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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.

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    $\begingroup$ 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). $\endgroup$ Jul 29, 2020 at 4:55

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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.

It would be great to hear others' input on other examples of anti-unitary operator in physics.

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