# Meaning of complex conjugate in $T$-symmetry

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.

• The Schroedinger equation is not going to be a priori time-reversal symmetric. It depends on whether the Hamiltonian is time-reversal symmetric. Nov 2, 2020 at 21:38
• Okay, assuming we have a T symmetry in the Hamiltonian then. Let me correct the question Nov 2, 2020 at 21:43
• You can define time-reversal to act however you want. Your choice may or may not be a symmetry of your system though, so some choices are more useful than other. You may want to have a look at this PSE post for more on this. Nov 2, 2020 at 22:36

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.