On the time-reversal invariance of the fundamental commutation relation

We check the time-reversal invariance of the fundamental commutation relation [p,x] = -ih. Although it has been checked in the active picture, where the transformation affects the states but not the operators, it is a puzzling issue in the passive picture, where the transformation affects the operators but not the states. It is puzzling because [p',x'] = [-p, x] = ih in the passive picture. At first sight, it seems that the fundamental commutation relation is not time-reversal invariant. In fact, in both older and more recent textbooks on quantum mechanics, the time-reversal invariance of the fundamental commutation relation has never been explicitly clarified in the passive picture. We point out that the claims in standard textbooks concerning the time-reversal invariance of the fundamental commutation relation in the passive picture are misleading. We show that [p, x], = [p', x'] = [-p, x] = ih, if the time-reversal operator is unitary. Hence the fundamental commutation relation is not time-reversal invariant if the time-reversal operator is unitary. On the other hand, [p, x]' = [x, p'] = [x, -p] = -ih, if the time-reversal operator is antiunitary. We conclude that the fundamental commutation relation is time-reversal invariant, provided that the time-reversal operator is antiunitary. The important fact [p, x]' = [x', p'], which is the origin of the difficulties, is not appreciated in standard textbooks on quantum mechanics. The present discussion provides a more fundamental justification for taking the time-reversal operator to be antiunitary, which applies to particles with arbitrary spin.