms for $p>n$ exist, so your series are actually finite. – ACuriousMind Nov 27 '16 at 16:12
• @ACuriousMind log series still has even powers of $F$ which are symmetric terms and infinite of them, which eventually have to be exponentiated. I can't figure out how it turns out be the simple expression. – levitt Jan 10 '17 at 14:25
• @levitt In four dimensions, there is a matrix identity which relates $F^4$ to expression at most of order $F^2$, cf. e.g. this paper. – ACuriousMind Jan 10 '17 at 15:08
• @ACuriousMind I am sorry I couldn't make anything out of that paper. I have been trying to relate traces of powers of $F$ to the two lorentz invariants, I couldn't make sense how the log and exponential power series reduce to such simple expression. If you could outline the steps in proving this, that would be helpful. Thanks. – levitt Jan 17 '17 at 13:27