This does not exactly answer your question but I think you want to justify $T^2=(-1)^N$

You best understand it by the action of Time-reversal operators on second quantized fermionic field operators. They act like

\begin{align}
\mathcal{T}\Psi_{\alpha}(\mathbf{k})\mathcal{T}^{-1} & =\sum_{\alpha'}(U^{\dagger}_{\operatorname{T}})_{\alpha,\alpha'}\Psi_{\alpha'}(-\mathbf{k}),\label{eq:-29}\\
\mathcal{T}\Psi_{\alpha}^{\dagger}(\mathbf{k})\mathcal{T}^{-1} & =\sum_{\alpha'}\Psi^{\dagger}_{\alpha'}(-\mathbf{k})(U_{\operatorname{T}})_{\alpha',\alpha},\label{eq: timereversal}
\end{align}

where $\alpha$ is some degree of freedom and $U_T$ is a unitary matrix. For example if I have a many-body state with $N$ electrons it can be given by
$$|N\rangle=\prod_{i}^{N}\Psi^{\dagger}_{\alpha}(k_i)|\mathrm{vac}\rangle,$$
Now it is clear that $$\mathcal{T}^2|N\rangle=(U_T^2)^N|N\rangle,
$$
finally if your system consists of pin one half particles $U_T$ squares to minus one.

This post imported from StackExchange Physics at 2020-10-30 22:41 (UTC), posted by SE-user physshyp