= Time reversal of a Dirac field
{title2=$\hat T\psi(x)\hat T^{-1}=B\psi(-x^0,\boldsymbol x)$}
The spinor matrix $B$ obeys $B^{-1}\gamma^{\mu*}B=t_\mu\gamma^\mu$, with $t_0=1$ and $t_i=-1$. Conjugating the explicit $i$ and three spatial matrices also gives $B^{-1}\gamma^{5*}B=\gamma^5$. With spin phases $f_s=(-1)^{1/2-s}$ and identities $f_su^{-s*}(p_T)=\gamma^5Cu^s(p)$, relabeling spin and momentum in the <mode expansion of a Dirac field> gives $B=-\gamma^5C$, because $f_{-s}=-f_s$. That last relation is phase-convention dependent; the <quantum time-reversal operator> itself is not just a spinor matrix.
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