The following is stated in (among others) the articles
- Topological insulators and superconductors: ten-fold way and dimensional hierarchy - Shinsei Ryu, Andreas Schnyder, Akira Furusaki, Andreas Ludwig
- Classification of topological quantum matter with symmetries - Ching-Kai Chiu, Jeffrey C.Y. Teo, Andreas P. Schnyder, Shinsei Ryu
The three discrete symmetries Time reversal symmetry ($T$), Particle-hole symmetry ($C$) and Chiral symmetry ($S$) all commute (are real symmetries) of the full second quantized Hamiltonian, $$\hat{H}=\sum_{A,B}\Psi_A^\dagger \mathcal{H}_{AB}\Psi_B. $$ That is, $$[\hat{T},\hat{H}]=[\hat{C},\hat{H}]=[\hat{S},\hat{H}]=0.\;\;\;\;\;\;\; (1)$$ They go on to claim that, $\hat{C}$ and $\hat{S}$ are not real symmetries in the "classical" sense of the first quantized Hamiltonian $\mathcal{H}$. In fact, the first quantized Hamiltonians obey,
$$[T,\mathcal{H}]=\{C,\mathcal{H}\}=\{S,\mathcal{H}\}=0. \;\;\;\;\;\; (2)$$ How can I prove that imposing Eq. (1) implies Eq. (2)?
