Expressing Covariant Derivative in Matrix Form

In summary, the given Lorentz four-vector in isospin space with three components can be used to express the covariant derivative explicitly in 2x2 matrix form, where the dot product is handled by using the Pauli matrices and rewriting the covariant derivative accordingly.
  • #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!
 
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  • #2


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.
 

Related to Expressing Covariant Derivative in Matrix Form

1. What is a covariant derivative?

A covariant derivative is a mathematical operator that describes how a vector field changes along a particular direction in a curved space. It takes into account the curvature of the space to accurately describe the vector field's behavior.

2. How is the covariant derivative expressed in matrix form?

The covariant derivative can be expressed in matrix form by using a set of basis vectors and a connection matrix. The connection matrix contains the information about the curvature of the space, and the basis vectors represent the directions in which the derivative is taken.

3. What is the advantage of expressing the covariant derivative in matrix form?

Expressing the covariant derivative in matrix form allows for a more compact and efficient representation of the derivative. It also makes it easier to perform calculations and apply the derivative to different vector fields in a curved space.

4. How does the covariant derivative differ from the partial derivative?

The covariant derivative takes into account the curvature of the space, while the partial derivative does not. This means that the covariant derivative is more accurate in describing the behavior of vector fields in a curved space, while the partial derivative is only suitable for flat spaces.

5. Can the covariant derivative be expressed in terms of the metric tensor?

Yes, the covariant derivative can be expressed in terms of the metric tensor. This involves using the Christoffel symbols, which are related to the metric tensor, in the matrix form of the covariant derivative. This allows for more efficient calculations and a better understanding of the behavior of vector fields in curved spaces.

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