I know that time reversal is realized as an anti-linear operator. Nevertheless I am quite bewildered by the realization of the $T$ reversal on $\gamma$-matrices.
We assume here a Minkowski metric $\eta_{\mu\nu}=diag(1,-1,-1,-1)$.
According to N. Beisert's QFT script (ETH) for time reversal $S_T=\gamma^1\gamma^3$ the following identity holds:
$$\Lambda^\mu_\nu S_T (\gamma^\nu)^\ast S^{-1}_T = - \gamma^\mu \tag{1}$$
whereas for transformations of the orthochrone Lorentz group $L^{\uparrow}$ the transformation rule is:
$$\Lambda^\mu_\nu S(L)\gamma^\nu S^{-1}(L) = \gamma^\mu \tag{2}$$
where $\Lambda^\mu_\nu$ symbolizes a Lorentz-transformation.
(2) is valid for parity transformations $P^\mu_\nu$ since we know according to the chosen metric $(\gamma^0)^2=1$:
$$ \gamma^{\dagger 1,2,3} = -\gamma^{1,2,3} \quad \text{and}\quad \gamma^{\dagger 0} =\gamma^{ 0}\quad \text{or shortly}\quad P^{\mu}_\nu\gamma^{\dagger\nu} = \gamma^{\mu}$$
$$ P^\mu_\nu \gamma^0\gamma^\nu(\gamma^0)^{-1} = P^\mu_\nu \gamma^0\gamma^{\nu} \gamma^0 = P^\mu_\nu \gamma^{\nu\dagger} = \gamma^\mu$$
where $P^\mu_\nu$ are the components of the parity Lorentz transformation.
Therefore we get $S_P = e^{i\alpha} \gamma^0$ because an additional arbitrary phase factor can be admitted. For reasons which do not matter in this context a non-zero number $z$ as additional factor with $|z|\neq 1$ can be usally be excluded.
Similarly we have $(PT)^\mu_\nu = -\delta^\mu_\nu$ as Lorentz transformation of the 4-dimensional reflection:
$$(PT)^\mu_\nu\gamma^5 \gamma^\nu (\gamma^5)^{-1} = -\gamma^5 \gamma^\mu (\gamma^5)^{-1}=\gamma^\mu$$
which is valid because the $\gamma^5$ matrix anticommutes with the usual $\gamma$ matrices. Therefore $S_{PT} = \gamma^5 e^{i\alpha}$. An additional arbitrary phase factor as above can be admitted again.
Actually in order concatenate a parity and a time-reversal transformation, naively I would assume in simplified notation:
$$\gamma^{\nu} = \Lambda(P) P \gamma^{\nu} P^{-1} = \Lambda(P)\Lambda(T) PT \gamma^\nu T^{-1} P^{-1} = \Lambda(PT) PT \gamma^\nu (PT)^{-1}$$
in order to make P-transformations compatible with PT-transformations.
However, according to N.Beisert the transformation law (1) the time-reversal is not
$$\gamma^{\mu}= T^\mu_\nu T \gamma^\nu T^{-1} \quad\text{but}\quad T^\mu_\nu T(\gamma^\nu)^\ast T^{-1} = -\gamma^\mu$$
Is it not a contradiction ?
