Meaning of complex conjugate in $T$-symmetry

Question

I have a question about the meaning of complex conjugation in time reversal symmetry in quantum mechanics.

$T$-symmetry in classical mechanics is defined simply by the substitution $t \to -t$. If I have an external magnetic field it is not enough and I have to substitute $ \textbf{B} \to - \textbf{B} $. This makes sense because reversing time makes the "external current" generating the magnetic field run backwards, therefore a time symmetry that acts on the system as a whole is indeed supposed to reverse the signs of the magnetic fields as well.

In quantum mechanics T symmetry is given by an operator that acts on a generic wave function as $ T \psi\left(\textbf{x}, t \right) = \psi^{*} \left(\textbf{x}, -t \right) $. The meaning of $t \to -t$ is clear but what about complex conjugation? I know it makes Schroedinger equation invariant if the Hamilton is invariant but what does it have to do with time reversal? Is there a way to justify it like we justify the correspondence $ \textbf{B} \to - \textbf{B} $ in electromagnetism? Because if not it appears to me that $T$ kind of has to do with reversing time but it isn't really a time reversal of the system.

Answer 1

Time reversal operator $T$, when acting on $i$, must result in $TiT^{-1} = -i$. This comes from the fact that $TxT^{-1} = x, TpT^{-1} = -p$, and commutation relation $[x, p] = i\hbar$.

Combine this fact with linearity of time reversal operator, we conclude that $T$ is antiunitary operator, and can be decomposed in the form $KU$, where $U$ is unitary and $K$ is complex conjugation operator. (For reference, it is proved by Wigner that all symmetries of quantum mechanics must be unitary or anti-unitary)

$U$ can of course vary depending on the system you are working on.