For vector Gauge fields we usually write the kinetic term:
$$ \mathcal{L} ~=~ - \frac{1}{4} F_{\mu \nu} F^{\mu \nu}$$
while for matter fields e.g. for a real scalar:
$$ \mathcal{L} ~=~ \frac{1}{2} D_\mu \phi D^\mu \phi $$
with Minkowski sign convention $(+,-,-,-)$. Why is the sign different for the two and what physical difference does it make? Is it only the relative sign important or the absolute one?
