TITLE: Properties of time reversal operation
QUESTION [0 upvotes]: During a course of mine time-reversal symmetry was introduced as an anti-linear operator. One property the lecturer pointed out is that
\begin{equation}
\langle\phi| A\psi\rangle \neq \langle A^\dagger \phi |\psi \rangle 
\end{equation}
but rather we have
\begin{equation}
\langle\phi| A\psi\rangle = \langle \psi | A^\dagger \phi \rangle = \langle A^\dagger \phi | \psi \rangle^*.
\end{equation}
I understand that time-reversal involves a complex conjugation, but I don't understand what makes the above equalities true.
The question is: what property of time-reversal itself makes the above hold. Is it only complex conjugation or is there anything else needed for the above to be true?

REPLY [1 votes]: This is a pedantic point, but in general, I would be very careful about using the language $A^\dagger$ when antihermitian operators are involved. The reason I say this is that antihermitian $A$ is not even a linear operator on the Hilbert space, so it's complete nonsense to talk about it as though it were. For the same reason, bra-ket notation is somewhat broken when taking the dual - it is never clear whether the complex conjugation inherent in an antiunitary acts on the bra, the ket or neither.
It is much easier to simply say $A$ with the understanding that if an operator is antiunitary, then
$$\langle A \psi| A \phi\rangle = \langle \psi| \phi \rangle^*$$
and derive all other results you need from the fact that $A^2$ is unitary. See here for a more in-depth discussion.
{"set_name": "stack_exchange", "score": 0, "question_id": 712147}