- #1

LesterTU

- 7

- 0

## 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!