# Examples of antiunitary operator other than time reversal operator

It is well-known that time reversal operation is implemented as an anti-unitary operator. I wonder what are some other examples of anti-unitary operators that appear in the context of quantum mechanics, or physics in general.

• Arguably, they always involve time-reversal, since time evolution is given by $U(t) = e^{i\hat H t}$, so any anti-unitary operator $T = A K$ (with $A$ being a unitary operator and $K$ being complex conjugation) that commutes with $H$ tells us that $A U(t) A^\dagger = U(-t)$ (i.e., the arrow of time has been reversed). Jul 29, 2020 at 4:55

Just for future reference, an example I encountered recently is the following: in the Bogoliubov-de Gennes (BdG) theory of superconductivity, the Hamiltonian has particle-hole symmetry. And the corresponding transformation is also implemented as an anti-unitary operator, as the following: $$\begin{pmatrix} 0 &\boldsymbol{1} \\ \boldsymbol{1} & 0 \end{pmatrix} K$$ where $$K$$ is the complex conjugation operator, and $$\boldsymbol{1}$$ has a dimension of $$(\text{number of bands})\cdot 2$$ . There could be an additional factor of 2 if including spin degeneracy.