of $\omega^2$ must be negative, representing an unstable mode. Since the momentum $q$ of the instability mode is in the range $0 < q < k$, we expect that the solution decays toward the lowest energy state with momentum $k=0$.

We have also performed numerical analysis of time dependent solutions with the initial configuration of constant electric field $E$ assuming that solutions depend only on the coordinates $t$ and $x_2$. We found that a small localized perturbation generates a domain where the electric and magnetic fields fluctuate and that the domain expands at the speed of light. The magnetic field in the domain carries a range of momenta, which tend to move to zero momentum state. The strength of the electric field decays as the domain expands, suggesting that the system will eventually settle down to the trivial state with $E=0$.

To summarize, the instability of the constant electric field in the Maxwell-Chern-Simons theory in the 5-dimensional Minkowski space, which we found in our previous paper [@Nakamura:2009tf], leads to the trivial vacuum state with no background field strength $F_{IJ}=0$. This reminds us of the Schwinger mechanism where a constant electric field is screened by virtual production of electron-positron pairs.

This 