How would you prove that $T^{-1}\gamma_\mu T=\gamma_\mu$? Being $T$ the time reversal operator defined as $T=\gamma_1\gamma_3 K$ with $K$ the complex conjugate operator and $\gamma$ the Dirac gamma matrices?
By substitution I wrote $K^{-1}\gamma_3 \gamma_1 \gamma_\mu \gamma_1 \gamma_3 K$ but then I don't know if the inverse of complex conjugator is equal to itself, if so I've tried to proceed as follows:
$K\gamma_3\gamma_1\gamma_\mu\gamma_1\gamma_3K$ and then I have to consider whether $\mu$ is a time index or a spatial index.
If $\mu=4$ then I have: $K\gamma_3\gamma_1\gamma_4\gamma_1\gamma_3K=-K\gamma_3\gamma_4\gamma_1\gamma_1\gamma_3K=-K\gamma_3\gamma_4\gamma_3K=K\gamma_3\gamma_3\gamma_4K=K\gamma_4K$ so I have to prove that $K\gamma_4K=\gamma_4$ which implies $[K, \gamma_4]=0$ and I don't know if this is true;
if $\mu=1$ I have $K\gamma_3\gamma_1\gamma_1\gamma_1\gamma_3K=K\gamma_3\gamma_1\gamma_3K=-K\gamma_3\gamma_3\gamma_1K=-K\gamma_1K$ and I'm left with $-K\gamma_1K=\gamma_1$
if $\mu=2$ then I have $K\gamma_3\gamma_1\gamma_2\gamma_1\gamma_3K=-K\gamma_3\gamma_1\gamma_1\gamma_2\gamma_3K=-K\gamma_3\gamma_2\gamma_3K=K\gamma_2K$
if $\mu=3$ I have $K\gamma_3\gamma_1\gamma_3\gamma_1\gamma_3K=-K\gamma_3K$
so it's a complete mess and I don't know how to proceed, maybe it is way simpler than what I did but I don't know how. It seems to me that I did something right since I get a minus sign only with $\gamma_1$ and $\gamma_3$ which are the only two matrices with imaginary numbers in them. Maybe this is the answer, a matrix filled with real numbers commutes with $K$ operator while a matrix filled with imaginary numbers anticommute with it?
I've used this relation to compute this: $\{\gamma_\mu, \gamma_\nu\}=2\delta_{\mu\nu}$