Expressing Covariant Derivative in Matrix Form

[LesterTU]

Homework Statement

We are given a Lorentz four-vector in "isospin space" with three components $\vec v^{\mu} = (v^{\mu}_1, v^{\mu}_2, v^{\mu}_3)$ and want to express the covariant derivative $$D^{\mu} = {\partial}^{\mu} - ig\frac {\vec \tau} {2}\cdot \vec v^{\mu}$$ explicitly in $2\times 2$ matrix form.

Homework Equations

This particular covariant derivative was introduced in the context of non-Abelian gauge theories for the SU(2) case, and I don't think the isospin space part of the question is relevant for this problem (hence why they put quotation marks), it's just to practice notation.

The ${\tau}_i$ are the Pauli matrices.

The Attempt at a Solution

I think I know how to handle the dot product: $$\vec \tau \cdot \vec v^{\mu} = {\sigma}_1v^{\mu}_1 + {\sigma}_2v^{\mu}_2 + {\sigma}_3v^{\mu}_3 = \begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix}v^{\mu}_1 +
\begin{pmatrix}
0 & -i \\
i & 0 \\
\end{pmatrix}v^{\mu}_2
+
\begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}v^{\mu}_3 =
\begin{pmatrix}
v^{\mu}_3 & v^{\mu}_1 - iv^{\mu}_2 \\
v^{\mu}_1 + iv^{\mu}_2 & -v^{\mu}_3 \\
\end{pmatrix}$$ Now I'm wondering if I'm just supposed to insert this in the expression for $D^{\mu}$ and leave it as it is, or should I express everything in terms of matrices? I don't know how if that's the case!

 

[mark726]

Your approach is correct so far. To express everything in terms of matrices, you can rewrite the covariant derivative as: $$D^{\mu} = \begin{pmatrix}
\partial^{\mu} - igv^{\mu}_3 & -ig(v^{\mu}_1 - iv^{\mu}_2) \\
-ig(v^{\mu}_1 + iv^{\mu}_2) & \partial^{\mu} + igv^{\mu}_3 \\
\end{pmatrix}$$ So now you have expressed the covariant derivative in terms of matrices.