On time reversal

Philip H. Butler

Ion I. Geru

On the basis of the group-theoretic approach, the existence of six new time reversal operators is proved, along with the well-known anti-unitary time-reversal operator introduced into quantum mechanics by Wigner. Among the new time-reversal operators, three are anti-unitary and three are unitary. A characteristic feature of the new time-reversal operators is that under their action the signs do not change for all three spin projection operators, but only for two or only for one of them. For this reason, such operators should be called operators of incomplete time reversal, in contrastIncomplete time reversal to the Wigner operator, which in this context is an operator of complete time reversal.

Hsiang-Shun Chou

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.

Xiao Mei

A more rational scheme of time reversal transformation is provided in the paper. In the new scheme, the transition probabilities are invariable under time reversal in the electromagnetic interaction processes of pure sates. But the evolutions of macro-processes can't keep unchanged under time reversal owing to the interference effects of probability amplitudes between non-pure state processes. In this way, the so-called reversibility paradox, i.e. time reversal is reversible in micro-processes but irreversible in the macro-processes, can eliminated completely.

Eugene Y. Wong