= Killing-form Yang-Mills Lagrangian
{c}
{title2=$\mathcal L_{\rm YM}$}
For a real compact semisimple algebra, $B=-\kappa$ is a positive internal metric. With signature $(+---)$ and connection $D_\mu=\partial_\mu+A_\mu$, the healthy Lagrangian is $\mathcal L=-B(F_{\mu\nu},F^{\mu\nu})/(4g^2)=\kappa(F_{\mu\nu},F^{\mu\nu})/(4g^2)$. <Gauge invariance> follows from Killing-form invariance and $\delta_XF=\epsilon[X,F]$. The physical <Hamiltonian> density is $[B(E_i,E_i)+B(B_i,B_i)]/(2g^2)$ after imposing the <Gauss law constraint in gauge theory> and treating the spatial boundary flux appropriately. An Abelian factor needs a separate positive invariant metric because its <Killing form> vanishes.
Back to article page
