i looked for time reversal operator,and i read in some quantum books that $T^2=+1$ for even number of fermions and $T^2=-1$ for odd number of fermions, so does $T^2$ depend on the hamiltonian or not? and what is $T$ explicitly for a system of bosons or fermions? my goal is to find out which symmetry class does one free fermion system belongs to in Altland-zirnbauer table of topological insulator. does particle-hole operator and chiral operator has a unique explicit form? or there is an arbitrariness in it? altland-zirnbauer symmetry classification of topological insulators
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I only know that $\hat{T}^2|j \,\text{half-integer}\rangle=-|j \,\text{half-integer}\rangle$ and $\hat{T}^2|j \,\text{integer}\rangle= |j \,\text{integer}\rangle$. $T^2$ do depend on the system hamiltonian for its statistics. For arbitarry $j$, $\hat T^2=(-1)^{2j}$, we can write $\hat T=\eta e^{-i\pi J_y/\hbar}\hat K$, where $\eta$ is an arbitrary phase. I recommand you the book of JJ Sakurai, Modern Quantum Mechanics, Chapter4.
