= Electromagnetic charge matrix of electroweak gauge fields
{title2=$Q_{\rm gauge}=\begin{pmatrix}0&-i&0&0\\i&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}$}
In the $(A^1,A^2,A^3,B)$ basis with <gauge transformations> $U=e^{i\alpha Q}$ and $Q=T_3+Y$, the non-Abelian connection transforms by conjugation. The <Lie brackets> $[T_3,T_1]=iT_2$ and $[T_3,T_2]=-iT_1$ give $\delta A^1=\alpha A^2$, $\delta A^2=-\alpha A^1$, with neutral $A^3,B$. Thus $\delta A=i\alpha Q_{\rm gauge}A$ gives the displayed Hermitian matrix. The fields $W^\pm=(A^1\mp iA^2)/\sqrt2$ transform as $e^{\pm i\alpha}W^\pm$, while both neutral combinations are uncharged. This separates the <Adjoint representation> on the gauge fields from the doublet charge matrix $\operatorname{diag}(1,0)$ on the <Higgs field>.
Back to article page
