Demanding that the time reversal operator leaves Q unchanged but reverses the sign of P (which is how time reversal in classical mechanics works) means that the time reversal operator has to be implemented by an anti-unitary operator. More hand-wavingly, since the Schrodinger equation gives e−iHt as the forward time evolution of a state ψ, e+iHt (flipping the sign of time) should give the backward time evolution. But that’s just the normal time evolution of ψ∗ as you can see if you just conjugate the Schrodinger equation.
See also this paper for more discussion on why the time reversal operator ought to behave in this way and on time reveral in general.
For the benefit of others reading this, in Shankar’s book referenced below, this is said to not be a full symmetry i.e. not always valid.
There is no reason to think that the evolution of the conjugate function will be the same as the evolution of the original function.
Also there is no time-reversed Copenhagen measurement process in the theory which he implicitly requires.
Think about a massive object like a planet moving from L to R. It is massive so quantum effects can be ignored. It is clearly not true that the planet would be measured as being in the same place 10 minutes ago and 10 minutes hence. So the statement “All possible futures are also possible pasts” is wrong.
Note, though, that time reversal is still an anti-unitary operator in quantum mechanics in spite of the hand-waving argument failing when time reversal isn’t a good symmetry. Even when time reversal symmetry fails, though, there’s still CPT symmetry (and CPT is also anti-unitary).
Demanding that the time reversal operator leaves Q unchanged but reverses the sign of P (which is how time reversal in classical mechanics works) means that the time reversal operator has to be implemented by an anti-unitary operator. More hand-wavingly, since the Schrodinger equation gives e−iHt as the forward time evolution of a state ψ, e+iHt (flipping the sign of time) should give the backward time evolution. But that’s just the normal time evolution of ψ∗ as you can see if you just conjugate the Schrodinger equation.
See also this paper for more discussion on why the time reversal operator ought to behave in this way and on time reveral in general.
For the benefit of others reading this, in Shankar’s book referenced below, this is said to not be a full symmetry i.e. not always valid.
There is no reason to think that the evolution of the conjugate function will be the same as the evolution of the original function.
Also there is no time-reversed Copenhagen measurement process in the theory which he implicitly requires.
Think about a massive object like a planet moving from L to R. It is massive so quantum effects can be ignored. It is clearly not true that the planet would be measured as being in the same place 10 minutes ago and 10 minutes hence. So the statement “All possible futures are also possible pasts” is wrong.
Note, though, that time reversal is still an anti-unitary operator in quantum mechanics in spite of the hand-waving argument failing when time reversal isn’t a good symmetry. Even when time reversal symmetry fails, though, there’s still CPT symmetry (and CPT is also anti-unitary).