Wledig
- 69
- 1
- Homework Statement
- Consider the following representations that satisfy the Dirac algebra:
$$ \gamma^0 =
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
$$
$$
\gamma^i=
\begin{pmatrix}
0 & \sigma^i \\
-\sigma^i & 0
\end{pmatrix}
$$
and
$$ \gamma^0 =
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
$$
$$ \gamma^i=
\begin{pmatrix}
0 & \sigma^i \\
-\sigma^i & 0
\end{pmatrix}
$$
Show that they are equivalent, that is write a 4x4 unitary matrix U such that:
$$\gamma^{\mu}_B = U\gamma^{\mu}_A U^\dagger$$
- Relevant Equations
- Dirac algebra: ##\{ \gamma^\mu , \gamma^\nu \} = 2\eta^{\mu \nu}##
Where ##\eta^{\mu \nu} ## is the metric tensor from special relativity.
One thing I was thinking about doing was to consider these representations as a basis for the gamma matrices vector space, then try to determine what the change of basis from one to the other would be. However I'm unsure if it's correct to treat the representations as a basis, or whether this is the right approach at all.