# Time reversal operator for general spin

Gross' QFT textbook states that spin-0, spin-1/2 and spin-1 fields transform as follows under a time reversal operation, \begin{align} \mathcal T \phi(t,\mathbf x) \mathcal T^{-1} &= \eta_\phi^T \;\phi(-t,\mathbf x) \\ \mathcal T \psi(t,\mathbf x) \mathcal T^{-1} &= \eta_\psi^T \;T\,\psi(-t,\mathbf x) \tag{8.94}\\ \mathcal T A^\mu(t,\mathbf x) \mathcal T^{-1} &= -{(\Lambda_T)^\mu}_\nu A^\nu(-t,\mathbf x) \end{align} where $$\eta^T_\phi=\pm 1=\eta^T_\psi$$, and $$T=C\gamma^5$$ is a $$4\times 4$$ matrix, acting on spinor indices.

How does this connect to another expression for a time reversal matrix, written as follows for any spin, $$T´´ = K\exp(\pm \text{i}\,\pi\, S_y)$$ where $$K$$ is complex conjugation?