This question concerns the Dirac equation and the $4\times4$ $\gamma$-matrices. The task is to prove that a similarity transformation of the standard $\gamma$-matrix conserves the commutation relation

$$
\{\gamma^\mu,\gamma^\nu\} ~=~ 2g^{\mu\nu},
$$

where $2g^{\mu\nu}$ is the metric tensor $\text{diag}(1,-1,-1,-1)$, and the similarity transformation is defined as

$$
\tilde{\gamma}^\mu = S \gamma^\mu S^\dagger,
$$

and $S$ is a unitary matrix. I will write down the start of my proof to show where I stop. First of all, we can use that $S$ is unitary and show that $\gamma^\mu = S^\dagger\tilde{\gamma}^\mu S$, and insert this into the commutator. This leaves us, again using that $SS^\dagger = I$, with

$$
S^\dagger\{\tilde{\gamma}^\mu,\tilde{\gamma}^\nu\}S = 2g^{\mu\nu}
$$

which again gives us

$$
\{\tilde{\gamma}^\mu,\tilde{\gamma}^\nu\} = 2Sg^{\mu\nu}S^\dagger.
$$

In order for the proof to hold, it requires that $g^{\mu\nu}$ and $S$ commute so that

$$
2Sg^{\mu\nu}S^\dagger = 2g^{\mu\nu}SS^\dagger = 2g^{\mu\nu}.
$$

So my question is: Do all unitary matrices commute with the metric tensor $g^{\mu\nu}$? If yes, how can I show this easily?

This post imported from StackExchange Physics at 2014-05-04 11:17 (UCT), posted by SE-user camzor00